On 5 and 7 Descents for Elliptic Curves

Thomas Anthony Fisher Clare College

A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge

August 2000

Preface

This dissertation is not substantially the same as any I have submitted for a degree or diploma or other qualification at any other university. It is the result of my own work and includes nothing which is the outcome of work done in collaboration.

I acknowledge with gratitude the guidance of my research supervisor Nick Shepherd-Barron. I also thank John Coates, John Cremona and Cathy O’Neil for valuable conversations along the way, and Karl Rubin for bringing to my attention the work of Cheryl Beaver [Be]. I am indebted to John Cremona for the use of his computer program mwrank. I also wish to thank my fellow students Paul Hacking and Jan Wierzba for many mathematical conversations. Finally I acknowledge the financial support of the Engineering and Physical Sciences Research Council.

Clare College August 2000

Summary

We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

In a special case, namely when the 5-torsion of our elliptic curve splits as μ_{5} ⊕ Z/5Z, we improve our estimate for the Mordell-Weil rank by using the Cassels-Tate pairing to perform a full 5-descent. We generalise our results to curves over Q(μ_{n}) and finally make some calculations for the curve X_{1}(11) over its field of 5-division points.

Contents

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Preliminaries 7 1.1 Weierstrass Equations and Reduction Types . . . . . . . . . . 7 1.2 Selmer Groups and Torsors . . . . . . . . . . . . . . . . . . . . 11 1.3 The Modular Curves X_{1}(n) and X(n) . . . . . . . . . . . . . . 15 1.4 Tate Local Duality and Reciprocity Laws . . . . . . . . . . . . 17

2 Descent Theorems over Q 21 2.1 Statement of the First Descent Theorem . . . . . . . . . . . . 21 2.2 Examples and Applications . . . . . . . . . . . . . . . . . . . 23 2.3 Exhibiting Large Tate-Shafarevich Groups . . . . . . . . . . . 27 2.4 Statement of the Second Descent Theorem . . . . . . . . . . . 30

3 Proof of the First Descent Theorem 33 3.1 Equations for Torsors in P^{n−1} . . . . . . . . . . . . . . . . . . 33 3.2 Local Solubility for Torsors . . . . . . . . . . . . . . . . . . . . 39 3.3 Further Hensel Lemma Calculations . . . . . . . . . . . . . . . 40 3.4 Computation of Selmer Groups . . . . . . . . . . . . . . . . . 44

4 On the Geometry of Elliptic Normal Curves 47 4.1 Elliptic Normal Curves . . . . . . . . . . . . . . . . . . . . . . 47 4.2 An Algebraic Construction of X(n) . . . . . . . . . . . . . . . 50 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.4 The Relationship Between A(n) and X(n) . . . . . . . . . . . 55 4.5 Alternative Criteria for Finding the Cusps . . . . . . . . . . . 57 4.6 Calculations with char(K) = n . . . . . . . . . . . . . . . . . . 58 5 Descent Theorems over Q(μ_{n}) 63

5.1 Statement of the First Descent Theorem . . . . . . . . . . . . 63 5.2 Examples and Applications . . . . . . . . . . . . . . . . . . . 65 5.3 Computation of Selmer Groups . . . . . . . . . . . . . . . . . 68 5.4 Proof of the First Descent Theorem over Q . . . . . . . . . . . 72 5.5 Proof of the First Descent Theorem over Q(μ_{n}) . . . . . . . . 74

6 Proof of the Second Descent Theorem 77 6.1 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Definition of the Cassels-Tate Pairing . . . . . . . . . . . . . . 79 6.3 Split Torsion and Local Pairings . . . . . . . . . . . . . . . . . 80 6.4 Computation of the Cassels-Tate Pairing . . . . . . . . . . . . 83

7 Application to the Curves of Conductor 11 87 7.1 The Field L = Q(A_{1}[5]) . . . . . . . . . . . . . . . . . . . . . 87 7.2 Estimating rank A(L) via First Descent . . . . . . . . . . . . . 89 7.3 Estimating rank A(L) via Second Descent . . . . . . . . . . . . 93

A Varieties defined by Pfaffians 97

B Descent by n-isogeny 101

C Explicit covering maps 105

D Tables 107

0.1 Introduction

Let E be an elliptic curve over a number field K. The Mordell-Weil theorem asserts that the group E(K) of K-rational points on E is a finitely generated abelian group. We call E(K) the Mordell-Weil group. Another important group associated to E is the Tate-Shafarevich group � (E/K). It is defined as the set of torsors under E which have points everywhere locally. Thus a non-zero element of the Tate-Shafarevich group corresponds to a smooth curve of genus 1 that violates the Hasse Principle, i.e. it has (local) K_{v}- rational points at all places v, but no (global) K-rational points. It is known that � (E/K) is torsion, and that the torsor corresponding to an element of order n admits a divisor class of degree n defined over K. On a curve of genus 1 a complete linear system of degree n also has dimension n. Thus for n = 2 we consider double covers of P^{1}, for n = 3 plane cubics, for n = 4 complete intersections of two quadrics in P^{3}, and so on.

There is no known algorithm guaranteed to compute the Mordell-Weil group, and the situation for the Tate-Shafarevich group is no better. However, for any integer n ≥ 2 the proof of the Mordell-Weil theorem gives an effective upper bound on the order of E(K)/nE(K). This yields an upper bound on the Mordell-Weil rank which is an overestimate precisely whenever � (E/K) contains non-trivial n-torsion. Such calculations are referred to as descent calculations. By performing an n-descent we obtain partial information concerning the groups E(K) and � (E/K). It is therefore of interest to make descent calculations practical for as many different values of n as possible.

A great deal of work has been done on 2- and 3-descents, the advantage of n being small outweighing the fact that the primes 2 and 3 often require special treatment in the study of elliptic curves. We give some examples of 5- and 7-descents. Unfortunately we work in a special case, so our results do not apply to an arbitrary elliptic curve. Firstly, our descent calculations assume the existence of an isogeny of degree n = 5 or 7. Secondly, we assume one of our pair of elliptic curves has a rational point of order n. Notice that by a result of Serre [Se1], the second condition is automatically satisfied if we restrict to semi-stable elliptic curves. Here we recall a curve is semi-stable if it has good or multiplicative reduction at all primes.

Accordingly we consider pairs of elliptic curves C and D defined over K, related by the exact sequences of Galois modules

0 → μ_{n} → C →^{ϕ} D → 0 and 0 → Z/nZ → D \underrightarrow{ϕ̂} C → 0.  (1)

In other words ϕ : C → D is an isogeny of degree n with kernel a labelled copy of μ_{n} inside C. The Weil pairing tells us that the dual isogeny ϕ̂ : D → C has kernel a labelled copy of Z/nZ inside D. We estimate the Mordell-Weil rank by bounding the groups D(K)/ϕC(K) and C(K)/ϕ̂D(K). This process is known as descent via n-isogeny. In many cases it enables us to compute the n-torsion of the Tate-Shafarevich groups � (C/K) and � (D/K).

Selmer wrote down the following example of a plane cubic which violates the Hasse principle.

T = {3x_{0}^{3} + 4x_{1}^{3} + 5x_{2}^{3} = 0} ⊂ P^{2}

As explained in [Ca2], T corresponds to an element of order 3 in � (C/Q) where the Jacobian C has equation x_{0}^{3} + x_{1}^{3} + 60x_{2}^{3} = 0. There is an action of μ_{3} on T given by x_{i} ↦ ζ_{3}^{i}x_{i}. The quotient is the elliptic curve D with Weierstrass equation

y^{2} = x^{3} + 30^{2}.

A 2-descent shows D(Q) ≅ Z/3Z. Writing down explicit equations for the quotient map it is then easy to check T (Q) = ∅.

We now give an example of an element of order 5 in the Tate-Shafarevich group.

T = \{ \begin{matrix}x_{0}^{2} + x_{1}x_{4} − 6x_{2}x_{3} & = & 0 \\ x_{1}^{2} + x_{0}x_{2} − 15x_{3}x_{4} & = & 0 \\ 2x_{2}^{2} + x_{1}x_{3} − 5x_{0}x_{4} & = & 0 \\ 3x_{3}^{2} + x_{2}x_{4} − x_{0}x_{1} & = & 0 \\ 5x_{4}^{2} + x_{0}x_{3} − 2x_{1}x_{2} & = & 0 \end{matrix} \} ⊂ P^{4}

To show T has points everywhere locally it suffices to consider the primes p = 2, 3, 5, 569 at all other primes the reduction being a smooth curve of genus 1. For p = 2, 3, 5 the reduction is a rational nodal curve of degree 5 and again local solubility is clear. Finally T meets the hyperplane {x_{4} = 0} in a point defined over Q(\sqrt[5]{12}) and p = 569 splits in this field. To show T has no Q-rational points we quotient out by the action of μ_{5} on T given by x_{i} ↦ ζ_{5}^{i}x_{i}. The quotient is an elliptic curve D with Weierstrass equation

y^{2} − 29xy − 30y = x^{3} − 30x^{2}.

A 2-descent, courtesy of Cremona’s program mwrank, shows D(Q) ≅ Z/5Z. The formulae in Appendix C give equations for the quotient map

(x : y : 1) = (30x_{0}x_{1}x_{4} : 30^{2}x_{2}x_{4}^{2} : −x_{0}^{3}).

It is then easy to check T (Q) = ∅.

The curve T corresponds to an element of order 5 in � (C/Q) where the Jacobian C is related to D by the exact sequences (1). In our example C and D have conductor N = 17070. We perform similar calculations for all pairs of elliptic curves C and D appearing in Cremona’s tables [Cr] of elliptic curves of conductor N ≤ 5300. We find the following examples of curves C with � (C/Q)[5] ≅ (Z/5Z)^{2}

570L3, 570L4, 870I3, 870I4, 1050O2, 1342C3, 1938J2, 1950Y2, 2370M2, 2550EE2, 3270H2.

We are able to treat the case n = 7 in the same explicit manner, and find the following examples of curves C with � (C/Q)[7] ≅ (Z/7Z)^{2}

546F2, 858K2, 1230K2, 5010H2.

It is pleasing to note that, with one exception, we have recovered Cremona’s list of all curves of conductor N ≤ 1000 and � of analytic order n^{2}. The exceptional curve 275B3, with � of analytic order 25, is the 5-twist of one of our curves C. We are also able to exhibit examples of non-trivial � for curves with positive Mordell-Weil rank.

The pairs of curves C and D are parametrised by the modular curve Y_{1}(n) = X_{1}(n) \ {cusps}. For n = 5 or 7 we have X_{1}(n) ≅ P^{1}. We fix a co-ordinate λ on P^{1}, essentially by listing the cusps, and write C_{λ} and D_{λ} for the pairs of curves above λ. In §1.1 we give Weierstrass equations for these curves and describe their reduction types. In §1.2 we recall the notion of torsor, or principal homogeneous space, and outline the methods we use to construct torsors under C_{λ}. In §1.3 and §1.4 we include some background material on modular curves and duality theorems respectively.

In Chapter 2 we state our main descent theorems over Q and show how they may be used both to compute ranks of elliptic curves and to exhibit elements of the Tate-Shafarevich group. Theorem 1, stated in §2.1, gives an explicit recipe for descent via n-isogeny on the curves C_{λ} and D_{λ} over Q parametrised by X_{1}(n). The idea is that λ determines sets of primes A and B, and the Selmer groups sought appear as the left and right kernel of a certain “reciprocity pairing” supported on these primes. We find that A ∪ B is precisely the set of primes with root number −1. Thus our estimates for the Mordell-Weil rank have the same parity as the analytic rank. This is a special case of a result of Monsky [Mo].

The elliptic curves E_{τ} over Q whose 5-torsion splits as μ_{5} ⊕ Z/5Z are parametrised by the modular curve X(5). For these curves we may perform two different descents via 5-isogeny. Applying Theorem 1 we show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large. Corresponding results for n = 2 and n = 3 have been established by Cassels [Ca1], Bölling [Bö] and Kramer [Kr]. Our method does not extend to n = 7 since there are no elliptic curves over Q whose 7torsion splits as μ_{7} ⊕ Z/7Z. The best result we can give is that the 7-Selmer group may become arbitrarily large.

Theorem 2, stated in §2.4, gives an explicit recipe for (full) 5-descent on the curves E_{τ} over Q parametrised by X(5). The idea is that we combine Theorem 1 with the work of Beaver [Be] on the Cassels-Tate pairing. We call Theorems 1 and 2 the first and second descent theorems respectively. A common theme is that the curves considered are parametrised by a modular curve, and we partition the primes of bad reduction by asking to which cusp our point on the modular curve reduces.

In Chapter 3 we prove the first descent theorem (Theorem 1). We begin by using the diagonal action of μ_{n} to help write down explicit equations for torsors in P^{n−1}. An understanding of the geometry of these curves leads to simple criteria for local solubility. We perform some further Hensel lemma calculations to treat the prime n itself, and then complete our proof of Theorem 1 by writing down the “reciprocity pairing” referred to above as a sum of local Tate pairings.

In Chapter 4 we discuss the geometric results required for the proof of our descent theorems. We work over an algebraically closed field and give equations in P^{n−1} for the elliptic curves parametrised by X(n). In §4.3 we relate these equations to the torsors written down in §3.1. This gives us a handle on the geometry of the latter. The case char(K) = n is left to the end of the chapter.

In Chapter 5 we generalise our first descent theorem to K = Q(μ_{n}). One reason for doing this is that an alternative, and perhaps more natural, proof of Theorem 1 is given by working over K and then descending back to Q. We give the details in §5.4. Although working over a field containing the nth roots of unity makes certain aspects of our proof simpler, we find that rather delicate arguments are required to establish conditions for local solubility at the prime above n. It turns out that our calculations are just what is required to give some interesting examples from the point of view of Iwasawa theory. We refer to [CS] for details.

In Chapter 6 we explain how the second descent theorem (Theorem 2) follows from our descent calculations and those of Beaver [Be]. Beaver works with the Tate parametrisation and therefore finds it necessary to make an assumption on the reduction type at 5. We remove this assumption and give a much more explicit description of the “switching exponents”. For these reasons we include rather full details.

In Chapter 7 we apply some of our methods to the elliptic curves over Q with conductor 11. From the point of view of Iwasawa theory there is some interest in the behaviour of the Mordell-Weil rank as we pass up various towers of number fields. For instance, the curves of conductor 11 are known to have rank 0 over Q(μ_{5^{∞}}). Another natural tower to consider is that defined by the 5-power division points of the curves themselves. As a first step in this direction we show that the rank is 0 over the field of 5-division points for X_{1}(11).

In Appendix A we relate the geometrical results in Chapter 4 to the work of Buchsbaum and Eisenbud [BE] on varieties defined by Pfaffians. In Appendix B we discuss the problem of extending our descent calculations to pairs of n-isogenous curves without the assumption that one of these curves has a rational point of order n. In Appendix C we give some explicit formulae which, although not required for our descent theorems, may be of some interest for the study of specific examples. In Appendix D we illustrate our descent theorems by giving tables of examples.

Let us make two remarks that may help the reader. Firstly, as the title of this thesis indicates, we often treat the cases n = 5 and n = 7 concurrently. It then become necessary, when giving formulae, to split into these two cases. At times we tire of reminding the reader that these are the cases n = 5 and n = 7. It is hoped that this will cause no confusion.

Secondly, we encounter many calculations that are straightforward to check with the aid of any computer algebra package, but which would be rather laborious to check by hand. The reader should either take these results on trust, or reach for his own computer. Whenever our computer calculations extend beyond these menial algebraic manipulations, we give explicit mention of the fact. For example in §2.2 we make use of mwrank to compute ranks of elliptic curves over Q via 2-descent, and in §7.1 we use pari to compute the unit group and class group of a certain number field of degree 20.

0.2 Notation

We summarise here some of our basic notation.

We work over a perfect field K and write \overline{K} for the algebraic closure. We write G_{K} = Gal(\overline{K}/K) for the absolute Galois group and H^{1}(K, −) for the Galois cohomology group H^{1}(G_{K}, −). Assuming char(K) ∤ n we write μ_{n} ⊂ \overline{K} for the nth roots of unity and fix ζ = ζ_{n} a generator. For M a finite G_{K}-module we write M^{∨} for the Cartier dual Hom(M, G_{m}).

Frequently we take K a number field and write O_{K}, U_{K} and I_{K} for the ring of integers, unit group and group of fractional ideals respectively. For v a place of K we write K_{v} for the completion in the v-adic topology. Whenever necessary we fix embeddings \overline{K} ↪ \overline{K}_{v} and so identify G_{v} = Gal(\overline{K}_{v}/K_{v}) as a subgroup of G_{K}. We reserve the term prime for 𝔭 a finite place, writing ord_{𝔭} : K_{𝔭}^{×} → Z for the normalised valuation and O_{𝔭} for the ring of 𝔭-adic integers.

An elliptic curve E/K is a smooth projective curve of genus 1, equipped with a specified rational point 0 ∈ E(K) taken as identity element for the group law. For n an integer we write [n] : E → E for the multiplication by n map. For P ∈ E we write τ_{P} for the map “translation by P ”. For ϕ : E → E′ an isogeny of degree n we write E[ϕ] for its kernel, ϕ̂: E′ → E for the dual isogeny and e_{ϕ} : E[ϕ] × E′[ϕ̂] → μ_{n} for the Weil pairing as defined in [Si1, Exercise 3.15].

The cardinality of a set S is denoted either |S| or #S. For A an abelian group we write A[n] for the n-torsion, and if n is prime write A(n) for the n-power torsion. Given elements a, b, … belonging to A we write ⟨a, b, … ⟩ for the subgroup they generate. For R a ring we write R^{+} for the underlying additive group, and R^{×} for the group of invertible elements under multiplication. The dual of a vector space V is denoted V^{∗}. For Ξ a pairing on vector spaces V and W we write ker_{L}(Ξ) ⊂ V and ker_{R}(Ξ) ⊂ W for the left and right kernels of Ξ. For v = (v_{i})_{i=1}^{n} a vector of length n we write Diag(v) for the n × n diagonal matrix with entries v_{i} down the diagonal. The trivial group shall always be denoted 0.

Chapter 1

Preliminaries

1.1 Weierstrass Equations and Reduction Types

Let K be a number field and n ≥ 4 an integer. We consider pairs of nisogenous elliptic curves C and D over K. We assume ϕ : C → D has kernel a labelled copy of μ_{n} and ϕ̂ : D → C has kernel a labelled copy of Z/nZ. The pairs of such curves C and D are parametrised by the modular curve Y_{1}(n). For n prime X_{1}(n) has n − 1 cusps, of which half are defined over Q and half are defined over Q(μ_{n}) ∩ R. Our interest is in the cases n = 5 and n = 7 when X_{1}(n) ≅ P^{1}. We shall shortly specify a co-ordinate λ on X_{1}(n) and write C_{λ} and D_{λ} for the pair of n-isogenous elliptic curves above λ.

Lemma 1.1 Let E/K be an elliptic curve and P ∈ E(K) a point of order at least 4.

(i) E has a Weierstrass equation y^{2} + uxy + vy = x^{3} + vx^{2} with P = (0, 0). (ii) The pair (E, P) uniquely determines (u, v) ∈ A^{2}(K). (iii) The pair (E, P) has no automorphisms.

Proof. [Si1, Exercise 8.13] or [Si2, Exercise 3.1]. □

We now compute mP = (x_{m}(u, v), y_{m}(u, v)) for small integers m. Equating some of these expressions we quickly write down some affine curves whose smooth projective model is X_{1}(n). In the cases n = 5 and n = 7 we find D_{λ} has Weierstrass equation

\begin{aligned}n = 5 & y^{2} + (1 − λ)xy − λy & = & x^{3} − λx^{2} \\ n = 7 & y^{2} + (1 + λ − λ^{2})xy + (λ^{2} − λ^{3})y & = & x^{3} + (λ^{2} − λ^{3})x^{2}. \end{aligned} (2)

Computing the discriminant we find the cusps of X_{1}(n).

n = 5

Rational cusps at λ = 0, ∞.

Irrational cusps at the roots of λ^{2} − 11λ − 1 = 0.

n = 7

Rational cusps at λ = 0, 1, ∞.

Irrational cusps at the roots of λ^{3} − 8λ^{2} + 5λ + 1 = 0.

The automorphisms of Y_{1}(n) are precisely the automorphisms of P^{1} that map cusps to cusps. There are (n − 1)/2 automorphisms preserving the rational cusps. These correspond to relabelling our copy of μ_{n} ↪ C respectively Z/nZ ↪ D. In addition there is an involution η that swaps over the rational cusps with the irrational cusps. We find η is defined over Q(μ_{n}) ∩ R and μ_{n} ↪ C_{λ} is isomorphic to Z/nZ ↪ D_{η(λ)} over Q(μ_{n}). Since there is no canonical choice of isomorphism μ_{n} ≅ Z/nZ there is no canonical choice of involution η. As far as possible we shall avoid making such a choice.

Remark 1.2 The proof of Lemma 1.1 involves writing down a Weierstrass equation and then making substitutions of the form

x ↦ π^{2}x + r, y ↦ π^{3}y + π^{2}sx + t.

By this method we may easily check, for n = 5 that (D_{λ}, 2P) ≅ (D_{−1/λ}, P), and for n = 7 that (D_{λ}, 2P) ≅ (D_{(λ−1)/λ}, P).

Vélu’s formulae [V1] tell us that C_{λ} has Weierstrass equation

\begin{aligned}y^{2} + (1 − λ)xy − λy & = & x^{3} − λx^{2} − 5tx − b_{2}t − 7w \\ y^{2} + (1 + λ − λ^{2})xy + (λ^{2} − λ^{3})y & = & x^{3} + (λ^{2} − λ^{3})x^{2} − 5tx − b_{2}t − 7w \end{aligned}(3)

where

\begin{matrix}n = 5 & \{ \begin{matrix}b_{2} & = & λ^{2} − 6λ + 1 \\ t & = & λ(λ^{2} + 2λ − 1) \\ w & = & λ^{2}(2λ^{2} + λ + 1) \end{matrix} \\ n = 7 & \{ \begin{matrix}b_{2} & = & λ^{4} − 6λ^{3} + 3λ^{2} + 2λ + 1 \\ t & = & λ(λ − 1)(λ^{2} − λ + 1)(λ^{3} + 2λ^{2} − 5λ + 1) \\ w & = & λ^{2}(λ − 1)^{2}(2λ^{6} − 2λ^{5} + λ^{4} − 8λ^{3} + 15λ^{2} − 9λ + 2). \end{matrix} \end{matrix}

The Weierstrass equations (2) and (3) have discriminant

\begin{matrix}n = 5 & \begin{matrix}Δ(C_{λ}) = λ(λ^{2} − 11λ − 1)^{5} \\ Δ(D_{λ}) = λ^{5}(λ^{2} − 11λ − 1) \end{matrix} \\ n = 7 & \begin{matrix}Δ(C_{λ}) = λ(λ − 1)(λ^{3} − 8λ^{2} + 5λ + 1)^{7} \\ Δ(D_{λ}) = λ^{7}(λ − 1)^{7}(λ^{3} − 8λ^{2} + 5λ + 1). \end{matrix} \end{matrix}

Although our descent calculations shall make remarkably little explicit use of the reduction types for C_{λ} and D_{λ}, it is good culture to describe them now. We recall that isogenous elliptic curves always have the same reduction type.

Lemma 1.3 Let K be a number field and let 𝔭 be a prime with 𝔭 ∤ n. Then for λ ∈ K_{𝔭} the curves C_{λ}/K_{𝔭} and D_{λ}/K_{𝔭} have either good or multiplicative reduction.

Proof. It suffices to show that if an elliptic curve E/K_{𝔭} has additive reduction then E(K_{𝔭})[n] = 0. We consider the filtration E(K_{𝔭}) ⊃ E_{0}(K_{𝔭}) ⊃ E_{1}(K_{𝔭}) as described in [Si1, Chapter VII]. By the theory of formal groups the multiplication by n map on E_{1}(K_{𝔭}) is invertible. The additive reduction tells us E_{0}(K_{𝔭})/E_{1}(K_{𝔭}) ≅ (O_{K}/𝔭)^{+} and the Tamagawa number [E(K_{𝔭}) : E_{0}(K_{𝔭})] is at most 4. The lemma follows. □

To learn more about the reduction types we work directly with the Weierstrass equation (2). Applying Tate’s algorithm, as described in [Si2] for example, we find

Lemma 1.4 Let K be a number field and let 𝔭 be a prime with 𝔭 ∤ n. Then for λ ∈ K_{𝔭} with ord_{𝔭}(λ) ≥ 0 the Weierstrass equation (2) is minimal and the reduction types are as follows.

(i) If λ reduces to a rational cusp, i.e. λ ≡ 0 (mod 𝔭), respectively λ(λ−1) ≡ 0 (mod 𝔭), then C_{λ} and D_{λ} have split multiplicative reduction. (ii) If λ reduces to an irrational cusp, i.e. λ^{2} − 11λ − 1 ≡ 0 (mod 𝔭), respectively λ^{3}−8λ^{2}+5λ+1 ≡ 0 (mod 𝔭), then C_{λ} and D_{λ} have multiplicative reduction and the reduction is split if and only if N𝔭 ≡ 1 (mod n). (iii) In the remaining cases C_{λ} and D_{λ} have good reduction.

The assumption ord_{𝔭}(λ) ≥ 0 here is no loss since we are always free to replace λ by −1/λ, respectively (λ−1)/λ. It remains to describe the reduction types for 𝔭|n. We find that cases (i) and (iii) go through as before. However if λ reduces to an irrational cusp it seems no longer possible to treat all number fields at once. In the case K = Q the Weierstrass equation (2) is still minimal and the reduction is additive.

We set up some notation to treat the case K = Q(μ_{n}). Let 𝔩 = (1 − ζ) be the unique prime of K above n. For λ ∈ K_{𝔩} we define

c(λ) = min{ ord_{𝔩}(λ − u) | u an irrational cusp }

if λ reduces to an irrational cusp, and c(λ) = 0 otherwise. We write c_{max} for ord_{𝔩}(u − v) where u and v are distinct irrational cusps.

Lemma 1.5 The cusp number c(λ) satisfies

(i) 0 ≤ c(λ) ≤ c_{max} (ii) c(λ) = c(−1/λ), respectively c(λ) = c((λ − 1)/λ) (iii) c(λ) + c(η(λ)) = c_{max}.

Proof. (i) and (ii) Straightforward. (iii) We have η(λ) = u(λ − v)/(λ − u) for suitable cusps u and v. Then

(λ − u)(η(λ) − u) = u(u − v)

and since u is a unit we deduce c(λ) + c(η(λ)) = c_{max} as required. □

To find the value of c_{max} it suffices to compute some discriminants

n = 5  Δ(λ^{2} − 11λ − 1) = 5^{3}  ⟹ c_{max} = 6
n = 7  Δ(λ^{3} − 8λ^{2} + 5λ + 1) = 7^{4} ⟹ c_{max} = 4.

Lemma 1.6 Let K = Q(μ_{n}). Let λ ∈ K_{𝔩} with c(λ) as defined above.

(i) If λ or η(λ) reduces to a rational cusp then C_{λ} and D_{λ} have split multiplicative reduction. In this case c(λ) = 0 or c(λ) = c_{max}. (ii) If 0 < c(λ) < c_{max} then C_{λ} and D_{λ} have additive potential good supersingular reduction. (iii) In the remaining cases C_{λ} and D_{λ} have good ordinary reduction.

Proof. As usual we may assume ord_{𝔩}(λ) ≥ 0. If c(λ) < c_{max} then

n = 5  ord_{𝔩}(λ^{2} − 11λ − 1) < 2c_{max} = 12
n = 7  ord_{𝔩}(λ^{3} − 8λ^{2} + 5λ + 1) < 3c_{max} = 12.

So in this case the Weierstrass equation (2) is minimal and the reduction types may be checked via computer algebra as before. Since C_{λ} ≅ D_{η(λ)} the general case follows by Lemma 1.5(iii).

It only remains to prove the statements concerning ordinary versus supersingular reduction. In case (iii) we may explicitly list the points Z/nZ ↪ D_{λ} and so learn that the reduction is ordinary. In case (ii) some rather messy calculations, the details of which we omit, show that ord_{𝔩}(j) > 0, respectively ord_{𝔩}(j−1728) > 0. We recall [Ha, IV.4.23] that the supersingular j-invariants in characteristic 5 and 7 are 0 and 1728 respectively. Thus we have potential good supersingular reduction as claimed. □

1.2 Selmer Groups and Torsors

Let E be an elliptic curve over a number field K. Let T/K be a smooth curve of genus 1. We say that T has the structure of torsor under E if there is a simple transitive action E × T → T defined over K. Since T has \overline{K}- rational points, it is clear that T is a twist of E. It is a general principle that the twists of an object X are parametrised by the Galois cohomology group H^{1}(K, Aut(X)).

Writing τ_{P} for translation by P ∈ E there is an exact sequence

0 → {translations τ_{P} } → Aut(E) → Aut(E, 0) → 0  (4)

where Aut(E) and Aut(E, 0) are the automorphism groups of E as a curve and as a group variety respectively. The translations make up the identity component of Aut(E) and may be characterised as the automorphisms without fixed points. More importantly for us they are the automorphisms of E as a torsor under itself. Thus the torsors under E are parametrised by the Weil-Châtelet group WC(E/K) := H^{1}(K, E). Concretely, T is a torsor under E if and only if there exists ψ : T ⟶^{∼} E an isomorphism over \overline{K} such that the cocycle σ ↦ σ(ψ)ψ^{−1} takes values in the translation subgroup of Aut(E). We remark that in this situation the curve T uniquely determines the curve E. In fact E is the Jacobian of T.

We write ξ_{T} ∈ WC(E/K) for the cohomology class determined by the torsor T/K. It is clear T (K) ≠ ∅ if and only if ξ_{T} = 0. Thus the Tate-Shafarevich group

� (E/K) := ker(WC(E/K) → ∏_{v} WC(E/K_{v}))

measures the failure of the Hasse Principle for torsors T under E.

Let ϕ : E → E′ be an isogeny defined over K. There is an exact sequence of Galois modules

0 → E[ϕ] → E → E′ → 0.

Taking Galois cohomology yields

0 → E′(K)/ϕE(K) → H^{1}(K, E[ϕ]) → WC(E/K)[ϕ] → 0.  (5)

If T is a torsor under E corresponding to ξ_{T} ∈ WC(E/K)[ϕ] then by (5) the isomorphism ψ : T ⟶^{∼} E may be chosen such that the cocycle σ ↦ σ(ψ)ψ^{−1} takes values in E[ϕ]. The Selmer group attached to the isogeny ϕ is

S^{(ϕ)}(E/K) := ker(H^{1}(K, E[ϕ]) → ∏_{v} WC(E/K_{v})).

It consists of those classes in H^{1}(K, E[ϕ]) which describe torsors under E with points everywhere locally. The exact sequence (5) now becomes

0 → E′(K)/ϕE(K) → S^{(ϕ)}(E/K) → � (E/K)[ϕ] → 0.  (6)

Thus the Selmer group S^{(ϕ)}(E/K) contains information about both the Mordell-Weil group E(K) and the Tate-Shafarevich group � (E/K). The aim of this thesis is to present some examples where S^{(ϕ)}(E/K) may be computed rather explicitly. Such calculations are referred to as descent calculations, since knowledge of E′(K)/ϕE(K) and E(K)/ϕ̂E′(K) enables one to compute the group E(K) via Fermat’s method of descent.

We now take n = 5 or 7 and consider the curves C_{λ} and D_{λ} introduced in §1.1. The exact sequence (5) becomes

0 → D_{λ}(K)/ϕC_{λ}(K) →^{δ} H^{1}(K, μ_{n}) → WC(C_{λ}/K)[ϕ] → 0.  (7)

By Hilbert’s Theorem 90 we identify H^{1}(K, μ_{n}) = K^{×}/(K^{×})^{n}. We write C_{λ,θ} for the torsor under C_{λ} described by θ ∈ K^{×}/(K^{×})^{n}. Since n is odd we may ignore the infinite places, so by definition

S^{(ϕ)}(C_{λ}/K) = { θ ∈ K^{×}/(K^{×})^{n} | C_{λ,θ}(K_{𝔭}) ≠ ∅ for all primes 𝔭 }. (8)

We give equations for the curves C_{λ,θ} via two different methods, which we refer to as the push-out method and the projective space method.

Equations via Push-Out. Let C_{λ} and D_{λ} be as above and write 1 for the generator of Z/nZ ↪ D_{λ}(K). We label our copy of μ_{n} ↪ C_{λ} such that e_{ϕ}(ζ, 1) = ζ for all ζ ∈ μ_{n}.

Lemma 1.7 Let f ∈ K(D_{λ}) have divisor (f) = n.1 − n.0 and be scaled, up to nth powers, such that f_{∘}ϕ = g^{n} for some g ∈ K(C_{λ}). Then the map δ in

0 → μ_{n}(K) → C_{λ}(K) →^{ϕ} D_{λ}(K) →^{δ} K^{×}/(K^{×})^{n}  (9)

is given by δ(P) = f(P) mod (K^{×})^{n}, for P ≠ 0, 1.

Proof. Pick P′ ∈ C_{λ} with ϕ(P′) = P. Then for σ ∈ G_{K} we compute

σ(P′) − P′ = e_{ϕ}(σ(P′) − P′, 1)
= g(X + σ(P′) − P′)/g(X)  for any X ∈ C_{λ}
= σ(g(P′))/g(P′)

Thus δ(P) = g(P′)^{n} = f_{∘}ϕ(P′) = f(P) mod (K^{×})^{n}. □

Taking D_{λ} with Weierstrass equation (2) and 1 = (0, 0) we claim that the rational function f ∈ K(D_{λ}) is given by

\begin{matrix}n = 5 & f(x, y) = xy + y − x^{2} \\ n = 7 & f(x, y) = (λ + 1)x^{3} − x^{2}y + λ^{2}x^{2} − (2λ + 1)xy − λ^{2}y. \end{matrix} (10)

Indeed on the affine piece of D_{λ} described, f only vanishes at 1 = (0, 0), as is seen by computing some resultants. It follows that f has the correct divisor. Since δ is a homomorphism it is easy to check that f is correctly scaled. To this end we compute f on the torsion Z/nZ ↪ D_{λ}(K).

x y f  x  y  f
2 λ λ^{2} λ^{3}  2 λ^{2}(λ − 1) λ^{3}(λ − 1)^{2} −λ^{6}(λ − 1)^{5}
3 λ 0 −λ^{2}  3 λ(λ − 1) λ(λ − 1)^{2} λ^{2}(λ − 1)^{4}
4 0 λ λ  4 λ(λ − 1) λ^{2}(λ − 1)^{2} −λ^{5}(λ − 1)^{3}
5 λ^{2}(λ − 1)  0  λ^{8}(λ − 1)^{2}
6  0  λ^{2}(λ − 1) −λ^{4}(λ − 1)

Thus C_{λ,θ} has affine model

\{ \begin{matrix}f(x, y) = θz^{n} \\ (x, y) ∈ D_{λ} \end{matrix} \} ⊂ A^{2} × G_{m}.  (11)

Remark 1.8 According to our table, the image of Z/nZ ↪ D_{λ}(K) under δ is generated by α(λ) := λ, respectively λ^{4}(λ − 1). It is clear from the exact sequence (9) that α(λ) is a Kummer generator for the field extension K(C_{λ}[n])/K(μ_{n}). In §1.3 we explain how we may use α(λ) to construct X(n) from X_{1}(n).

Equations in Projective Space. We recall [Si1, Chapter III] that for D a divisor on the elliptic curve E

D ∼ 0 ⟺ deg D = 0 and sum D = 0.  (12)

Thus if we embed E ↪ P^{n−1} by means of a complete linear system of degree n then the translation map τ_{P} lifts to an automorphism of P^{n−1} if and only if P ∈ E[n].

Now take T ↪ P^{n−1} a smooth curve of genus 1 and degree n. In this situation we always take it as read that T is embedded via a complete linear system. It is equivalent to demand that T is contained in no hyperplane. We claim that the action of Jac(T) on T determines an inclusion of G_{K}-modules Jac(T)[n] ↪ PGL_{n}. Indeed, from a geometric point of view this situation is no different from that described in the last paragraph.

Suppose given T a torsor under C_{λ} with ξ_{T} ∈ WC(C_{λ}/K)[ϕ]. We know μ_{n} ↪ C_{λ} acts on T with quotient U a torsor under D_{λ}. Then

ϕ : WC(C_{λ}/K) → WC(D_{λ}/K); ξ_{T} ↦ ξ_{U} = 0.

Choosing a K-point on U gives a divisor of degree n on T and so an embedding T ↪ P^{n−1}. As explained above, μ_{n} ↪ C_{λ} lifts to an action on P^{n−1}. Geometrically there are exactly n hyperplanes fixed under this action. They correspond to our original choice of K-point on U and its translates under Z/nZ ↪ D_{λ}. In particular these hyperplanes are defined over K, so for a suitable choice of co-ordinates on P^{n−1} the action μ_{n} ↪ PGL_{n} is given by

ζ ↦ Diag(1 : ζ : …: ζ^{n−1}).  (13)

Thus the torsors C_{λ,θ} all arise as smooth curves of genus 1, degree n in P^{n−1} invariant under (13). In §3.1 we shall give equations for all such curves. In the meantime, let us reverse the above argument to show that all the curves we obtain are of the form C_{λ,θ} for suitable λ and θ.

So let T be a smooth curve of genus 1, degree n in P^{n−1} invariant under the diagonal action of μ_{n} given by (13). Since n is prime to 6 we know μ_{n} ↪ Jac(T) and so Jac(T) ≅ C_{λ} for some λ ∈ K. Now T is a torsor under C_{λ} and U := T/μ_{n} is a torsor under D_{λ}. But the intersection of T with one of the co-ordinate hyperplanes is an orbit under μ_{n} globally defined over K. Thus U is trivial as a torsor under D_{λ} and ξ_{T} ∈ WC(C_{λ}/K)[ϕ]. From the exact sequence (7) it follows T ≅ C_{λ,θ} for some θ ∈ K^{×}/(K^{×})^{n}.

1.3. THE MODULAR CURVES X_{1}(n) AND X(n) 15

Remark 1.9 The action (13) of μ_{n} on P^{n−1} is the unique one that lifts to GL_{n}. By unique here we mean unique up to change of co-ordinates defined over K. In general the obstruction to lifting μ_{n} ↪ PGL_{n} to μ_{n} ↪ GL_{n} is given by H^{1}(K, Z/nZ).

Remark 1.10 The above arguments show that the torsors parametrised by WC(C/K)[ϕ] may be embedded in P^{n−1}. This is also known for the torsors parametrised by � (C/K)[n], and we suspect the local assumptions here are essential. We refer to [CM] where it is explained how this result is a consequence of the Hasse Principle for Brauer-Severi varieties.

1.3 The Modular Curves X_{1}(n) and X(n)

In this section we recall the definitions of the modular curves X_{1}(n) and X(n) and record some of their most basic properties.

Definition 1.11 Let n ≥ 4. The pairs (E, P) where E is an elliptic curve and P ∈ E[n] is a point of order n, are parametrised by the smooth curve Y_{1}(n) → Spec Z[1/n].

Definition 1.12 Let n ≥ 3. The triples (E, i, Q) where E is an elliptic curve, i : μ_{n} ↪ E, Q ∈ E[n] and e_{n}(i(ζ), Q) = ζ for ζ ∈ μ_{n}, are parametrised by the smooth curve Y (n) → Spec Z[1/n].

We write X_{1}(n) and X(n) for the smooth projective curves containing as an open subset Y_{1}(n) and Y (n) respectively. For a proof of the existence of these moduli spaces, and the interpretation of X_{1}(n) and X(n) in terms of a moduli problem, we refer to [DR, Chapter IV]. We note that our restrictions on n are those required for the objects parametrised to have no automorphisms.

For simplicity we work over a field K with char(K) ∤ n. We do not assume μ_{n} ⊂ K. The relationship between X(n) and X_{1}(n) is as follows. There is an action of μ_{n} on X(n) given by

ζ : (E, i, Q) ↦ (E, i, Q + i(ζ))  (14)

with quotient X_{1}(n). In terms of function fields the map X(n) → X_{1}(n) is a Kummer extension of degree n. Taking α : X_{1}(n) → P^{1} a Kummer generator we see X(n) has affine model

{(λ, τ) ∈ X_{1}(n) × G_{m} : α(λ) = τ^{n}}.  (15)

In §1.1 we gave a construction of the curve X_{1}(n), and in §1.2 we computed the rational function α in the cases n = 5 and 7. This gives one possible construction of the modular curve X(n).

In Chapter 4 we discuss another construction of X(n). We work over an algebraically closed field so that Y (n) may be simply described as parametrising the triples (E, P, Q) with P, Q ∈ E[n] and e_{n}(P, Q) = ζ_{n}. There is no difficulty extending our calculations to a non-algebraically closed field, but we prefer not to carry round the notation required. We define an action of PSL_{2}(Z/nZ) on X(n) via

(\begin{matrix}a & b \\ c & d \end{matrix}) : (E, P, Q) ↦ (E, aP + cQ, bP + dQ).  (16)

The quotient by this action is the j-map j : X_{1}(n) → P^{1}. From the analytic theory we know that the j-map is ramified at 0, 1728 and ∞ with stabilisers of order 3, 2 and n respectively. We arrive at the classical formulae

# cusps on X(n) = (1/n) #PSL_{2}(Z/nZ)
genus X(n) = (n − 6)/(12n) #PSL_{2}(Z/nZ) + 1

(17)

where for n ≥ 3 we recall #PSL_{2}(Z/nZ) = (n^{3}/2) ∏_{p|n}(1 − 1/p^{2}).

We describe X(n) for some small values of n. For n = 3, 4, 5 the cusps are arranged as the vertices of one of the Platonic solids. The curve X(7) is the Klein quartic [K1] whose automorphism group is the simple group of order 168.

n  X(n)  PSL_{2}(Z/nZ)  cusps
3  P^{1}  A_{4}  tetrahedron
4  P^{1}  S_{4}  octrahedron
5  P^{1}  A_{5}  icosahedron
6  {y^{2} = x^{3} + 1}  S_{3} × A_{4}  E[1 − ω] ⊕ E[2]
7 {a^{3}b + b^{3}c + c^{3}a = 0}  G_{168}  points of inflection
⋮  ⋮  ⋮  ⋮

For p a prime we record the formulae

genus X(p) = (p + 2)(p − 3)(p − 5)/24
genus X_{1}(p) = (p − 5)(p − 7)/24. (18)

1.4 Tate Local Duality and Reciprocity Laws

In this section we review some results from class field theory and arithmetic duality theory. References for this are [CF], [Se2] and [Mi].

Local Results. Let K be a finite extension of Q_{p}. We write ord : K^{×} → Z for the normalised valuation and k for the residue field.

Let M be a finite G_{K}-module and let A be an abelian variety over K. We write M^{∨} for the Cartier dual Hom(M, G_{m}) and A^{∨} for the dual abelian variety Ext^{1}(A, G_{m}). We recall from [Mi] that there are naturally defined pairings

H^{r}(K, M^{∨}) × H^{2−r}(K, M) → Q/Z  (19)
H^{r}(K, A^{∨}) × H^{1−r}(K, A) → Q/Z.  (20)

Theorems 2.1 and 2.3 in [T1] state that (19) and (20) are non-degenerate for all r. One way to define (19) is as the pairing induced by cup product from M^{∨} ×M → G_{m} composed with the isomorphism Br(K) →^{inv} Q/Z. We discuss three examples with r = 1.

Example I Let M = Z/nZ. Then (19) becomes

(·, ·) : H^{1}(K, μ_{n}) × H^{1}(K, Z/nZ) → Z/nZ.  (21)

We identify H^{1}(K, μ_{n}) = K^{×}/(K^{×})^{n} and H^{1}(K, Z/nZ) = Hom(G_{K}, Z/nZ).

Lemma 1.13 For a ∈ K^{×}/(K^{×})^{n} and χ ∈ Hom(G_{K}, Z/nZ)

(a, χ) = −χ(ρ_{K}(a))

where ρ_{K} : K^{×} → G_{K}^{ab} is the local reciprocity map.

Proof. See [Se2, Chapter XIV]. Notice that we do not assume μ_{n} ⊂ K. □

Lemma 1.14 Let L/K be an unramified abelian extension of local fields and write Frob(L/K) ∈ Gal(L/K) for the Frobenius element. Then

ρ_{L/K}(a) = Frob(L/K)^{ord(a)}

where ρ_{L/K} : K^{×} → Gal(L/K) is the local reciprocity map.

Proof. [CF, Chapter VI, §2.5]. □

We write ℑ_{K} ⊂ G_{K} for the inertia subgroup.

Proposition 1.15 The “unit” and “not ramified” subgroups

H_{un}^{1}(K, μ_{n}) := { a ∈ K^{×}/(K^{×})^{n} | ord(a) ≡ 0 (mod n) }
H_{nr}^{1}(K, Z/nZ) := Hom(G_{K}/ℑ_{K}, Z/nZ)

are exact annihilators with respect to the Tate pairing (21).

Proof. Let a ∈ K^{×}/(K^{×})^{n} and χ ∈ Hom(G_{K}/ℑ_{K}, Z/nZ). Then χ factors through Gal(L/K) where L/K is an unramified extension. The last two lemmas tell us

(a, χ) = − ord(a) χ(Frob(L/K)).

Thus H_{un}^{1}(K, μ_{n}) is the exact annihilator of H_{nr}^{1}(K, Z/nZ). □

Remark 1.16 More generally for B a G_{K}-module it is usual to define

H_{nr}^{1}(K, B) := H^{1}(G_{K}/ℑ_{K}, B^{ℑ_{K}}) = ker(H^{1}(G_{K}, B) →^{res} H^{1}(ℑ_{K}, B)).

If char(k) ∤ n then the groups H_{un}^{1}(K, μ_{n}) and H_{nr}^{1}(K, μ_{n}) are identical.

Example II Let ϕ : C → D be an isogeny of elliptic curves defined over K with deg ϕ = n. The Weil pairing e_{ϕ} : C[ϕ] × D[ϕ̂] → μ_{n} induces a non-degenerate pairing

(·, ·) : H^{1}(K, C[ϕ]) × H^{1}(K, D[ϕ̂]) → Z/nZ.  (22)

Proposition 1.17 The images of the connecting maps

δ_{ϕ} : D(K) → H^{1}(K, C[ϕ]) and δϕ̂ : C(K) → H^{1}(K, D[ϕ̂])

are exact annihilators with respect to the Tate pairing (22).

Proof. The pairings (19) and (20) yield the following commutative diagram.

0 ⟶ D(K)/ϕC(K) ⟶^{δ_{ϕ}} H^{1}(K, C[ϕ]) ⟶ H^{1}(K, C)[ϕ] ⟶ 0
↓  ↓  ↓
0 ⟶ H^{1}(K, D)[ϕ̂]^{∗} ⟶ H^{1}(K, D[ϕ̂])^{∗} ⟶^{δϕ̂^{∗}} (C(K)/ϕ̂D(K))^{∗} ⟶ 0

Since the vertical maps are isomorphisms the exact annihilation is nothing more than the statement that the rows are exact. □

Remark 1.18 The non-degeneracy of the pairing (20) is in fact proved from the non-degeneracy of (19) by considering the above diagram in the case ϕ = ϕ̂ = [n] and employing some counting arguments.

Example III Assume μ_{n} ⊂ K. We fix a choice of isomorphism Z/nZ ≅ μ_{n}. Then the pairing (21) becomes the Hilbert norm residue symbol

(·, ·) : K^{×}/(K^{×})^{n} × K^{×}/(K^{×})^{n} → μ_{n}.

This pairing is skew-symmetric. By Lemma 1.13 an equivalent definition is

ρ_{K}(b)\sqrt[n]{a} = (a, b)\sqrt[n]{a}.  (23)

In particular the norm residue symbol does not depend upon our choice of isomorphism Z/nZ ≅ μ_{n}.

Global Results. Let L/K be a finite abelian extension of number fields. Let S be the set of primes ramifying in L/K. For 𝔭 ∉ S we write (𝔭, L/K) for the Frobenius element in Gal(L/K). Extending linearly gives the global reciprocity map

( · , L/K) : I_{K}^{S} → Gal(L/K).

Assuming μ_{n} ⊂ K we define the power residue symbol (a/b) ∈ μ_{n} via

(b, K(\sqrt[n]{a})/K)\sqrt[n]{a} = (a/b)\sqrt[n]{a}  (24)

where a, b ∈ K^{×} and ord_{𝔭}(b) = 0 for all 𝔭 ramifying in K(\sqrt[n]{a})/K. Thus (a/𝔭) = 1 if and only if a ∈ (K_{𝔭}^{×})^{n}. The aim of the nth power reciprocity law is to relate the symbols (a/b) and (b/a). The next two lemmas indicate how this may be achieved using the local Hilbert norm residue symbols (·, ·)_{𝔭}.

Lemma 1.19 Whenever the symbol (a/b) is defined we have

(a/b) = ∏_{𝔭∈B}(a, b)_{𝔭}

where B is any set of primes unramified in K(\sqrt[n]{a})/K and containing all primes 𝔭 with ord_{𝔭}(b) ≠ 0.

Proof. We compare (23) and (24) using Lemma 1.14. □

20 CHAPTER 1. PRELIMINARIES Lemma 1.20 Let a, b ∈ K^{×} and assume n > 2. Then

∏_{𝔭}(a, b)_{𝔭} = 1.

Proof. This follows from the exact sequence

0 ⟶ Br(K) ⟶ ⊕_{v} Br(K_{v}) \underrightarrow{∑ inv} Q/Z ⟶ 0.

We assume n > 2 here so as to ignore the infinite places. □

Likewise there is a “product formula” for the local Tate pairings (·, ·)_{𝔭} defined by (21) and (22).

Descent Theorems over Q

2.1 Statement of the First Descent Theorem

Let n = 5 or 7. We consider the elliptic curves C_{λ} and D_{λ} over Q with μ_{n} ↪ C_{λ} and Z/nZ ↪ D_{λ}. For λ ∈ Q we recall D_{λ} has equation

\begin{aligned}y^{2} + (1 − λ)xy − λy & = & x^{3} − λx^{2} & λ ≠ 0 \\ y^{2} + (1 + λ − λ^{2})xy + (λ^{2} − λ^{3})y & = & x^{3} + (λ^{2} − λ^{3})x^{2} & λ ≠ 0, 1. \end{aligned}

In each case Z/nZ ↪ D_{λ} is generated by (x, y) = (0, 0) and the isogenous curve C_{λ} is defined as the quotient by this subgroup. We write ϕ : C_{λ} → D_{λ} for the n-isogeny with C[ϕ] ≅ μ_{n} and D[ϕ̂] ≅ Z/nZ. In the case n = 5 we put

A = { p prime | ord_{p}(λ) < 0 or λ ≡ 0 (mod p) }
B = \{p prime | \begin{matrix}λ^{2} − 11λ − 1 ≡ 0 (mod p) and p ≡ 1 (mod 5) \\ or p = 5 and λ ≡ 18 (mod 25) \end{matrix} \}

and in the case n = 7 we put

A = { p prime | ord_{p}(λ) < 0 or λ ≡ 0, 1 (mod p) }
B = \{p prime | \begin{matrix}λ^{3} − 8λ^{2} + 5λ + 1 ≡ 0 (mod p) and p ≡ 1 (mod 7) \\ or p = 7 and λ ≡ 5 (mod 7) \end{matrix} \}.

For S a set of rational primes, we write [S] for the subspace of Q^{×}/(Q^{×})^{n} generated by S.

Theorem 1 Let n = 5 or 7 and let λ ∈ Q with λ ≠ 0, respectively λ ≠ 0, 1. Let A and B be the disjoint sets of rational primes defined above. Then the Selmer group attached to the n-isogeny ϕ : C_{λ} → D_{λ} is given by

S^{(ϕ)}(C_{λ}/Q) ≅ { θ ∈ [A] | θ ∈ (Q_{p}^{×})^{n} for all p ∈ B }.

The contribution to S^{(ϕ)}(C_{λ}/Q) coming from Z/nZ ↪ D_{λ}(Q) is generated by λ, respectively λ^{4}(λ−1). Furthermore there is a pairing Ξ : [A]×[B] → Z/nZ such that S^{(ϕ)}(C_{λ}/Q) ≅ ker_{L}(Ξ) and S^{(ϕ̂)}(D_{λ}/Q) ≅ ker_{R}(Ξ).

To help us carry the cases n = 5 and n = 7 together we shall from now on write α(λ) and β(λ) for the polynomials

α(λ) = \{ \begin{matrix}λ \\ λ^{4}(λ − 1) \end{matrix} β(λ) = \{ \begin{matrix}λ^{2} − 11λ − 1 \\ λ^{3} − 8λ^{2} + 5λ + 1. \end{matrix}

Thus the condition for p ≡ 1 (mod n) to belong to B is β(λ) ≡ 0 (mod p). In view of the identity β(λ) = (λ + 7)(λ − 18) + 125, respectively β(λ) = (λ+2)(λ−5)^{2}−49, the condition for n to belong to B is β(λ) ≡ 0 (mod 125), respectively β(λ) ≡ 0 (mod 49).

Lemma 2.1 For p a rational prime, the congruence β(λ) ≡ 0 (mod p) is soluble if and only if p ≡ 0, ±1 (mod n).

Proof. Consider how p factors in Q(μ_{n}) ∩ R. □

Remark 2.2 The pairing Ξ is straightforward to compute. For each prime q ∈ B we choose a non-trivial character χ_{q} : (Z/qZ)^{×} → Z/nZ, respectively χ_{n} : (Z/n^{2}Z)^{×} → Z/nZ. Then Ξ is represented by the matrix

[A, B] := (χ_{q}(p))_{p∈A,q∈B}.  (25)

We sketch the proof of Theorem 1. The definition (8) reads

S^{(ϕ)}(C_{λ}/Q) = { θ ∈ Q^{×}/(Q^{×})^{n} | C_{λ,θ}(Q_{p}) ≠ ∅ for all primes p }.

In §3.1 we give equations for the C_{λ,θ} as curves in P^{n−1}. We then describe the geometry of the reductions mod p and so derive some simple criteria for the existence of local points. We find that S^{(ϕ)}(C_{λ}/Q) is the subspace of [A] consisting of nth powers modulo the primes in B. To treat the dual isogeny ϕ̂ the rough idea is that the involution η swaps over the sets of primes A and B, so S^{(ϕ̂)}(D_{λ}/Q) is the subspace of [B] consisting of nth powers modulo the primes in A. The existence of the pairing Ξ is then nothing more than a statement of the nth power reciprocity law. To make these ideas precise we must work over Q(μ_{n}). Since [Q(μ_{n}) : Q] = n − 1 is prime to n, there is no difficulty in descending back to Q.

In Chapter 3 we give a proof of Theorem 1 working entirely over Q. This proof suffers from not appearing as natural as the method sketched above, but involves considerably less work. In Chapter 5 we generalise our descent calculations to K = Q(μ_{n}) and give an alternative proof of Theorem 1 much closer to the above description.

We should mention that in the case p = n the projective space method fails to give complete criteria for the existence of local points. Our solution is to revert to the push-out method and perform some rather delicate Hensel lemma calculations. It is these calculations which make it difficult to work over a general number field.

2.2 Examples and Applications

Let n = 5 or 7 and let λ ∈ Q with λ ≠ 0, respectively λ ≠ 0, 1. Theorem 1 gives us information about both the Mordell-Weil groups C_{λ}(Q) and D_{λ}(Q) and the Tate-Shafarevich groups � (C_{λ}/Q) and � (D_{λ}/Q). We now make this explicit. Following Mazur’s classification of torsion groups for elliptic curves over Q we find

D_{λ}(Q)_{tors} ≅ \{ \begin{matrix}Z/5Z or Z/10Z & if n = 5 \\ Z/7Z & if n = 7. \end{matrix}

Let r = rank C_{λ}(Q) = rank D_{λ}(Q) and ι = dim_{n} C_{λ}(Q)[n]. Then ι = 0 unless α(λ) ∈ (Q^{×})^{n} and this can only happen in the case n = 5. Now Theorem 1 together with the exact sequences

0 → D_{λ}(Q)/ϕC_{λ}(Q) → S^{(ϕ)}(C_{λ}/Q) →^{ϕ} � (C_{λ}/Q)[ϕ] → 0

0 → C_{λ}(Q)/ϕ̂D_{λ}(Q) → S^{(ϕ̂)}(D_{λ}/Q) \underrightarrow{ϕ̂} � (D_{λ}/Q)[ϕ̂] → 0

tells us r = r_{1} + r_{2} with r_{1}, r_{2} ≥ 0 and

|A| − rank(Ξ) = r_{1} + 1 − ι + dim_{n} � (C_{λ}/Q)[ϕ]
|B| − rank(Ξ) = r_{2} + ι + dim_{n} � (D_{λ}/Q)[ϕ̂].

(26)

Thus the upper bound for the rank provided by first descent is

|A| + |B| − 1 − 2 rank(Ξ).  (27)

In the following examples we make use of the trivial exact sequences

\begin{matrix}0 → � (C_{λ}/Q)[ϕ] → � (C_{λ}/Q)[n] → � (D_{λ}/Q)[ϕ̂] \\ 0 → � (D_{λ}/Q)[ϕ̂] → � (D_{λ}/Q)[n] → � (C_{λ}/Q)[ϕ]. \end{matrix}

Example 2.3 Let n = 5 and λ = 38. Then A = {2, 19} and B = {41}. Since 2^{8} ≢ 1 (mod 41) it is clear that Ξ has rank 1. By Theorem 1 we deduce

\begin{matrix}C_{38}(Q) = 0 & � (C_{38}/Q)(5) = 0 \\ D_{38}(Q) ≅ Z/5Z & � (D_{38}/Q)(5) = 0. \end{matrix}

Example 2.4 Let n = 7 and λ = 8. Then A = {2, 7} and B = ∅. A computer search yields the point of infinite order (x, y) = (30, 198) on D_{8}. By Theorem 1 we deduce

C_{8}(Q) ≅ Z  � (C_{8}/Q)(7) = 0
D_{8}(Q) ≅ Z/7Z ⊕ Z  � (D_{8}/Q)(7) = 0.

We may use Theorem 1 together with Cremona’s program mwrank, which performs a 2-descent, to exhibit some elements of the Tate-Shafarevich group. As explained in the introduction these elements are explicit in the sense that we are able to give equations in P^{n−1} for the corresponding curves of genus 1 that violate the Hasse principle.

Example 2.5 Let n = 5 and λ = −60, −42, −30, 30, 60 or 90. In all these examples we have |A| = 3 and B = ∅. According to mwrank rank D_{λ}(Q) = 0. By Theorem 1 we deduce

� (C_{λ}/Q)[5] ≅ (Z/5Z)^{2}.

Example 2.6 Let n = 7 and λ = −6, −5, 6, 7, 10 or 11. In all these examples |A| = 3 and B = ∅. According to mwrank rank D_{λ}(Q) = 0. By Theorem 1 we deduce

� (C_{λ}/Q)[7] ≅ (Z/7Z)^{2}.

Example 2.7 Let n = 5 and λ = ±30/7, ±35/6 or 14/15. In all these examples A = {2, 3, 5, 7} and B = ∅. According to mwrank rank D_{λ}(Q) = 1. By Theorem 1 we deduce

� (C_{λ}/Q)[5] ≅ (Z/5Z)^{2}.

So far we have exhibited elements of � by comparing our n-descent with a 2-descent. We could equally compare our n-descent with values of L-functions and then cite the work of Kolyvagin [Ko]. This allows us to give examples beyond the reach of mwrank. We tabulate many further examples in Appendix D.

Let us assume � (C_{λ}/Q)(n) and � (D_{λ}/Q)(n) are finite. Since the maps ϕ and ϕ̂ on Tate-Shafarevich groups are adjoints with respect to the Cassels-Tate pairing it follows

dim_{n} � (C_{λ}/Q)[ϕ] + dim_{n} � (D_{λ}/Q)[ϕ̂]

is even. In other words our estimate (27) for the rank differs from the true rank by an even integer. This leads to the following parity result.

Corollary 1 Let E/Q be an elliptic curve with a rational point of order n = 5 or 7. Assume � (E/Q)(n) is finite. Then the parity of the rank of E(Q) is as predicted by the Birch Swinnerton-Dyer conjecture.

Proof. Theorem 1 and the finiteness assumption on � tells us

rank E(Q) ≡ |A| + |B| − 1 (mod 2).

On the other hand the Birch Swinnerton-Dyer conjecture predicts

(−1)^{rank E(Q)} = ∏_{v} W_{v}

where W_{v} = ±1 are the local root numbers. Since W_{∞} = −1 for any elliptic curve over Q it suffices for us to prove

{p prime | W_{p} = −1} = A ∪ B.  (28)

The following explicit description of the local root numbers for an elliptic curve over Q is taken from [Ro]. See also [Con].

Proposition 2.8 Let E be an elliptic curve over Q and let p be a prime.

(i) If E has good reduction at p then W_{p} = +1. (ii) If E has multiplicative reduction at p then

W_{p} = \{ \begin{matrix}−1 & if E has split reduction \\ +1 & if E has non-split reduction. \end{matrix}

(iii) If E has additive reduction at p and p ≥ 5 then

W_{p} = \{ \begin{matrix}(−1/p) & if e = 2 or 6 \\ (−3/p) & if e = 3 \\ (−2/p) & if e = 4 \end{matrix}

where (a/p) is the quadratic Legendre symbol and e = 12/ gcd(ord_{p}(Δ), 12).

Our claim (28) now follows from the description of the reduction types in Lemma 1.4. The following table summarises the calculations required in the case of additive reduction.

p = n  λ  ord_{p}(β(λ)) e  W_{p}
5 ord_{5}(λ − 18) = 1  2  6  +1
5 ord_{5}(λ − 18) ≥ 2  3  4  −1
7  ord_{7}(λ − 5) ≥ 1  2  6  −1
□

We may use Theorem 1 to exhibit elliptic curves with large n-Selmer group. To do this we choose λ ∈ Q such that |A| is large and |B| is small, or vice versa. I am grateful to Ed Schaefer for bringing to my attention the following result.

Proposition 2.9 Let F (x) be an irreducible polynomial of degree d with integer coefficients. Suppose that for all primes p the congruence F (x) ≡ 0 (mod p) has fewer than p solutions. Then there exist infinitely many integers y such that F (y) consists of at most d + 1 prime factors.

Proof. [HR, Theorem 9.7]. □

Corollary 2 For n = 5 or 7 the n-Selmer group of an elliptic curve over Q may become arbitrarily large.

Proof. Let M be the product of the first m primes. Applying Proposition 2.9 to the polynomial F (x) = β(Mx) gives infinitely many integers λ = My with |A| ≥ m and |B| ≤ (n + 1)/2. The corollary is now immediate from Theorem 1. □

We could equally prove Corollary 2 by taking |A| small and |B| large. Lemma 2.1 shows how we may force |B| to be large by imposing linear congruences on λ. In the case n = 5 we then appeal to Dirichlet’s theorem on primes in arithmetic progression to give |A| = 1. (In fact this is all we shall need in the next section to show that the 5-torsion of the Tate-Shafarevich group may become arbitrarily large.) In the case n = 7 we appeal to [HR, Theorem 10.11] which generalises Proposition 2.9 to F (x) a product of distinct irreducible polynomials. It seems we must take λ = (integer)/2 in order to satisfy the hypothesis at p = 2.

2.3 Exhibiting Large Tate-Shafarevich Groups

Let E/Q be an elliptic curve with E[5] ≅ μ_{5}⊕Z/5Z. Then E belongs to both the families C_{λ} and D_{λ}. We may estimate rank E(Q) by applying Theorem 1 for two different values of λ. Arranging for our estimates to differ we prove

Corollary 3 The 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

We define polynomials

\begin{aligned}f(τ) & = & τ^{4} + 3τ^{3} + 4τ^{2} + 2τ + 1 & = & ((τ + 1)^{5} + 1)/(τ + 2) \\ g(τ) & = & τ^{4} − 2τ^{3} + 4τ^{2} − 3τ + 1 & = & ((τ − 1)^{5} + τ^{5})/(2τ − 1). \end{aligned}

Lemma 2.10 The elliptic curves E/Q with E[5] ≅ μ_{5}⊕Z/5Z are the curves E_{τ} := C_{τ^{5}} ≅ D_{τf(τ)/g(τ)} for τ ∈ Q with τ ≠ 0.

Proof. We know C_{λ}(Q)[5] ≠ 0 if and only if λ ∈ (Q^{×})^{5}. So as explained in §1.3 the required universal family is E_{τ} = C_{τ^{5}}. Here τ is a co-ordinate on X(5) ≅ P^{1} and the cusps are at τ = 0, ∞, ζ^{i}ϕ, ζ^{i}\overline{ϕ}, where ϕ = 1 + ζ + ζ^{4} is the golden ratio and \overline{ϕ} = 1 + ζ^{2} + ζ^{3} is its conjugate. There are involutions

\begin{matrix}η : X_{1}(5) → X_{1}(5) ; & λ ↦ (ϕ^{5}λ + 1)/(λ − ϕ^{5}) \\ ε : X(5) → X(5) ; & τ ↦ (ϕτ + 1)/(τ − ϕ). \end{matrix}

We know C_{λ} ≅ D_{η(λ)} over Q(μ_{5}). In Chapter 4 we explicitly describe the action (16) of PSL_{2}(Z/5Z) on X(5). We find that the involution ε swaps over the subgroups μ_{5} and Z/5Z. The following isomorphisms are defined over Q(μ_{n}).

C_{τ^{5}} ≅ E_{τ} ≅ E_{ε(τ)} ≅ C_{ε(τ)^{5}} ≅ D_{η(ε(τ)^{5})}  (29)

We claim η(ε(τ)^{5}) = τf(τ)/g(τ). Indeed both sides have zeros at τ = 0, ζ^{2}ϕ, ζ^{3}ϕ, ζ\overline{ϕ}, ζ^{4}\overline{ϕ} and poles at τ = ∞, ζϕ, ζ^{4}ϕ, ζ^{2}\overline{ϕ}, ζ^{3}\overline{ϕ}. It suffices to show equality at the single value τ = ϕ. But f(τ) = τ^{4}g(−1/τ) and g(τ) = g(1−τ), so f(ϕ) = ϕ^{4}g(\overline{ϕ}) = ϕ^{4}g(ϕ) and η(ε(ϕ)^{5}) = ϕ^{5} = ϕf(ϕ)/g(ϕ) as required. The isomorphism (29) maps

Z/5Z ↪ C_{τ^{5}}  onto  Z/5Z ↪ D_{τf(τ)/g(τ)}.

By Lemma 1.1(iii) we deduce C_{τ^{5}} ≅ D_{τf(τ)/g(τ)} over Q. □

For τ ∈ Q with τ ≠ 0 we define disjoint sets of rational primes

P = { p prime | ord_{p}(τ) ≠ 0 }
Q = { p prime | f(τ)g(τ) ≡ 0 (mod p) and p ≡ 1 (mod 5) }
R = { p prime | τ^{2} − τ − 1 ≡ 0 (mod p) and p ≡ 0, 1 (mod 5) }.

Lemma 2.11 For p a rational prime

(i) f(τ) ≡ 0 (mod p) is soluble ⟺ p = 5 or p ≡ 1 (mod 5) (ii) g(τ) ≡ 0 (mod p) is soluble ⟺ p = 5 or p ≡ 1 (mod 5) (iii) τ^{2} − τ − 1 ≡ 0 (mod p) is soluble ⟺ p = 5 or p ≡ ±1 (mod 5).

Proof. Consider how p factors in Q(μ_{5}). □

Lemma 2.12 We apply Theorem 1 for λ = τ^{5} and λ = τf(τ)/g(τ).

(i) If λ = τ^{5} then A = P and B = Q ∪ R. (ii) If λ = τf(τ)/g(τ) then A = P ∪ Q and B = R.

Proof. These claims follow from the identities

\begin{aligned}τ^{10} − 11τ^{5} − 1 & = & (τ^{2} − τ − 1)f(τ)g(τ) \\ τ^{2}f(τ)^{2} − 11τf(τ)g(τ) − g(τ)^{2} & = & (τ^{2} − τ − 1)^{5}. \end{aligned}

For τ prime to 5 we are assisted by the observation

ord_{5}(τ^{2} − τ − 1) = ord_{5}(f(τ)) = ord_{5}(g(τ)) = 0 or 1.
□

Proof of Corollary 3. Let τ ∈ Q with τ ≠ 0. Theorem 1 gives two estimates for rank E_{τ}(Q). In the notation of Remark 2.2 these are

|P| + |Q| + |R| − 1 − 2 rank[P, Q ∪ R]

|P| + |Q| + |R| − 1 − 2 rank[P ∪ Q, R]. (30)

By repeated use of Dirichlet’s theorem on primes in arithmetic progression we choose Q_{0} and R_{0} sets of primes p ≡ 1 (mod 5) with rank[Q_{0}, R_{0}] arbitrarily large. With one final application of Dirichlet we choose τ prime such that Q ⊃ Q_{0} and R ⊃ R_{0}. Then |P| = 1 and

rank[P, Q ∪ R] ≪ rank[P ∪ Q, R].

The difference in the estimates (30) now proves the corollary. □

A closer inspection of the proof of Corollary 3 shows that we are forcing elements of order 5 in the Tate-Shafarevich group of the elliptic curves isogenous to E_{τ} rather than for E_{τ} itself.

Example 2.13 For τ = 2 we find P = {2}, Q = {11, 61}, R = ∅. Taking λ = τ^{5} = 32 in Theorem 1 gives rank E_{2}(Q) = 0 and � (E_{2}/Q)(5) = 0. Taking λ = τf(τ)/g(τ) = 122/11 gives

� (C_{122/11}/Q)(5) ≅ (Z/5Z)^{2}.

Example 2.14 For τ = 8 we find P = {2}, Q = {661, 1181}, R = {5, 11}. Taking λ = τf(τ)/g(τ) = 9448/661 in Theorem 1 gives rank E_{8}(Q) = 0 and � (E_{8}/Q)(5) = 0. Taking λ = τ^{5} = 2^{15} gives

� (D_{2^{15}}/Q)(5) ≅ (Z/5Z)^{2}.

By computer search we find many more examples where one descent calculation shows the rank is zero and the other exhibits elements in the Tate-Shafarevich group. For example, writing E′_{τ} = D_{τ^{5}} and E′′_{τ} = C_{τf(τ)/g(τ)}

\begin{aligned}� (E′_{2003}/Q)(5) & ≅ & (Z/5Z)^{6} \\ � (E′′_{6930}/Q)(5) & ≅ & (Z/5Z)^{10}. \end{aligned}

2.4 Statement of the Second Descent Theorem

We continue to consider the elliptic curves E = E_{τ} whose 5-torsion splits as μ_{5} ⊕Z/5Z. By comparing our first descent calculations with a description of the Cassels-Tate pairing due to Beaver [Be] we arrive at the following second descent theorem.

Theorem 2 Let τ ∈ Q with τ ≠ 0 and let P, Q, R be the disjoint sets of rational primes defined in §2.3. For p ∈ Q ∪ R choose non-trivial characters χ_{p} : (Z/pZ)^{×} → Z/5Z, respectively χ_{5} : (Z/25Z)^{×} → Z/5Z and suppose the characters χ_{p} for p ∈ Q are compatible in a sense to be explained below. Then S^{(5)}(E/Q) ≅ ker(Ψ) where

Ψ = (\begin{matrix}0 & [P, Q] & [P, R] \\ −[P, Q]^{T} & [Q, Q] − [Q, Q]^{T} & [Q, R] \\ −[P, R]^{T} & −[Q, R]^{T} & 0 \end{matrix})

and we use the notation [P, Q] for the matrix (χ_{q}(p))_{p∈P,q∈Q}.

Remark 2.15 For p ∈ Q the quantity χ_{p}(p) is undefined. It is to be understood that Ψ is a skew-symmetric matrix, and that the diagonal entries are therefore zero.

Remark 2.16 At each prime of bad reduction for E, our point τ on X(5) reduces to one of the cusps τ = 0, ∞, ζ^{i}ϕ, ζ^{i}\overline{ϕ}. In fact

P = { p prime | τ ≡ 0, ∞  (mod p) }
Q ⊂ \{ p prime | τ ≡ ζ^{i}ϕ, ζ^{i}\overline{ϕ} (mod p) \}
R ⊂ \{ p prime | τ ≡ ϕ, \overline{ϕ}  (mod p) \}

If we fix the value of i, then each prime p ∈ Q comes equipped with a distinguished element ζ of order 5 in (Z/pZ)^{×}. We then take χ_{p} to be the composite of the maps x ↦ x^{(p−1)/5} and ζ ↦ 1, and say that the characters χ_{p} are compatible if chosen in this way.

Remark 2.17 The statement of Theorem 2 leaves a certain amount of freedom in choosing our characters χ_{p}. It is clear by row and column operations that these choices do not alter the rank of Ψ.

The estimate for rank E_{τ}(Q) provided by Theorem 2 is

|P| + |Q| + |R| − 1 − rank Ψ.  (31)

This estimate is at most the minimum of the two estimates (30) provided by first descent. We give some examples where it is strictly smaller. For many further examples see Table 5 in Appendix D.

Example 2.18 Let τ = 7. Then P = {7}, Q = {11, 31, 61, 331} and R = {41}. After making a compatible choice of characters the matrix Ψ is

7 11 31 61 331 41
7 0 4 3 2 1 1
11 1 0 1 2 2 2
31 2 4 0 2 1 2
61 3 3 3 0 2 1
331 4 3 4 3 0 0
41 4 3 3 4 0 0

This matrix has rank 4 so by Theorem 2, rank E_{7}(Q) ≤ 1. This improves on the estimate rank E_{7}(Q) ≤ 3 given by Theorem 1 with λ = 7^{5} or λ = 25487/1891. Running Cremona’s program findinf for the Weierstrass equation (2) with λ = 7^{5} we exhibit the point of infinite order

(x, y) = (−149525/923^{2}, 29379375/923^{3}).

Thus rank E_{7}(Q) = 1 and

\begin{aligned}� (E_{7}/Q)(5) & = & 0 \\ � (E′_{7}/Q)(5) & ≅ & (Z/5Z)^{2} \\ � (E′′_{7}/Q)(5) & ≅ & (Z/5Z)^{2}. \end{aligned}

Example 2.19 Let τ = 12. Then P = {2, 3}, Q = {11, 71, 251, 2411} and R = {131}. After making a compatible choice of characters the matrix Ψ is

2 3 11 71 251 2411 131
2 0 0 1 1 0  0  1
3 0 0 3 1 1  4  2
11 4 2 0 2 0  2  1
71 4 4 3 0 0  0  0
251 0 4 0 0 0  1  1
2411 0 1 3 0 4  0  1
131 4 3 4 0 4  4  0

This matrix has rank 6, so by Theorem 2, rank E_{12}(Q) = 0. This improves on the estimates rank E_{12}(Q) ≤ 4 and rank E_{12}(Q) ≤ 2 given by Theorem 1 for λ = 12^{5} and λ = 318252/17821 respectively. We deduce

\begin{aligned}� (E_{12}/Q)(5) & = & 0 \\ � (E′_{12}/Q)(5) & ≅ & (Z/5Z)^{4} \\ � (E′′_{12}/Q)(5) & ≅ & (Z/5Z)^{2}. \end{aligned}

These examples show how we may explicitly exhibit elements of � for the curves E′_{τ} and E′′_{τ}. Our methods are less suited to finding elements of � for E_{τ} itself. However, for τ = 9 the L-value listed in Table 5 is non-zero, so by Kolyvagin’s theorem such elements certainly do exist.

For τ ∈ Z with |τ| ≤ 10000 we estimate rank E_{τ}(Q) using Theorem 1 with λ = τ and λ = τf(τ)/g(τ) and then using Theorem 2. We record how these estimates are distributed.

estimate for  Theorem 1  Theorem 2
rank E_{τ}(Q)  λ = τ^{5} λ = τf(τ)/g(τ)
0  635  66  7676
1  2199  297  9948
2  3696  845  2252
3  4456  1731  117
4  3984  2875  7
5  2819  3854  0
6  1442  3890  0
7  547  3164  0
8  173  1891  0
9  43  951  0
10  5  352  0
11  1  64  0
12  0  16  0
13  0  4  0

We conclude that the elliptic curves isogenous to E_{τ} tend to have large 5torsion of � , whereas E_{τ} itself does not. Thus a full 5-descent on E_{τ} gives a much better estimate for the rank than does descent via 5-isogeny.

Proof of the First Descent Theorem

3.1 Equations for Torsors in P^{n−1}

Let n = 5 or 7. As explained in §1.2 we aim to give explicit equations for the torsors C_{λ,θ} as curves in P^{n−1}. We work over a perfect field K and until further notice assume char(K) ≠ n.

Let T ↪ P^{n−1} be a smooth curve of genus 1 and degree n which is invariant under the action μ_{n} ↪ Aut(P^{n−1}) given by

ζ ↦ Diag(1 : ζ : …: ζ^{n−1}).  (32)

We write (x_{0} : …: x_{n−1}) for our co-ordinates on P^{n−1} and agree that all subscripts are to be read mod n. The hyperplanes fixed by (32) are the co-ordinate hyperplanes H_{i} := {x_{i} = 0}. As explained in §1.2 the action of Jac(T)[n] on T lifts to P^{n−1}. Since n is prime to 6 we know μ_{n} ↪ Jac(T) and so this action is generated by

M_{1} = (\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ & 0 & ⋯ & 0 \\ 0 & 0 & ζ^{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ^{n−1} \end{matrix}) M_{2} = ( \begin{matrix}0 & 0 & ⋯ & 0 & ∗ \\ ∗ & 0 & ⋯ & 0 & 0 \\ 0 & ∗ & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & ∗ & 0 \end{matrix} )

where ∗ denotes a non-zero element of \overline{K}. In Chapter 4 we shall see that the curve T is defined by 5 quadrics, respectively 14 quadrics. The action of M_{1} divides these quadrics up into n-eigenspaces, and the action of M_{2} tells us that these eigenspaces all have the same dimension. Concretely

H^{0}(P^{n−1}, I_{T} (2)) = ⊕_{i}V_{i} with V_{i} ⊂ ⟨x_{i}^{2}, x_{i−1}x_{i+1}, … ⟩.

Lemma 3.1 Let T ↪ P^{n−1} satisfy the above hypotheses.

(i) The curve T meets the hyperplanes H_{i} in a total of n^{2} points. (ii) Every non-zero quadric containing T has at least three non-zero terms.

Proof. (i) Let P ∈ T ∩ H_{0}. Then the translates of P under M_{1} and M_{2} give the required n^{2} distinct points. (ii) Suppose V_{0} contains a quadric with exactly two non-zero terms. Then the n^{2} points in (i) fail to be distinct. □

Remark 3.2 We sometimes refer to these points of intersection with the coordinate hyperplanes as flexes, since they are the points P with n.P linearly equivalent to the hyperplane section.

We recall from §2.1 our notation

α(λ) = \{ \begin{matrix}λ \\ λ^{4}(λ − 1) \end{matrix} β(λ) = \{ \begin{matrix}λ^{2} − 11λ − 1 \\ λ^{3} − 8λ^{2} + 5λ + 1. \end{matrix}

Definition 3.3 Let λ, τ_{0}, …, τ_{n−1} ∈ K satisfy α(λ) = ∏ τ_{i} ≠ 0. We define T [λ; τ_{0}, …, τ_{n−1}] to be the subscheme of P^{n−1} with equations

n = 5  τ_{0}x_{0}^{2} + x_{1}x_{4} − τ_{2}τ_{3}x_{2}x_{3} = 0 & cyclic permutes
n = 7 \{ \begin{matrix}τ_{0}x_{0}^{2} + x_{1}x_{6} − (1/λ^{2})τ_{2}τ_{3}τ_{4}τ_{5}x_{2}x_{5} = 0 \\ τ_{0}x_{0}^{2} + λx_{1}x_{6} − (1/λ^{3})τ_{2}τ_{3}^{2}τ_{4}^{2}τ_{5}x_{3}x_{4} = 0 \end{matrix} & cyclic permutes.

Proposition 3.4 Every curve T ↪ P^{n−1} satisfying the above hypotheses is equal to T [λ; τ_{0}, …, τ_{n−1}] for some λ, τ_{0}, …, τ_{n−1} ∈ K.

Proof. (i) Case n = 5. Lemma 3.1(ii) tells us that T has equations

τ_{0}x_{0}^{2} + x_{1}x_{4} + a_{0}x_{2}x_{3} = 0 & cyclic permutes

for some non-zero constants τ_{i}, a_{i} ∈ K. By Lemma 3.1(i) we may choose (0 : z_{1} : z_{2} : z_{3} : z_{4}) ∈ T ∩ H_{0} with all z_{i} ≠ 0. Then

\begin{matrix}z_{1}z_{4} + a_{0}z_{2}z_{3} & = & 0 \\ τ_{2}z_{2}^{2} + z_{1}z_{3} & = & 0 \\ τ_{3}z_{3}^{2} + z_{2}z_{4} & = & 0 \end{matrix} \} ⟹ a_{0} = −τ_{2}τ_{3}

3.1. EQUATIONS FOR TORSORS IN P^{n−1} 35

and by symmetry this completes the proof in the case n = 5. (ii) Case n = 7. Lemma 3.1(ii) tells us that T has equations

\begin{matrix}τ_{0}x_{0}^{2} + x_{1}x_{6} + a_{0}x_{2}x_{5} = 0 \\ τ_{0}x_{0}^{2} + λ_{0}x_{1}x_{6} + b_{0}x_{3}x_{4} = 0 \end{matrix}  & cyclic permutes

for some non-zero constants τ_{i}, λ_{i}, a_{i}, b_{i} ∈ K. The action of M_{2} tells us λ_{0} = λ_{1} = … = λ_{n−1} and we write λ for the common value. Similarly, the action of M_{2} suggests we write a_{0} = aτ_{2}τ_{3}τ_{4}τ_{5} and b_{0} = bτ_{2}τ_{3}^{2}τ_{4}^{2}τ_{5} since these relations will still hold when we cyclically permute the subscripts. It remains to see how the quantities a, b, λ and ∏ τ_{i} are related. By Lemma 3.1(i) we may choose (0 : z_{1} : …: z_{6}) ∈ T ∩ H_{0} with all z_{i} ≠ 0. We find

τ_{2}z_{2}^{2} + z_{1}z_{3} = τ_{3}z_{3}^{2} + λz_{2}z_{4} = τ_{4}z_{4}^{2} + λz_{3}z_{5} = τ_{5}z_{5}^{2} + z_{4}z_{6} = 0.  (33)

If ∏ z_{i}^{μ_{i}} and ∏ z_{i}^{ν_{i}} are two monomials with ∑ μ_{i} = ∑ ν_{i} and ∑ iμ_{i} = ∑ iν_{i} then (33) allows us to express their ratio in terms of λ and the τ_{i}. In particular

\begin{aligned}z_{1}z_{6} + aτ_{2}τ_{3}τ_{4}τ_{5}z_{2}z_{5} = 0 & ⟹ & a = −1/λ^{2} \\ λz_{1}z_{6} + bτ_{2}τ_{3}^{2}τ_{4}^{2}τ_{5}z_{3}z_{4} = 0 & ⟹ & b = −1/λ^{3} \\ \begin{matrix}(λ − 1)z_{1}z_{3} + bτ_{4}τ_{5}^{2}τ_{6}^{2}τ_{0}z_{5}z_{6} = 0 \\ τ_{1}z_{1}^{2} + bτ_{3}τ_{4}^{2}τ_{5}^{2}τ_{6}z_{4}z_{5} = 0 \end{matrix} \} & ⟹ & λ^{4}(λ − 1) = ∏ τ_{i}. \end{aligned}

This completes the proof in the case n = 7. □

We proceed to study the schemes T [λ; τ_{0}, …, τ_{n−1}].

Lemma 3.5 Up to rescaling co-ordinates, the schemes T [λ; τ_{0}, …, τ_{n−1}] over K are uniquely determined by λ ∈ K and θ = ∏ τ_{i}^{−i} ∈ K^{×}/(K^{×})^{n}.

Proof. Rescaling the co-ordinate x_{1} by u ∈ K^{×} gives

T [λ; τ_{0}, τ_{1}, τ_{2}, …, τ_{n−1}] ≅ T [λ; uτ_{0}, u^{−2}τ_{1}, uτ_{2}, …, τ_{n−1}].

Repeating for the other co-ordinates we find

T [λ; τ_{0}, τ_{1}, τ_{2}, …, τ_{n−1}] ≅ T [λ; u^{−n}τ_{0}, u^{n}τ_{1}, τ_{2}, …, τ_{n−1}].

The lemma follows easily from these observations. □

An important consequence of Lemma 3.5 is that geometrically, i.e. over \overline{K}, the schemes T = T [λ; τ_{0}, …, τ_{n−1}] are determined up to isomorphism by λ. Proposition 3.10 below tells us that T is 1-dimensional and that if β(λ) ≠ 0 then it is a smooth curve of genus 1. We originally proved the next two results by direct computation using mathematica. A little more thought enables us to treat the cases n = 5 and 7 together.

Lemma 3.6 The scheme T = T [λ; τ_{0}, …, τ_{n−1}] meets the co-ordinate hyperplanes H_{i} in a total of n^{2} points. The points of intersection with H_{i} are defined over K(ζ, \sqrt[n]{α(λ)^{i}θ}) where θ = ∏ τ_{i}^{−i}.

Proof. Directly from the equations we may check T ∩ H_{i} ∩ H_{j} = ∅ for all i ≠ j. Then the action of μ_{n} tells us that T ∩ H_{i} spans the hyperplane H_{i}. Since this is true for all co-ordinate hyperplanes, the map

H^{0}(P^{n−1}, I_{T} (2)) → H^{0}(H_{i}, I_{T ∩H_{i}}(2))

is injective and by counting dimensions we learn that T meets each coordinate hyperplane H_{i} in a single orbit under the action of μ_{n}.

By symmetry it suffices to show that the points T ∩ H_{0} are defined over K(ζ, \sqrt[n]{θ}). But if T = T [λ; α(λ), 1, …, 1] then T ∩ H_{0} has a unique fixed point under

[−1] : (x_{0} : x_{1} : …: x_{n−1}) ↦ (x_{0} : x_{n−1} : …: x_{1}).

The uniqueness statement tells us this point is K-rational. We are now done by Lemma 3.5. □

Proposition 3.7 The family of elliptic curves C_{λ} defined in §1.1 has equations

T [λ; α(λ), 1, …, 1] with 0 = \{ \begin{matrix}(0 : 1 : 1 : −1 : −1) \\ (0 : 1 : −λ : −λ^{2} : λ^{2} : λ : −1). \end{matrix}

Proof. We temporarily write T_{λ} := T [λ; α(λ), 1, …, 1]. For all but finitely many λ we know T_{λ} is a smooth curve of genus 1. We take as identity element the K-rational point constructed in the proof of Lemma 3.6. An easy calculation shows that this is the point 0 appearing in the statement of the proposition.

Let C/K be an elliptic curve with μ_{n} ↪ C. We claim that μ_{n} ↪ C is isomorphic to μ_{n} ↪ T_{λ} for some unique λ ∈ K. Equivalently there is a unique λ ∈ K such that C and T_{λ} are isomorphic as curves and this isomorphism respects the actions of μ_{n} via translation. By Lemma 1.1(iii) we may assume K is algebraically closed. We embed C ↪ P^{n−1} via a complete linear system and choose co-ordinates such that the action of μ_{n} on C via translation is the diagonal action (32). Then Proposition 3.4 and Lemma 3.5 tell us C ≅ T_{λ} for some λ ∈ K. To check the uniqueness statement we consider the automorphisms of P^{n−1} that commute with (32). These are given by rescaling and cyclically permuting our co-ordinates. In particular they do not alter λ and this proves our claim.

To show that the families C_{λ} and T_{λ} are equal, it only remains to check that we have chosen the same co-ordinate λ on X_{1}(n) ≅ P^{1} here as in §1.1. But in both cases our co-ordinate λ has been chosen such that the cusps are as listed in §1.1. This completes the proof of the proposition. □

In §1.2 we considered the exact sequence

… → C_{λ}(K) → D_{λ}(K) →^{δ} K^{×}/(K^{×})^{n} → WC(C_{λ}/K) → …  (34)

and defined C_{λ,θ} to be the torsor under C_{λ} described by θ ∈ K^{×}/(K^{×})^{n}.

Lemma 3.8 Let λ ∈ K not a cusp of X_{1}(n).

(i) If τ_{0}, …, τ_{n−1} ∈ K with α(λ) = ∏ τ_{i} and θ = ∏ τ_{i}^{−i} then

T [λ; τ_{0}, …, τ_{n−1}] ≅ C_{λ,θ}.

(ii) The image of Z/nZ ↪ D_{λ}(K) under δ is generated by α(λ).

Proof. (i) By Lemma 3.5 there is an isomorphism defined over \overline{K}

ψ : T [λ; τ_{0}, τ_{1}, …, τ_{n−1}] ⟶^{∼} C_{λ} = T [λ; α(λ), 1, …, 1]  (35)

given by rescaling co-ordinates. But the only automorphisms of C_{λ} given by rescaling co-ordinates belong to our diagonal action of μ_{n}. Thus the cocycle σ ↦ σ(ψ)ψ^{−1} takes values in μ_{n}. We compute σ(ψ)ψ^{−1} = σ(\sqrt[n]{θ})/\sqrt[n]{θ}. (ii) Since

T [λ; τ_{0}, τ_{1}, …, τ_{n−1}] ≅ T [λ; τ_{n−1}, τ_{0}, …, τ_{n−2}]

it is clear from (i) and the exact sequence (34) that α(λ) ∈ im δ. Taking τ_{0} = … = τ_{n−2} = 1 and τ_{n−1} = α(λ) in (35) we construct ψ : C_{λ} → C_{λ}, translation by an element of C_{λ}[n], such that σ(ψ)ψ^{−1} = σ(\sqrt[n]{α(λ)} )/\sqrt[n]{α(λ)}. Thus α(λ) ∈ im δ is in fact the image of the torsion Z/nZ ↪ D_{λ}(K). □

Remark 3.9 Relabelling our copy of μ_{n} gives isomorphisms C_{λ,θ} ≅ C_{−1/λ,θ^{2}} respectively C_{λ,θ} ≅ C_{(λ−1)/λ,θ^{2}}. Lemma 3.8 allows us to explicitly write down these isomorphisms. For example in the case n = 5 we have

\begin{matrix}T [λ; τ_{0}, τ_{1}, τ_{2}, τ_{3}, τ_{4}] & ≅ & T [−1/λ; −τ_{0}/(τ_{2}τ_{3}), …, −τ_{3}/(τ_{0}τ_{1})] \\ (x_{0} : x_{1} : x_{2} : x_{3} : x_{4}) & ↦ & (x_{0} : x_{2} : x_{4} : x_{1} : x_{3}). \end{matrix}

We now drop our assumption char(K) ≠ n. The next two propositions describe the geometry of the schemes T [λ; τ_{0}, …, τ_{n−1}] and the degenerations that occur when λ becomes a cusp.

Proposition 3.10 Let λ, τ_{0}, …, τ_{n−1} ∈ K with α(λ) = ∏ τ_{i} ≠ 0. We describe the geometry of the scheme T = T [λ; τ_{0}, …, τ_{n−1}].

(a) If β(λ) ≠ 0 then T is a smooth curve of genus 1.

(b) If β(λ) = 0 and char(K) ≠ n then T is a collection of n distinct lines.

(c) If β(λ) = 0 and char(K) = n then T is a line with multiplicity n.

Proof. We postpone the proof to Chapter 4, but note here that it suffices to work over an algebraically closed field, and to treat the case τ_{0} = τ_{1} = … = τ_{n−1}. □

The equations appearing in the statement of the next proposition are obtained from those for T [λ; τ_{0}, …, τ_{n−1}] by manipulating under the assumption α(λ) = ∏ τ_{i} ≠ 0 and then setting λ = 0, respectively λ = 1. We write P_{0} = (1 : 0 : 0 : …), P_{1} = (0 : 1 : 0 : …), . . ..

Proposition 3.11 Let τ_{0}, …, τ_{n−1} ∈ K with I = {i | τ_{i} = 0} ≠ ∅. Then the subscheme of P^{n−1} with equations

n = 5  τ_{0}x_{0}^{2} + x_{1}x_{4} − τ_{2}τ_{3}x_{2}x_{3} = 0 & cyclic permutes
n = 7 \{ \begin{matrix}τ_{0}x_{0}^{2} + x_{1}x_{6} − τ_{2}τ_{3}τ_{4}τ_{5}x_{2}x_{5} = 0 \\ τ_{0}^{2}τ_{1}τ_{6}x_{0}^{2} − x_{2}x_{5} + τ_{3}τ_{4}x_{3}x_{4} = 0 \end{matrix} & cyclic permutes

is the union of |I| rational curves. More precisely for each consecutive pair i, i + d ∈ I we have a rational curve of degree d joining P_{i} and P_{i+d}.

Proof. Rescaling co-ordinates, over \overline{K} if necessary, we may suppose that the non-zero τ_{i} are all 1. We split into cases according as the τ_{i} are 0 or 1. In each case a straightforward calculation proves the proposition. Notice that in the most degenerate case, namely when all the τ_{i} are 0, we obtain a collection of n lines. □

3.2 Local Solubility for Torsors

In this section we work over a field which is complete with respect to a discrete valuation and has finite residue field. We use the results of §3.1 to give some criteria for the existence of rational points on the torsors C_{λ,θ}. We adopt the following notation.

K  a field complete with respect to the valuation ord : K^{×}↠ Z
O  the ring of integers of K = {x ∈ K | ord(x) ≥ 0}
𝔭  the maximal ideal of O = {x ∈ K | ord(x) > 0}
k  the residue field of O = O/𝔭 (assumed finite).

We define ord^{∗}(λ) to be the non-negative integer r with

n = 5 {ord(λ), ord(−1/λ)} = {−r, r}
n = 7  {ord(λ), ord((λ − 1)/λ), ord(1/(1 − λ))} = {−r, 0, r}.

Proposition 3.12 Let λ ∈ K not a cusp of X_{1}(n). For θ ∈ K^{×}/(K^{×})^{n}

C_{λ,θ}(K) ≠ ∅ ⟺ θ ∈ im (δ : D_{λ}(K) → K^{×}/(K^{×})^{n}).

(a) If ord^{∗}(λ) > 0 then im δ = K^{×}/(K^{×})^{n}.

(b) If ord^{∗}(λ) = 0 and β(λ) ≢ 0 (mod 𝔭) then im δ = O^{×}/(O^{×})^{n}.

(c) If ord^{∗}(λ) = 0 and β(λ) ≡ 0 (mod 𝔭) and char(k) ≠ n then im δ = 0.

Proof. The first statement is immediate from the exact sequence (34).

(a) Let θ ∈ K^{×}/(K^{×})^{n}. By Remark 3.9 we may assume ord(λ) > 0, respectively ord(λ − 1) > 0. We write C_{λ,θ} in the form T [λ; τ_{0}, …, τ_{n−1}] with ord(τ_{i}) ≥ 0 for all i. By Proposition 3.11 the reduction of this curve is a collection of rational curves each defined over k. At least one of these curves has odd degree and so is isomorphic to P^{1} over k. We pick a smooth point on the reduction and lift using Hensel’s lemma to give C_{λ,θ}(K) ≠ ∅ as required.

(b) and (c) Suppose θ ∈ K^{×}/(K^{×})^{n} with ord(θ) ≡ 1 (mod n). We write C_{λ,θ} in the form T [λ; τ_{0}, …, τ_{n−1}] with ord(τ_{0}) = −1, ord(τ_{n−1}) = 1 and ord(τ_{i}) = 0 for all other i. We take (x_{0} : …: x_{n−1}) a K-point with min{ord(x_{i})} = 0. Examining the equations for T [λ; τ_{0}, …, τ_{n−1}] we successively deduce ord(x_{i}) > 0 for i = 0, 1, …, n − 1. This contradiction tells us C_{λ,θ}(K) = ∅. Thus im δ ⊂ O^{×}/(O^{×})^{n}.

Let θ ∈ O^{×}/(O^{×})^{n} and write C_{λ,θ} in the form T [λ; τ_{0}, …, τ_{n−1}] with ord(τ_{i}) = 0 for all i. The reduction of this curve is described by Proposition 3.10. In case (b) the reduction is a smooth curve of genus 1. By [Ca2, Chapter 25] we have a rational point which we lift using Hensel’s lemma. Thus im δ = O^{×}/(O^{×})^{n} as required. In case (c) the reduction is a collection of n distinct lines. If C_{λ,θ}(K) ≠ ∅ then one of these lines must be defined over k. This line meets each of the hyperplanes H_{i} so by Lemma 3.6 we deduce both α(λ) and θ are nth powers mod 𝔭. Since char(k) ≠ n it follows θ ∈ (K^{×})^{n}. Thus im δ = 0 as required. □

The above proof turns up a useful observation.

Corollary 3.13 Let F be any field. Then β(u) = 0 ⟹ α(u) ∈ (F^{×})^{n}.

Proof. By Proposition 3.10 we know that T [u; α(u), 1, …, 1] is a collection of lines. Considering intersections with the hyperplanes H_{i} we deduce by Lemma 3.6 that α(u) ∈ (F^{×})^{n}. □

Proposition 3.12 is incomplete in the sense that we do not treat the case char(k) = n when λ reduces to an irrational cusp. Indeed our equations for T [λ; τ_{0}, …, τ_{n−1}] reduce to give a line with multiplicity n, making it difficult to apply Hensel’s lemma.

3.3 Further Hensel Lemma Calculations

We keep our notation for K a local field. In §3.2 we gave some criteria for C_{λ,θ}(K) ≠ ∅. These are sufficient to compute Selmer groups in many cases and it is tempting to restrict attention to such examples. However we consider it worthwhile to investigate the case char(k) = n more carefully. We recall that in §1.2 we gave equations for C_{λ,θ} via push-out from D_{λ}. Indeed

δ : D_{λ}(K) → K^{×}/(K^{×})^{n}

is given by δ(P) = f(P) mod (K^{×})^{n} where f ∈ K(D_{λ}) is an explicit rational function. It is our task to identify the image of δ when char(k) = n and λ reduces to an irrational cusp. The latter is equivalent to λ ≡ 3 (mod 𝔭), respectively λ ≡ 5 (mod 𝔭).

We use the formal group defined by D_{λ} as a vehicle for our Hensel lemma calculations. We recall from [Si1, Chapter IV] that the Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

is satisfied by the power series

x(z) = \frac{1}{z^{2}} − \frac{a_{1}}{z} − a_{2} − a_{3}z − (a_{4} + a_{1}a_{3})z^{2} − …
y(z) = −\frac{1}{z^{3}} + \frac{a_{1}}{z^{2}} + \frac{a_{2}}{z} + a_{3} + (a_{4} + a_{1}a_{3})z + …

and that the invariant differential has expansion

ω(z) = (1 + a_{1}z + (a_{1}^{2} + a_{2})z^{2} + (a_{1}^{3} + 2a_{1}a_{2} + 2a_{3})z^{3} + …)dz.

To study the curves D_{λ} we take

(a_{1}, a_{2}, a_{3}, a_{4}, a_{6}) = \{ \begin{matrix}(1 − λ, −λ, −λ, 0, 0) & if n = 5 \\ (1 + λ − λ^{2}, λ^{2} − λ^{3}, λ^{2} − λ^{3}, 0, 0) & if n = 7. \end{matrix}

The series x(z), y(z) and ω(z) now have coefficients in Z[λ]. The composite

𝔭O ⟶ D_{λ}(K) ⟶^{δ} K^{×}/(K^{×})^{n}
z ↦ (x(z), y(z))
(x, y)  ↦  f(x, y)

is given by z ↦ F (z) where F (z) := ±z^{n}f(x(z), y(z)) is again a power series with coefficients in Z[λ]. Explicitly in the cases n = 5 and 7 we find

F (z) = 1 + (2λ − 1)z + λ(λ + 2)z^{2} + … + λ(λ + 1)z^{5} + …
F (z) = 1 + (3λ^{2} − 2λ − 2)z + (λ − 1)(3λ^{3} + 3λ^{2} − 3λ − 1)z^{2} + …
… + λ^{4}(λ − 1)(λ^{5} − λ^{4} + 1)z^{7} + ….

The leading terms here are sufficient to determine the image of δ in the case K = Q_{p}.

Lemma 3.14 Let p = n. Let λ ∈ Z_{p} with β(λ) ≡ 0 (mod p). Then the image of the map δ : D_{λ}(Q_{p}) → Q_{p}^{×}/(Q_{p}^{×})^{n} is generated by α(λ).

Proof. As seen in §1.1 our Weierstrass equation (2) for D_{λ} is minimal. We write E = D_{λ} and recall from [Si1, Chapter VII] that there is a filtration

E(Q_{p}) ⊃ E_{0}(Q_{p}) ⊃ E_{1}(Q_{p}).

We are given λ ≡ 3 (mod 5), respectively λ ≡ 5 (mod 7). By inspection of the above series expansions we find F′(0) ≡ 0 (mod p). If c ∈ pZ_{p} then F (c) ≡ 1 (mod p^{2}). But p^{2}Z_{p} ⊂ (Q_{p}^{×})^{n} so δ restricted to E_{1}(Q_{p}) is the zero map.

Since E_{0}(Q_{p})/E_{1}(Q_{p}) ≅ Ẽ_{ns}(Z/nZ) ≅ Z/nZ the table of §1.2 tells us that the image of δ restricted to E_{0}(Q_{p}) is generated by α(λ). Finally in the case of additive reduction the Tamagawa number [E(Q_{p}) : E_{0}(Q_{p})] is at most 4, and so prime to n. The lemma follows. □

Proposition 3.15 Let p = n and let N = 125, respectively 49. Let λ ∈ Q_{p} not a cusp of X_{1}(n). For θ ∈ Q_{p}^{×}/(Q_{p}^{×})^{n}

C_{λ,θ}(Q_{p}) ≠ ∅ ⟺ θ ∈ im (δ : D_{λ}(Q_{p}) → Q_{p}^{×}/(Q_{p}^{×})^{n}).

(a) If ord^{∗}(λ) > 0 then im δ = Q_{p}^{×}/(Q_{p}^{×})^{n}.

(b) If ord^{∗}(λ) = 0 and β(λ) ≢ 0 (mod N) then im δ = Z_{p}^{×}/(Z_{p}^{×})^{n}.

(c) If ord^{∗}(λ) = 0 and β(λ) ≡ 0 (mod N) then im δ = 0.

Proof. If β(λ) ≢ 0 (mod p) then we are already done by Proposition 3.12. Otherwise Lemma 3.14 tells us that im δ is generated by α(λ). So the condition for im δ to be trivial is α(λ) ∈ (Q_{p}^{×})^{n}. We compute

\{ \begin{matrix}λ ≡ 3 (mod 5) \\ λ ∈ (Q_{5}^{×})^{5} \end{matrix} \} ⟺ λ ≡ 18 (mod 25) ⟺ β(λ) ≡ 0 (mod 125)
\{ \begin{matrix}λ ≡ 5 (mod 7) \\ λ^{4}(λ − 1) ∈ (Q_{7}^{×})^{7} \end{matrix} \} ⟺ λ ≡ 5 (mod 7) ⟺ β(λ) ≡ 0 (mod 49).
□

The reader only interested in descent calculations over Q should now turn to §3.4. Our next task is to develop some identities that will be crucial to our Hensel lemma calculations.

We return to considering K a local field with char(k) = n and write e = ord(n). We take λ ∈ K with ord^{∗}(λ) = 0 and write E = D_{λ}. The points tabulated in §1.2 reduce to give Z/nZ ↪ Ẽ_{ns}(k) generated by 1 = (0, 0). The rational function f ∈ K(E) reduces to f̃ ∈ k(Ẽ). Since char(k) = n we find

( f̃) = n.1 − n.0 and ( df̃) = n.1 − n.0.

If Ẽ is smooth then df̃/f̃ is a multiple of the invariant differential. If Ẽ is singular, hence rational, then df̃/f̃ = 0. These observations give some explanation for the congruences appearing in the next lemma.

Let g(x, y) be the Weierstrass cubic (2) defining D_{λ}. The invariant differential is ω = (∂g/∂y)^{−1}dx = −(∂g/∂x)^{−1}dy.

Lemma 3.16 Let I ◁ Z[λ, x, y] be the ideal generated by n and g(x, y). Then

df/ω = | \begin{matrix}∂f/∂x & ∂f/∂y \\ ∂g/∂x & ∂g/∂y \end{matrix} | ≡ β′(λ)f (mod I).

Proof. The first equality is a tautology. For the congruence we compute

n = 5 df/ω = (−3λ − 1)f(x, y) − 3g(x, y) + 5y^{2}
n = 7 df/ω = (−4λ^{2} + 5λ − 2)f(x, y)

+ (3x + 3λ − 2)g(x, y) + 7y(λx^{2} − xy − λy).

□

Lemma 3.17 Let λ ∈ K with ord^{∗}(λ) = 0. Let F (z) be the power series constructed above and put t = ord(β′(λ)).

(i) If t < e then ord(F′(0)) = t. (ii) If t ≤ e then F′(z) ≡ 0 (mod 𝔭^{t}).

Proof. (i) This follows from the observation F′(0) ≡ β′(λ) (mod n). (ii) We have power series

df/dz = (df/ω)(ω/dz).

Since λ ∈ O we know f(x(z), y(z)) ∈ O[[z]] and ω(z)/dz ∈ O[[z]]. Then by Lemma 3.16, df/ω ≡ 0 (mod 𝔭^{t}). Hence F′(z) ≡ df/dz ≡ 0 (mod 𝔭^{t}). □

We end this section by giving the version of Hensel’s lemma we have been working towards.

Proposition 3.18 Let λ ∈ K with ord^{∗}(λ) = 0. Let t = ord(β′(λ)) and suppose 0 < t < e. If ν ≥ t/(n − 1) then

{ F (c) | c ∈ 𝔭^{ν}O } = { θ ∈ K^{×} | ord(θ − 1) ≥ t + ν }.

Proof. “⊂” Let c ∈ 𝔭^{ν}O. Since F′(z) ≡ 0 (mod 𝔭^{t}) and ord(c^{n}) ≥ nν ≥ t+ν, we deduce F (c) ≡ 1 (mod 𝔭^{t+ν}) as required.

“⊃” Let θ ≡ 1 (mod 𝔭^{t+ν}). We claim that for all m ≥ t + ν there exists c_{m} ∈ 𝔭^{ν}O with

\begin{matrix}F (c_{m}) ≡ θ (mod 𝔭^{m}) \\ ord(F′(c_{m})) = t. \end{matrix}  (36)

For the case m = t + ν we take c_{m} = 0 and appeal to Lemma 3.17(i). We now suppose (36) is true for m and let c_{m+1} = c_{m} + aπ^{m−t} where ord(π) = 1 and a ∈ O is yet to be determined. We consider the Taylor series expansion of F about c_{m}. Lemma 3.17(ii) and the inequality n(m − t) > m give

F (c_{m+1}) ≡ F (c_{m}) + aπ^{m−t}F′(c_{m}) (mod 𝔭^{m+1}).

Since ord(F′(c_{m})) = t we may solve for a such that F (c_{m+1}) ≡ θ (mod 𝔭^{m+1}). Further F′(c_{m+1}) ≡ F′(c_{m}) (mod 𝔭^{m}) so certainly ord(F′(c_{m+1})) = t. This proves (36) for m + 1 and our claim follows by induction. Taking c = lim c_{m} gives F (c) = θ as required. □

3.4 Computation of Selmer Groups

Let n = 5 or 7. Let λ ∈ Q not a cusp of X_{1}(n) and let A, B be the sets of rational primes defined in §2.1. In this section we prove that the Selmer group S^{(ϕ)}(C_{λ}/Q) is as described in Theorem 1. We then complete the proof of Theorem 1 by explaining how the groups S^{(ϕ)}(C_{λ}/Q) and S^{(ϕ̂)}(D_{λ}/Q) are related.

Proposition 3.19 Let λ ∈ Q not a cusp of X_{1}(n). The connecting maps

\begin{matrix}δ_{ϕ} : D_{λ}(Q_{p}) → H^{1}(Q_{p}, μ_{n}) \\ δϕ̂ : C_{λ}(Q_{p}) → H^{1}(Q_{p}, Z/nZ) \end{matrix}

have images

im δ_{ϕ} = \{ \begin{matrix}H^{1}(Q_{p}, μ_{n}) & if p ∈ A \\ H_{un}^{1}(Q_{p}, μ_{n}) & if p ∉ A ∪ B \\ 0 & if p ∈ B \end{matrix}

and

im δϕ̂ = \{ \begin{matrix}0 & if p ∈ A \\ H_{nr}^{1}(Q_{p}, Z/nZ) & if p ∉ A ∪ B \\ H^{1}(Q_{p}, Z/nZ) & if p ∈ B \end{matrix}

where H_{un}^{1} and H_{nr}^{1} are the subgroups of H^{1} defined in Proposition 1.15.

Proof. (i) The description of im δ_{ϕ} is given by combining Propositions 3.12 and 3.15. Note that H_{un}^{1}(Q_{p}, μ_{n}) = Z_{p}^{×}/(Z_{p}^{×})^{n} and this group is trivial unless p = n or p ≡ 1 (mod n). (ii) Tate local duality, specifically Proposition 1.17, tells us im δ_{ϕ} and im δϕ̂ are exact annihilators with respect to the Tate pairing

H^{1}(Q_{p}, μ_{n}) × H^{1}(Q_{p}, Z/nZ) → Z/nZ.  (37)

The description of im δϕ̂ now follows from the description of im δ_{ϕ}. □

The first statement of Theorem 1 is an immediate consequence.

S^{(ϕ)}(C_{λ}/Q) = { θ ∈ Q^{×}/(Q^{×})^{n} | C_{λ,θ}(Q_{p}) ≠ ∅ for all primes p }
= { θ ∈ [A] | θ ∈ (Q_{p}^{×})^{n} for all p ∈ B }

We have already seen two proofs of the statement concerning torsion, in §1.2 using the push-out method, and in §3.1 using the projective space method. So to prove Theorem 1 it only remains to explain the relationship between S^{(ϕ)}(C_{λ}/Q) and S^{(ϕ̂)}(D_{λ}/Q).

For S a finite set of rational primes we define

[S] := ker (H^{1}(Q, μ_{n}) → ∏_{p∉S} H^{1}(Q_{p}, μ_{n})/H_{un}^{1}(Q_{p}, μ_{n}))
⟨S⟩ := ker (H^{1}(Q, Z/nZ) → ∏_{p∉S} H^{1}(Q_{p}, Z/nZ)/H_{nr}^{1}(Q_{p}, Z/nZ)).

Since we are identifying H^{1}(Q, μ_{n}) = Q^{×}/(Q^{×})^{n}, the notation [S] agrees with that used to state Theorem 1.

We define Ξ : [A] × ⟨B⟩ → Z/nZ via

Ξ(a, b) = ∑_{p∈B}(a, b)_{p} = − ∑_{p∈A}(a, b)_{p}  (38)

where (·, ·)_{p} is the Tate pairing (37). If p ∉ A ∪ B then a ∈ H_{un}^{1}(Q_{p}, μ_{n}) and b ∈ H_{nr}^{1}(Q_{p}, Z/nZ). So by Proposition 1.15 together with the “product formula” the two sums given in (38) are indeed equal. To complete the proof of Theorem 1 it suffices to show ⟨B⟩ ≅ [B] and to identify the left and right kernels of Ξ as Selmer groups.

For each prime p ≡ 0, 1 (mod n) we write F^{(p)} for the degree n cyclic extension of Q unramified outside p. Concretely F^{(p)} is the degree n subfield of Q(μ_{p}), respectively Q(μ_{n^{2}}). For each p ∈ B we make an arbitrary choice of isomorphism ξ^{(p)} : Gal(F^{(p)}/Q) ≅ Z/nZ. By class field theory there is an isomorphism [B] ≅ ⟨B⟩ via p ↦ ξ^{(p)}.

We relate our Selmer groups to the pairing Ξ. For any element (∗) we write (∗)_{p} for the corresponding local element.

S^{(ϕ)}(C_{λ}/Q) ≅ { a ∈ H^{1}(Q, μ_{n}) | a_{p} ∈ im δ_{ϕ} for all primes p }
≅ { a ∈ [A] | a_{p} = 0 for all p ∈ B }
⊂ ker_{L}(Ξ)
S^{(ϕ̂)}(D_{λ}/Q) ≅ { b ∈ H^{1}(Q, Z/nZ) | b_{p} ∈ im δϕ̂ for all primes p }
≅ { b ∈ ⟨B⟩ | b_{p} = 0 for all p ∈ A }
⊂ ker_{R}(Ξ)

To show that these inclusions are in fact equalities we need a lemma.

Lemma 3.20 The natural maps

[A] → ⊕_{p∈A}H^{1}(Q_{p}, μ_{n})/H_{un}^{1}(Q_{p}, μ_{n})
⟨B⟩ → ⊕_{p∈B}H^{1}(Q_{p}, Z/nZ)/H_{nr}^{1}(Q_{p}, Z/nZ)

are isomorphisms.

Proof. (i) The map is injective since

{ θ ∈ Q^{×}/(Q^{×})^{n} | ord_{p}(θ) ≡ 0 (mod n) for all p }

is trivial. Both spaces have dimension |A|. (ii) The map is injective since Q has no unramified (abelian) extensions. For p ≡ 0, 1 (mod n) we know H_{un}^{1}(Q_{p}, μ_{n}) = Z_{p}^{×}/(Z_{p}^{×})^{n} is cyclic of order n. Thus both sides have dimension |B|. □

Now take a ∈ [A] and suppose a ∈ ker_{L}(Ξ), i.e. ∑_{p∈B}(a, b)_{p} = 0 for all b ∈ ⟨B⟩. Then a_{p} ∈ H_{un}^{1}(Q_{p}, μ_{n}) for all p ∉ A. So by Proposition 1.15 and the description of ⟨B⟩ in Lemma 3.20 it follows a_{p} = 0 for all p ∈ B. Thus S^{(ϕ)}(C_{λ}/Q) ≅ ker_{L}(Ξ) and by the same reasoning S^{(ϕ̂)}(D_{λ}/Q) ≅ ker_{R}(Ξ). This completes the proof of Theorem 1 subject to Proposition 3.10. We have postponed the proof of this geometrical result to the next chapter.

On the Geometry of Elliptic Normal Curves

This chapter is intended as a survey of certain geometrical results relevant to our descent calculations. In particular we prove Proposition 3.10 which is an essential ingredient in the proofs of our descent theorems. The equations (41) and (42), which we derive from an elementary point of view, go back to Klein [K2] as relations among theta functions. They also feature prominently in [AR] and [V2]. In Appendix A we discuss the relationship of these equations with the work of Buchsbaum and Eisenbud [BE].

4.1 Elliptic Normal Curves

Let K be an algebraically closed field and n ≥ 3 an integer. Until further notice we assume char(K) ∤ n and fix ζ = ζ_{n} a primitive nth root of unity. We make further restrictions on n in due course.

Definition 4.1 (i) An elliptic normal curve of degree n is an elliptic curve E ↪ P^{n−1} of degree n contained in no hyperplane. (ii) A Néron polygon of degree n is a collection of n lines ℓ_{0}, …, ℓ_{n−1} in P^{n−1}, contained in no hyperplane and arranged such that ℓ_{i} meets ℓ_{j} if and only if i − j ≡ ±1 (mod n).

Remark 4.2 Let E be an elliptic normal curve. For d ≥ 1 the hypersurfaces of degree d in P^{n−1} cut out a complete linear system on E. The case d = 1, i.e. linear normality, is an immediate consequence of Riemann-Roch. For the general case, i.e. projective normality, see [Ha, IV Exercises 4.1, 4.2] for a proof using Weierstrass equations, or [Hu, IV Lemma 1.2] for a proof that involves reducing to a similar statement about finite sets of points in general position.

Lemma 4.3 Let E ↪ P^{n−1} be either (i) an elliptic normal curve or (ii) a Néron polygon. Then E lies on n(n − 3)/2 quadrics and for n ≥ 4 these quadrics suffice to define E.

Proof. (i) There is a short exact sequence

0 → H^{0}(P^{n−1}, I_{E}(2)) → H^{0}(P^{n−1}, O_{P^{n−1}}(2)) → H^{0}(E, O_{E}(2)) → 0

where exactness on the right is part of the statement of projective normality. Thus h^{0}(P^{n−1}, I_{E}(2)) = n(n + 1)/2 − 2n = n(n − 3)/2 as required. That the quadrics suffice to define E may be proved by induction on n. We refer to [Hu, IV Theorem 1.3] where a stronger result is proved. (ii) We move the n points of intersection to the points (1 : 0 : 0 : …), (0 : 1 : 0 : …), … and the result is clear. □

In order to construct the modular curve X(n) defined in §1.3 we consider triples (E, P, Q) where E is an elliptic curve and P, Q ∈ E[n] satisfy e_{n}(P, Q) = ζ_{n}.

Proposition 4.4 Let (E, P, Q) be as above. If we embed E ↪ P^{n−1} by means of a complete linear system, then we may choose co-ordinates on P^{n−1} such that the translation maps τ_{P} and τ_{Q} are given by

M_{P} := (\begin{matrix}1 & 0 & 0 & ⋯ & 0 \\ 0 & ζ & 0 & ⋯ & 0 \\ 0 & 0 & ζ^{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & ⋯ & ζ^{n−1} \end{matrix}) M_{Q} := (\begin{matrix}0 & 0 & ⋯ & 0 & 1 \\ 1 & 0 & ⋯ & 0 & 0 \\ 0 & 1 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & 0 \end{matrix}).

Proof. As explained in §1.2, τ_{P} and τ_{Q} lift to elements of Aut(P^{n−1}) = PGL_{n}. Since no hyperplane may contain more than n points of E we deduce M_{P} has distinct eigenvalues and M_{Q} cyclically permutes the hyperplanes fixed by M_{P}. Thus we may choose co-ordinates on P^{n−1} such that M_{P} and M_{Q} are as given, at least if we replace ζ by ζ^{r} for some r prime to n. We observe that M_{P} and M_{Q} commute as elements of PGL_{n}, but do not commute when lifted to GL_{n}. Lemma 4.5 below tells us that their commutator in GL_{n} gives one possible definition of the Weil pairing. It follows r = 1 as required. □

We write x_{0}, x_{1}, …, x_{n−1} for our co-ordinates on P^{n−1} and agree that all subscripts are to be read mod n.

Lemma 4.5 Let E ↪ P^{n−1} be an elliptic normal curve and let P, Q ∈ E[n]. Suppose τ_{P} and τ_{Q} lift to matrices M̃_{P} and M̃_{Q} in GL_{n}. Then

e_{n}(P, Q) = M̃_{P} M̃_{Q}M̃_{P}^{−1}M̃_{Q}^{−1}.

Proof. It suffices to work with the triple (E, P, Q) of Proposition 4.4. We are also free to suppose 0 ∈ E ∩ {x_{0} = 0}. Let g be the rational function x_{1}/x_{0} on E. Then (g) = ∑(rP + Q) − ∑(rP) where the sums run over r ∈ Z/nZ. For ϕ : E → E′ the isogeny with kernel generated by P we find

e_{ϕ}(P, ϕQ) = g(X + P)/g(X) = ζ = M̃_{P} M̃_{Q}M̃_{P}^{−1}M̃_{Q}^{−1}.  (39)

The dual isogeny is ϕ̂: E′ → E. We take P′ ∈ E′ with ϕ̂P′ = P and compute

e_{ϕ}(P, ϕQ) = e_{ϕ}(ϕ̂P′, ϕQ) = e_{n}(P′, ϕQ) = e_{n}(ϕ̂P′, Q) = e_{n}(P, Q). (40)

Combining (39) and (40) proves the lemma. □

Proposition 4.6 Let n be odd and let (E, P, Q) be as above. If we embed E ↪ P^{n−1} by means of a complete linear system |D| with [−1]^{∗}D ∼ D then there is a unique choice of co-ordinates on P^{n−1} such that τ_{P}, τ_{Q} and [−1] are given by M_{P}, M_{Q} and

[−1] := (\begin{matrix}1 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & 0 & 1 \\ 0 & 0 & ⋯ & 1 & 0 \\ ⋮ & ⋮ & & ⋮ & ⋮ \\ 0 & 1 & ⋯ & 0 & 0 \end{matrix}).

Proof. We first choose co-ordinates as described in Proposition 4.4. Since the subgroup generated by M_{P} and M_{Q} is its own centraliser inside PGL_{n} our co-ordinates are uniquely determined up to the action of this subgroup. We require

[−1]M_{P} [−1] = M_{P}^{−1} [−1]M_{Q}[−1] = M_{Q}^{−1} and [−1]^{2} = I.

These commutator relations determine [−1] up to multiplication by elements of ⟨M_{P} , M_{Q}⟩. Since n is odd, the n^{2} possible matrices representing the map [−1] form a single orbit under the action of ⟨M_{P} , M_{Q}⟩ via conjugation. Taking [−1] as given now fixes our choice of co-ordinates. □

4.2 An Algebraic Construction of X(n)

For simplicity we restrict to n ≥ 5 odd. Let (E, P, Q) be a triple as above. We embed E ↪ P^{n−1} via the linear system |n.0| and choose co-ordinates as in Proposition 4.6. We then ask the question “where is 0?”. This determines an embedding Y (n) ↪ P^{n−1}. Indeed, if we know the co-ordinates of 0 ∈ P^{n−1} then M_{P} and M_{Q} allow us to write down n^{2} points on E. Since E is defined by quadrics, it follows by Bézout’s theorem that these n^{2} points suffice to determine E.

Our assumption that n is odd tells us

n.0 ∼ 0 + P + 2P + … + (n − 1)P.

Therefore 0 belongs to exactly one of the hyperplanes fixed by M_{P}. But 0 is fixed by [−1] so we have either

\begin{aligned}& 0 & = & (0 : a_{1} : a_{2} : …: a_{2} : a_{1}) & (+) \\ or & 0 & = & (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) & (−) \end{aligned}

for some a_{1}, a_{2}, …, a_{(n−1)/2} non-zero. We shall shortly see that (+) never happens. For a proof over C using Weierstrass σ-functions we refer to [Hu, Chapter 1]. Working with (−) we make the natural inclusions

Y (n) ↪ P^{(n−3)/2} ↪ P^{n−1}.

The action of PSL_{2}(Z/nZ) on Y (n) extends to a projective representation \overline{ρ} : PSL_{2}(Z/nZ) → PGL_{(n−1)/2}. In order to give an explicit description of \overline{ρ} we first construct ρ : SL_{2}(Z/nZ) → PGL_{n} such that

ρ (\begin{matrix}a & b \\ c & d \end{matrix})^{−1} M_{P}^{u}M_{Q}^{v} ρ (\begin{matrix}a & b \\ c & d \end{matrix}) = M_{P}^{au+bv}M_{Q}^{cu+dv}

4.2. AN ALGEBRAIC CONSTRUCTION OF X(n) 51

and the image of ρ commutes with [−1]. Then ρ and \overline{ρ} are given by

S = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , ρ(S) = (ζ^{ij})_{i,j=0}^{n−1},  \overline{ρ}(S) = (ζ^{ij} − ζ^{−ij})_{i,j=1}^{(n−1)/2}
T = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , ρ(T) = Diag(ζ^{−i^{2}/2})_{i=0}^{n−1}, \overline{ρ}(T) = Diag(ζ^{−i^{2}/2})_{i=1}^{(n−1)/2}.

Here all exponents are to be interpreted as elements of Z/nZ. A further useful observation is that ρ and \overline{ρ} map diagonal matrices to permutation matrices.

We consider the vector spaces

V := H^{0}(P^{n−1}, I_{E}(2)) ⊂ W := H^{0}(P^{n−1}, O(2)).

The action of M_{P} allows us to write these as direct sums V = ⊕_{i}V_{i} and W = ⊕_{i}W_{i} with V_{i} ⊂ W_{i} = ⟨x_{i}^{2}, x_{i−1}x_{i+1}, … ⟩. Since n is odd we deduce via the action of M_{Q} that dim V_{i} = (n − 3)/2 and dim W_{i} = (n + 1)/2. The point 0 = (0 : a_{1} : a_{2} : …: −a_{2} : −a_{1}) and its translates under M_{Q} impose some linear conditions on the coefficients of the quadrics in V_{0}. We are led to make the following definition.

Definition 4.7 For n ≥ 5 odd, let A(n) ⊂ P^{n−1} be the subscheme defined by a_{0} = 0, a_{n−i} = −a_{i} and

rank(a_{i−j}a_{i+j})_{i,j=0}^{n−1} ≤ 2.  (41)

More precisely A(n) is defined by the leading 4 × 4 Pfaffians.

The above construction shows Y (n) ⊂ A(n) and it is natural to ask whether these quartics suffice to define X(n). We shall discuss below the theorem of Vélu that for p a prime A(p) = X(p).

Remark 4.8 Suppose char(K) ≠ 2 and that (+) holds for some triple (E, P, Q). Then the matrix (41) is symmetric and by considering the top left 3 × 3 minor we learn that some of the a_{i} are zero. This contradicts our earlier observations. For char(K) = 2 there is of course no distinction between (+) and (−).

We know that the triple (E, P, Q) determines the point 0. Conversely the quadrics defining E are precisely the ones containing 0 and its translates under M_{P} and M_{Q}. We are led to make the following definition.

Definition 4.9 For n ≥ 5 odd, let A(n) ⊂ A(n) × P^{n−1} be the subscheme defined by

rank(a_{i−j}x_{i+j})_{i,j=0}^{n−1} ≤ 2.  (42)

More precisely, A(n) is defined by the leading 4 × 4 Pfaffians.

If we take 0 = (0: a_{1} : a_{2} : …) corresponding to the triple (E, P, Q) then the 4 × 4 Pfaffians here are quadrics vanishing on E. A little combinatorial checking shows that we obtain all such quadrics.

Let Y(n) ⊂ Y (n) × P^{n−1} be defined by the equations (42). Then Y(n) → Y (n) is a family of elliptic curves, and indeed is the universal family over Y (n). We write X (n) ⊂ X(n) × P^{n−1} for the scheme-theoretic closure of Y(n). Then X (n) → X(n) is a flat family of curves each of degree n and arithmetic genus 1.

It is easy to see that PSL_{2}(Z/nZ) acts on each of the families Y(n) → Y (n), X (n) → X(n) and A(n) → A(n). For A(n) we check ρ(S) and ρ(T) do not alter the rank of the matrix (a_{i−j}a_{i+j})_{ij}. The same calculation shows PSL_{2}(Z/nZ) acts on A(n).

4.3 Examples

We examine Klein’s equations (41) and (42) for some small values of n. In particular we prove Proposition 3.10 under our assumption char(K) ≠ n. We postpone the case char(K) = n to §4.6.

Equations for X(5)

Let 0 = (0 : a : b : −b : −a). A(5) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} \\ a^{2} & 0 & ab \\ b^{2} & −ab & 0 \end{matrix}) ≤ 2.

Thus X(5) = A(5) is a copy of P^{1} and V_{0} is spanned by

abx_{0}^{2} + b^{2}x_{1}x_{4} − a^{2}x_{2}x_{3} = 0.  (43)

There are rational cusps at (a : b) = (1 : 0), (0 : 1). The remaining cusps are found using the action of PSL_{2}(Z/5Z) generated by

\begin{matrix}S : & (a : b) & ↦ & (ϕa + b : a − ϕb) \\ T : & (a : b) & ↦ & (a : ζb) \end{matrix}

where ϕ = (ζ − ζ^{4})/(ζ^{2} − ζ^{3}) is the golden ratio. The cusps

a/b = 0, ∞, ζ^{i}ϕ, ζ^{i}\overline{ϕ}

may be viewed under stereographic projection as the vertices of an icosahedron. By inspection of the equations (43), the fibre of A(5) → A(5) at each rational cusp is a Néron polygon. The action of PSL_{2}(Z/5Z) tells us that the same is true at all the cusps.

In the notation of §3.1 the curve (43) is T [λ; τ, …, τ] where τ = a/b and λ = τ^{5}. To prove Proposition 3.10 we must find the condition on λ for T [λ; τ_{0}, …, τ_{4}] to degenerate to a Néron polygon. We see that the condition is λ = ϕ^{5} or \overline{ϕ^{5}}. These are the zeros of β(λ) = λ^{2} − 11λ − 1 as required.

Equations for X(7)

Let 0 = (0 : a : b : −c : c : −b : −a). A(7) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −c^{2} \\ a^{2} & 0 & ac & −bc \\ b^{2} & −ac & 0 & ab \\ c^{2} & bc & −ab & 0 \end{matrix}) ≤ 2 ⟺ a^{3}b + b^{3}c + c^{3}a = 0.

Thus X(7) = A(7) is the Klein quartic [K1] and V_{0} is spanned by

\begin{aligned}& & abx_{1}x_{6} & + & bcx_{2}x_{5} & + & cax_{3}x_{4} & = & 0 \\ abx_{0}^{2} & & & + & c^{2}x_{2}x_{5} & − & b^{2}x_{3}x_{4} & = & 0 \\ bcx_{0}^{2} & − & c^{2}x_{1}x_{6} & & & + & a^{2}x_{3}x_{4} & = & 0 \\ cax_{0}^{2} & + & b^{2}x_{1}x_{6} & − & a^{2}x_{2}x_{5} & & & = & 0. \end{aligned}  (44)

We have rational cusps at (a : b : c) = (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1). The remaining cusps are found using the action of PSL_{2}(Z/7Z). They are at

(a : b : c) = (ζ − ζ^{6} : ζ^{2} − ζ^{5} : ζ^{4} − ζ^{3})

and its translates under (a : b : c) ↦ (b : c : a) and (a : b : c) ↦ (ζ^{3}a : ζb : c). Again we find the fibre of A(7) → A(7) at each cusp is a Néron polygon.

In the notation of §3.1 the curve (44) is T [λ; τ, …, τ] where λ = −ac^{2}/b^{3} and τ = ac/b^{2}. So the following birational map comes for free.

\begin{matrix}\{a^{3}b + b^{3}c + c^{3}a = 0\} & → & \{λ^{4}(λ − 1) = τ^{7}\} \\ (a : b : c) & ↦ & (−ac^{2}/b^{3}, ac/b^{2}) \\ (τ^{3} : −λτ : λ^{2}) & ↦ & (λ, τ) \end{matrix}

To prove Proposition 3.10 we must find the condition on λ for T [λ; τ_{0}, …, τ_{6}] to degenerate to a Néron polygon. We see that the condition is λ = −(ζ − ζ^{6})(ζ^{4} − ζ^{3})^{2}/(ζ^{2} − ζ^{5})^{3} or one of its conjugates. These are the zeros of β(λ) = λ^{3} − 8λ^{2} + 5λ + 1 as required.

Equations for X(9)

Let 0 = (0 : a : −b : d : c : −c : −d : b : −a). A(9) is defined by

rank (\begin{matrix}0 & −a^{2} & −b^{2} & −d^{2} & −c^{2} \\ a^{2} & 0 & −ad & bc & cd \\ b^{2} & ad & 0 & ac & −bd \\ d^{2} & −bc & −ac & 0 & −ab \\ c^{2} & −cd & bd & ab & 0 \end{matrix}) ≤ 2 ⟺ \{ \begin{matrix}(a^{2}b + b^{2}c + c^{2}a)d & = & 0 \\ ab^{3} − ac^{3} − bd^{3} & = & 0 \\ bc^{3} − ba^{3} − cd^{3} & = & 0 \\ ca^{3} − cb^{3} − ad^{3} & = & 0. \end{matrix}

Adding together the last three equations and factoring we find

X(9) = \{ \begin{matrix}a^{2}b + b^{2}c + c^{2}a & = & 0 \\ ab^{2} + bc^{2} + ca^{2} & = & d^{3} \end{matrix} \} ⊂ P^{3}

whereas A(9) contains four additional isolated points

(0 : 0 : 0 : 1), (1 : 1 : 1 : 0), (1 : ζ^{3} : ζ^{6} : 0), (1 : ζ^{6} : ζ^{3} : 0).

We use the action of PSL_{2}(Z/9Z) to find the cusps of X(9). They are at

\begin{aligned}(a : b : c : d) & = & (1 : 0 : 0 : 0), (1 : 1 : ζ^{3} : 0), (1 : 1 : ζ^{6} : 0), \\ & & (ζ − ζ^{8} : ζ^{7} − ζ^{2} : ζ^{4} − ζ^{5} : ζ^{3} − ζ^{6}) \end{aligned}

and their translates under

\begin{matrix}(a : b : c : d) & ↦ & (b : c : a : d), \\ (a : b : c : d) & ↦ & (ζ^{6}a : ζ^{3}b : c : ζd). \end{matrix}

The fibres of A(9) → A(9) at the cusps are Néron polygons of degree 9. The fibres of A(9) → A(9) at the isolated points consist of three disjoint copies of P^{2}.

The curve X(11) was described by Klein as the singular locus of the Hessian of the cubic threefold

{v^{2}w + w^{2}x + x^{2}y + y^{2}z + z^{2}v = 0} ⊂ P^{4}.

We refer to Adler [AR] for further details.

4.4. THE RELATIONSHIP BETWEEN A(n) AND X(n) 55

4.4 The Relationship Between A(n) and X(n)

We continue to take n ≥ 5 odd and consider the family A(n) → A(n).

Definition 4.10 Let c_{r} ∈ A(n) be the point with a_{i} = 0 for i ≢ ±r (mod n). We call the points c_{r} rational cusps, even if they do not lie on X(n).

Lemma 4.11 The fibre of A(n) → A(n) above the rational cusp c_{r} is a collection of k Néron polygons of degree n/k where k = gcd(r, n) and we assume n/k ≥ 5. If n/k = 3 we obtain k copies of P^{2} instead.

Proof. Let i, j ∈ Z/nZ with i − j ≢ 0, ±2r (mod n). Then

#{(i ± r)/2, (j ± r)/2} = 4

and by considering the corresponding leading 4 × 4 submatrix of (42) we find x_{i}x_{j} = 0. The lemma follows. □

Lemma 4.11 leads us to suspect that for composite n the scheme A(n) contains copies of X(m) for m|n. More precisely we have

Lemma 4.12 A(n) contains k ∏_{p|k}(1 + 1/p) copies of the curve X(n/k) where k|n and n/k ≥ 5. If n/k = 3 we obtain isolated points instead.

Proof. There is an inclusion A(n/k) ↪ A(n) given by

0 = (0 : a_{1} : a_{2} : …) ↦ 0′ = (0 : a′_{1} : a′_{2} : …)

where a′_{i} = a_{i/k} if i ≡ 0 (mod k) and a′_{i} = 0 otherwise. Then the fibre of A(n) → A(n) above 0′ consists of k copies of the fibre of A(n/k) → A(n/k) above 0. To see this we need only reorder rows and columns such that the matrix (a′_{i−j}x_{i+j})_{ij} is in block diagonal form, with each block corresponding to a congruence class mod k.

This construction gives a copy of X(n/k) ↪ A(n) with M_{P}^{n/k} acting trivially on the fibres. By means of the action of PSL_{2}(Z/nZ) we obtain further copies for each subgroup Z/kZ ↪ ⟨M_{P} , M_{Q}⟩. So the lemma follows by counting the cyclic subgroups of order k in (Z/nZ)^{2}. Naturally the case n/k = 3 is badly behaved since a plane cubic cannot be defined by quadrics. □

We suspect that the connected components of A(n) are precisely the curves and points described in Lemma 4.12. In the case n = p is prime this is a theorem due to Vélu [V2].

Theorem 4.13 For p a prime the families A(p) → A(p) and X (p) → X(p) are the same. In particular the quartics (41) suffice to define X(p).

We recall an important step in the proof of the theorem.

Proposition 4.14 Let C be the fibre of A(n) → A(n) above 0_{C} = (0 : a_{1} : a_{2} : …) and suppose a_{i} ≠ 0 for all i ≠ 0. Then ⟨M_{P} ⟩ acts freely on C with quotient ϕ : C → D; (x_{0} : …: x_{n−1}) ↦ (X : Y : Z) where

X = x_{0}x_{1}x_{n−1}, Y = x_{n−1}^{2}x_{2}, Z = x_{0}^{3}

and D has Weierstrass equation

Y^{2}Z − \frac{a_{2}^{5} + a_{1}^{4}a_{4}}{a_{1}^{3}a_{2}a_{3}} XY Z + \frac{a_{2}^{2}}{a_{1}^{2}} Y Z^{2} = −\frac{a_{2}^{2}}{a_{1}^{2}} X^{3} + \frac{a_{3}}{a_{1}} X^{2}Z.  (45)

Proof. [V2, Chapitre 9]. Vélu gives explicit formulae showing that C is isomorphic to a curve defined by push-out from D. □

In the setting of Proposition 4.14 we deduce C is either a collection of k elliptic normal curves each of degree n/k or a collection of k Néron polygons each of degree n/k. These correspond to the cases where D is a smooth cubic, respectively a nodal cubic. In the latter case the existence of a singular point on D shows that there exists Z/nZ ↪ ⟨M_{P} , M_{Q}⟩ a cyclic subgroup of order n fixing some point on C. Up to the action of PSL_{2}(Z/nZ) we may suppose this subgroup is ⟨M_{P} ⟩. Lemma 4.15 below tells us that we are at a rational cusp and we conclude by Lemma 4.11.

Lemma 4.15 Let (0 : a_{1} : a_{2} : …) ∈ A(n) and let C be the corresponding fibre of A(n) → A(n). If C has a fixed point under M_{P}^{n/k} then a_{i}a_{j} = 0 whenever i ≢ ±j (mod k).

Proof. There exists (x_{0} : x_{1} : …: x_{n−1}) ∈ C with x_{i} = 0 for i ≢ i_{0} (mod k). We choose r such that x_{r} ≠ 0 and suppose i ≢ ±j (mod k). Then

#{(r ± i)/2, (r ± j)/2} = 4

and by considering the corresponding leading 4 × 4 submatrix of (42) we find a_{i}a_{j} = 0 as required. □

We now sketch the proof of Theorem 4.13. For p a prime, Vélu [V2, Lemme 8.18] works directly from the equations (41) to show that every point on A(p) is either a rational cusp or satisfies the hypotheses of Proposition 4.14. It follows A(p) consists of Y (p) together with a single orbit under the action of PSL_{2}(Z/pZ). Thus A(p) = X(p) and the fibres of A(p) → A(p) at the cusps are Néron polygons. Finally A(p) = X (p) as claimed.

4.5 Alternative Criteria for Finding the Cusps

We restrict to n ≥ 5 prime. The family X (n) → X(n) consists of elliptic normal curves above the points of Y (n) and Néron polygons above the cusps. One way to find the irrational cusps is to start with the rational cusps and then use the explicit description of \overline{ρ} : PSL_{2}(Z/nZ) → PGL_{(n−1)/2} given in §4.2. In this section we give some alternative criteria. The action (14) of μ_{n} on X(n) is generated by \overline{ρ}(T). Up to this action we find the condition for (0 : a_{1} : a_{2} : …) to be an irrational cusp is

rank(a_{i−j})_{i,j=0}^{n−1} = 2.  (46)

Let C be the fibre of X (n) → X(n) above 0_{C} = (0 : a_{1} : a_{2} : …).

Fixed point criterion. The curve C degenerates if and only if some matrix in ⟨M_{P} , M_{Q}⟩ fixes a point on C.

Proof. If C does not degenerate ⟨M_{P} , M_{Q}⟩ consists of translations and so has no fixed points. If C degenerates then M_{P} and M_{Q} act by rotating the Néron polygon. Thus some element of ⟨M_{P} , M_{Q}⟩ fixes the lines and so also fixes the points of intersection. □

If M_{P} fixes a point then Lemma 4.15 tells us we are at a rational cusp. Otherwise up to the action of μ_{n} on X(n) we may suppose M_{Q} fixes a point. But the fixed points under M_{Q} are (1 : 1 : …: 1) and its translates under M_{P}. The condition (46) follows immediately from (42).

Flex alignment criterion. The curve C degenerates if and only if there exist collinear points R_{0}, …, R_{n−1} with R_{i} ∈ C ∩ {x_{i} = 0}.

Proof. If C is a Néron polygon then clearly such points R_{0}, …, R_{n−1} exist. Conversely if the flexes line up on a line ℓ then C, being defined by quadrics, must contain ℓ. □

Some matrix in ⟨M_{P} , M_{Q}⟩ preserves the line ℓ. As before we may suppose this matrix is M_{Q}. Then R_{0}, …, R_{n−1} are the points 0_{C} and its translates under M_{Q}. Again the condition (46) follows.

Lemma 4.16 Let x_{0}, x_{1}, …, x_{n−1} ∈ K with all subscripts read mod n. Then the matrix rank(x_{i−j})_{i,j=0}^{n−1} ≤ 2 has rank 2 if and only if there exist non-zero A, B ∈ K and distinct ζ_{A}, ζ_{B} ∈ μ_{n} such that x_{i} = Aζ_{A}^{i} + Bζ_{B}^{i}.

Proof. If the matrix has rank 2 then the x_{i} satisfy a second order recurrence relation x_{i+1} + bx_{i} + cx_{i−1} = 0. This may be written

(\begin{matrix}x_{i} \\ x_{i+1} \end{matrix}) = (\begin{matrix}0 & 1 \\ −c & −b \end{matrix}) (\begin{matrix}x_{i−1} \\ x_{i} \end{matrix}) .

The 2 × 2 matrix has minimal polynomial m(X) = X^{2} + bX + c. We require m(X)|(X^{n}−1). Since char(K) ∤ n it follows m(X) = (X −ζ_{A})(X −ζ_{B}) where ζ_{A}, ζ_{B} are distinct nth roots of unity. Thus x_{i} = Aζ_{A}^{i} + Bζ_{B}^{i} as required. □

We use the lemma to interpret the condition (46). We have a_{i} = Aζ_{A}^{i} + Bζ_{B}^{i}. But a_{0} = 0 and a_{n−i} = −a_{i} forcing A + B = 0 and ζ_{A}ζ_{B} = 1. So the solutions to (46) are

(0 : a_{1} : a_{2} : …) = (0 : ζ − ζ^{−1} : ζ^{2} − ζ^{−2} : …)  (47)

where ζ is a primitive nth root of unity. This description of the cusps is in agreement with our earlier calculations.

4.6 Calculations with char(K) = n

We continue to take n ≥ 5 prime, but now work over K an algebraically closed field with char(K) = n. We consider the family of curves A(n) → A(n) defined by (41) and (42), but do not attempt to relate these to any moduli problem. Let C be the fibre of A(n) → A(n) above 0_{C} = (0 : a_{1} : a_{2} : …). If 0_{C} is a rational cusp then C is a Néron polygon of degree n, exactly as before. It is easy to check A(n) contains the point ϖ := (0 : 1 : 2 : 3 : …).

Proposition 4.17 Let C be as above with 0_{C} not a rational cusp.

(i) If 0_{C} ≠ ϖ then C is an elliptic curve of degree n. (ii) If 0_{C} = ϖ then C is a line with multiplicity n.

Proof. Since a_{1}, a_{2}, …, a_{(n−1)/2} are non-zero, we may follow Vélu’s proof of Proposition 4.14. We deduce that C ↪ P^{n−1} is an irreducible curve of degree n and that the (non-reduced) group scheme μ_{n} acts on C with quotient ϕ : C → D. Here D is the plane cubic defined by (45).

4.6. CALCULATIONS WITH CHAR(K) = n 59

(i) If D is a smooth cubic then we recognise ϕ : C → D as an inseparable isogeny of degree n. In particular C is an elliptic curve. (ii) If D has a singular point, then a priori this could be either a node or a cusp. Either way C has a fixed point under M_{Q}. But the only fixed point under M_{Q} is (1 : 1 : …: 1) and the condition for C to contain this point is

rank(a_{i−j})_{i,j=0}^{n−1} = 2.

Adapting Lemma 4.16 to the case char(K) = n gives a_{i} = A + Bi for some A, B ∈ K. Since a_{0} = 0 it follows 0_{C} = ϖ.

It only remains to describe C in the case 0_{C} = ϖ. Then C contains ϖ and its translates under M_{Q} and these lie on a line ℓ. Since C is defined by quadrics it must contain ℓ. We now know that C ↪ P^{n−1} is an irreducible curve of degree n which is set-theoretically a line. Although these conditions certainly do not suffice to determine C as a scheme, we shall be content to call C a line with multiplicity n. □

Remark 4.18 In the above proof ϕ : C → D is set-theoretically an isomorphism. So in case (ii) the plane cubic D must have a cusp rather than a node. Substituting a_{i} = i into the Weierstrass equation (45) gives

(Y − 3X + 2Z)^{2}Z = −4(X − Z)^{3}  (48)

and this is indeed the equation of a cuspidal cubic.

Remark 4.19 The linear subspaces preserved by M_{Q} are

Π^{r} := { (f(0) : f(1) : …: f(n − 1)) | f(x) ∈ K[x] with deg f ≤ r }.

If 0_{C} = ϖ then C is set-theoretically equal to the line Π^{1} and contains the point Π^{0} = (1 : 1 : …: 1). This point is distinguished on C by its Zariski tangent space Π^{3}. All other points on C have tangent space Π^{2}.

Proposition 4.17 allows us to complete our proof of Proposition 3.10. In the case n = 5 we have A(5) = P^{1} and ϖ = (1 : 2). Thus the condition for T [λ, τ_{0}, …, τ_{4}] to degenerate is β(λ) = (λ − 3)^{2} = 0. In the case n = 7 we have A(7) = {a^{3}b + b^{3}c + c^{3}a = 0} and ϖ = (1 : 2 : 4). Thus the condition for T [λ, τ_{0}, …, τ_{6}] to degenerate is β(λ) = (λ − 5)^{3} = 0.

We suspect the curves C appearing in Proposition 4.17(i) are elliptic normal curves. We are able to prove this in the cases n = 5 and 7.

Lemma 4.20 Let n = 5 or 7 and let (0 : a_{1} : a_{2} : …) ∈ A(n). Then the matrix (a_{i−j})_{i,j=0}^{n−1} has either rank 2 or rank n − 1.

Proof. In the case n = 5 this is immediate. In the case n = 7 all the 6 × 6 Pfaffians are equal to (a + 2b + 4c)^{3}. It is easy to check that the line a + 2b + 4c = 0 meets A(7) only at the point ϖ. □

The elliptic curve C passes through the points 0_{C} = (0 : a_{1} : a_{2} : …) and its translates under M_{Q}. By Lemma 4.20 these points span the hyperplane

H := {x_{0} + x_{1} + … + x_{n−1} = 0}.

Thus to prove C is (projectively) normal we are reduced to showing C is not contained in H. We give three proofs of this in the case n = 5, the last of which generalises to n = 7.

The first proof uses the fact that C is defined by quadrics. If C is contained in H ≅ P^{3}, then it lies in the intersection of two quadrics and so has degree at most 4. But we know C has degree 5 and this gives the required contradiction.

For our second proof we use computer algebra to give equations for the tangent line to C at 0 = (0 : a : b : −b : −a). The tangent line passes through the point

(10a^{3}b^{3} : a(a^{5} − 3b^{5}) : −b(3a^{5} + b^{5}) : −b(3a^{5} + b^{5}) : a(a^{5} − 3b^{5})).

Adding up the co-ordinates we find that the condition for C to lie in H is 2(a^{2} − ab − b^{2})^{3} = 0. Since char(K) = 5 this is equivalent to (a : b) = ϖ. Similar calculations for n = 7 are possible but rather messy.

For our third proof we take n = 5 or 7 and recall C meets the hyperplane H_{0} = {x_{0} = 0} at a single point, and likewise for the other co-ordinate hyperplanes, thanks to the action of M_{Q}. It therefore suffices to prove the cases n = 5 and 7 of the following conjecture.

Conjecture 4.21 Let n ≥ 3 prime, and let E/K be an elliptic curve with E[n] ≅ Z/nZ. Then the divisors {nP | P ∈ E[n]} span a complete linear system.

We may prove this conjecture for n = 5 and 7 by means of a direct calculation. Indeed the curves in question are the D_{λ} with Weierstrass equation (2). We recall that 1 = (0, 0) generates Z/nZ ↪ D_{λ}(K). In the case n = 5 we write down rational functions f_{r} ∈ ⟨x^{2}, xy, x, y, 1⟩ with (f_{r}) = 5.(r1) − 5.0. We obtain a 5 × 5 matrix

x^{2}  xy  x  y  1
f_{0}  0  0  0  0  1
f_{1} −1  1  0  1  0
f_{2} 2λ  −1 λ^{3} − 3λ^{2} −λ^{2} + λ λ^{3}
f_{3} −λ − 1 −1 λ^{2} + 2λ −λ^{2} + λ −λ^{2}
f_{4} λ − 2 −1 2λ − 1  −1  λ

with determinant (λ^{2} − λ − 1)^{2}(λ^{2} − 6λ − 1). Since char(K) = 5, this matrix is non-singular whenever D_{λ} is non-singular. In the case n = 7 the rational functions f_{r} ∈ ⟨x^{3}, x^{2}y, x^{2}, xy, y^{2}, x, y, 1⟩ for 0 ≤ r ≤ 6 together with the equation for D_{λ} itself give an 8 × 8 matrix. In characteristic 7 this matrix is non-singular whenever D_{λ} is non-singular. Computing the f_{r} here is straightforward. Indeed, f_{1} = f is given by (10) and the remaining f_{r} follow from the substitutions envisaged in Remark 1.2.

To end this chapter we sketch an alternative proof of Proposition 4.17(i), specific to the cases n = 5 and 7, but which does not rely on Vélu’s explicit formulae. First we express K as the residue field for some local field of characteristic zero, see [Se2, Chapter II, Theorem 3]. Then we deduce from our work in the case char(K) ≠ n that C contains a curve C_{1} of degree n and arithmetic genus 1. Since 0_{C} ≠ ϖ, Lemma 4.20 tells us that M_{Q} acts on C without fixed points and that the points of intersection of C with the hyperplanes H_{i} span H. From these observations it follows C_{1} is irreducible and smooth. Then the above proof of Conjecture 4.21 in the cases n = 5 and 7 tells us C_{1} is an elliptic normal curve. Finally C and C_{1} are both defined by n(n − 3)/2 quadrics and so C = C_{1} as required.

Descent Theorems over Q(μ_{n})

Let n = 5 or 7. In this chapter we generalise our first descent theorem to K = Q(μ_{n}). It is well known that K has class number 1 and ring of integers Z[ζ] where ζ is a primitive nth root of unity. The unit group U_{K} is generated modulo torsion by the ratios (ζ^{i} − ζ^{−i})/(ζ^{j} − ζ^{−j}). For 𝔭 a prime we recall dim_{n} O_{𝔭}^{×}/(O_{𝔭}^{×})^{n} = 1 + ord_{𝔭}(n), and indeed for 𝔭 ∤ n this is immediate by Hensel’s lemma. At 𝔩 = (1 − ζ), the unique prime above n, we work with the filtration O_{𝔩}^{×} ⊃ U_{1} ⊃ U_{2} ⊃ …, where U_{r} = {x ∈ O_{𝔩}^{×} | x ≡ 1 (mod 𝔩^{r})} and for r ≥ 1 the quotient U_{r}/U_{r+1} is cyclic of order n.

5.1 Statement of the First Descent Theorem

Let λ ∈ K not a cusp of X_{1}(n). Let C_{λ} and D_{λ} be the n-isogenous elliptic curves introduced in §1.1. We define disjoint sets of primes

A = { 𝔭 prime | ord_{𝔭}^{∗}(λ) > 0 }
B = { 𝔭 prime | ord_{𝔭}^{∗}(η(λ)) > 0 }

(49)

where ord_{𝔭}^{∗} is as defined in §3.2 and η is the involution of X_{1}(n) swapping over the rational and irrational cusps. We define d = d(λ) as a function of the cusp number c = c(λ) introduced in §1.1.

n = 5 \begin{matrix}c & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ d & 0 & 0 & 1 & 2 & 3 & 4 & 4 \end{matrix} n = 7 \begin{matrix}c & 0 & 1 & 2 & 3 & 4 \\ d & 0 & 1 & 3 & 5 & 6 \end{matrix}

64 CHAPTER 5. DESCENT THEOREMS OVER Q(μ_{n})

Theorem 3 Let n = 5 or 7 and let λ ∈ K not a cusp of X_{1}(n). For A, B defined by (49) and d = d(λ) we write

[A] = { θ ∈ K^{×}/(K^{×})^{n} | ord_{𝔭}(θ) ≡ 0 (mod n) for all 𝔭 ∉ A }
I(B, d) = (subgroup of I_{K}/I_{K}^{n} generated by B) ⊕ U_{n−d}/U_{n}.

Then the Selmer group attached to the n-isogeny ϕ : C_{λ} → D_{λ} is given by

S^{(ϕ)}(C_{λ}/K) ≅ \{ θ ∈ [A] | \begin{matrix}θ ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ B \\ and θ^{n−1} ≡ 0, 1 (mod 𝔩^{d+1}) \end{matrix} \}.

The contribution to S^{(ϕ)}(C_{λ}/K) coming from Z/nZ ↪ D_{λ}(K) is generated by α(λ). Furthermore there is a pairing Ξ : [A] × I(B, d) → μ_{n} such that S^{(ϕ)}(C_{λ}/K) ≅ ker_{L}(Ξ) and S^{(ϕ̂)}(D_{λ}/K) ≅ ker_{R}(Ξ).

Remark 5.1 The pairing Ξ is again straightforward to compute. First we give a basis for [A] consisting of units and generators for the primes in A. Then for each prime 𝔭 ∈ B with 𝔭 ≠ 𝔩 we choose a non-trivial character χ_{𝔭} : (O_{K}/𝔭)^{×} → Z/nZ and compute χ_{𝔭} on our basis for [A]. To treat the conditions at 𝔩 we define characters χ_{i} : O_{𝔩}^{×} → Z/nZ via

θ ≡ (1 − γ)^{χ_{1}(θ)}(1 − γ^{2})^{χ_{2}(θ)} … (1 − γ^{n})^{χ_{n}(θ)} mod (K_{𝔩}^{×})^{n}  (50)

where γ = 1 − ζ. The matrix representing Ξ has columns corresponding to χ_{1}, …, χ_{d}. If d = n − 1 then we include the character χ_{n} precisely whenever the prime 𝔩 belongs to B.

Remark 5.2 Replacing λ by η(λ) in Theorem 3 leads to pairings Ξ represented by matrices of different sizes, even though the Selmer groups computed must be the same. When computing examples by hand we naturally choose to work with the smaller matrix.

Remark 5.3 Let G := Gal(K/Q). In many cases of interest the sets A and B are G-stable. For example this happens whenever C_{λ} and D_{λ} admit models over Q. In these cases the pairing Ξ is G-equivariant, i.e.

Ξ(σa, σb) = σ Ξ(a, b) for all σ ∈ G.

We may thus simplify our calculations by first decomposing the G-modules [A] and I(B, d) into 1-dimensional submodules. We write ψ for the cyclotomic character G ⟶^{∼} (Z/nZ)^{×} and do not distinguish in our notation between characters and the 1-dimensional G-modules they describe. Thus U_{K}/U_{K}^{n} ≅ ψ ⊕ ψ^{2}, respectively ψ ⊕ ψ^{2} ⊕ ψ^{4}, and U_{r}/U_{r+1} ≅ ψ^{r}. The contribution from each rational prime p merely depends on how p splits in K.

5.2 Examples and Applications

Let n = 5 or 7 and let λ ∈ K not a cusp of X_{1}(n). Theorem 3 gives us information about both the Mordell-Weil groups C_{λ}(K) and D_{λ}(K) and the Tate-Shafarevich groups � (C_{λ}/K) and � (D_{λ}/K). We now make this explicit. Let R = rank C_{λ}(K) = rank D_{λ}(K) and define s, t ≥ 1 via |C_{λ}(K)(n)| = n^{s} and |D_{λ}(K)(n)| = n^{t}. Theorem 3 together with the exact sequences

\begin{matrix}0 → D_{λ}(K)/ϕC_{λ}(K) → S^{(ϕ)}(C_{λ}/K) → � (C_{λ}/K)[ϕ] → 0 \\ 0 → C_{λ}(K)/ϕ̂D_{λ}(K) → S^{(ϕ̂)}(D_{λ}/K) → � (D_{λ}/K)[ϕ̂] → 0 \end{matrix}

tells us R = R_{1} + R_{2} with R_{1}, R_{2} ≥ 0 and

|A| + (n − 1)/2 − rank(Ξ) = R_{1} + t + 1 − s + dim_{n} � (C_{λ}/K)[ϕ]
|B| + d − rank(Ξ) = R_{2} + s + 1 − t + dim_{n} � (D_{λ}/K)[ϕ̂].

(51)

Thus the upper bound for the rank provided by first descent is

|A| + |B| + (n − 1)/2 + d − 2 − 2 rank(Ξ).  (52)

Example 5.4 Let n = 5 and λ = 2. Then A = {2}, B = {𝔭_{1}, 𝔭_{2}} and d = 0 where (19) = 𝔭_{1}𝔭_{2}. Since [A] ≅ 1 ⊕ ψ ⊕ ψ^{2} and I(B, d) ≅ 1 ⊕ ψ^{2} as G-modules, we know rank(Ξ) ≤ 1. But 5^{2} ∤ 19^{2} − 1, so ζ ∉ (F_{19^{2}}^{×})^{5} and rank(Ξ) = 1. Reducing modulo a few primes we learn D_{2}(K)_{tors} ≅ Z/5Z and so (x, y) = (1, (3 + \sqrt{5})/2) is a point of infinite order. By Theorem 3 we deduce

\begin{matrix}C_{2}(K) ≅ Z/5Z ⊕ Z & � (C_{2}/K)(5) = 0 \\ D_{2}(K) ≅ Z/5Z ⊕ Z & � (D_{2}/K)(5) = 0. \end{matrix}

Example 5.5 Let n = 7 and λ = η(5). Then A = ∅, B = {𝔭_{1}, 𝔭_{2}, 5} and d = 0 where (2) = 𝔭_{1}𝔭_{2}. Since [A] ≅ ψ ⊕ ψ^{2} ⊕ ψ^{4} and I(B, d) ≅ 1 ⊕ 1 ⊕ ψ^{3} as G-modules, we know rank(Ξ) ≤ 2. But 7^{2} ∤ 2^{3} − 1, so ζ ∉ (F_{2^{3}}^{×})^{7} and rank(Ξ) ≥ 1. If rank(Ξ) = 1 then ζ + ζ^{−1} ≡ 1 (mod 𝔭_{1}) and ζ + ζ^{−1} ≡ 1 (mod 𝔭_{2}). This gives ζ + ζ^{−1} ≡ 1 (mod 2) which is absurd. So rank(Ξ) = 2 and by Theorem 3 we deduce

\begin{matrix}C_{5}(K) ≅ Z/7Z & � (C_{5}/K)(7) = 0 \\ D_{5}(K) ≅ Z/7Z & � (D_{5}/K)(7) = 0. \end{matrix}

Let Σ(λ) = {λ, −1/λ}, respectively {λ, (λ − 1)/λ, 1/(1 − λ)}. For λ ∈ K the curves C_{λ} and D_{λ} admit models over Q if and only if Σ(λ) is G-stable. We obtain elliptic curves over Q with a labelled copy of the Tate twist Z/nZ(k). For each 0 ≤ k < n−1 these curves are parametrised by some twist of Y_{1}(n). For E an elliptic curve over Q we write E^{∗} for the 5- respectively −7-twist of E. Then the families under discussion are

n = 5 \begin{aligned}k & 0 & 1 & 2 & 3 \\ & D_{λ} & C_{λ} & D_{λ}^{∗} & C_{λ}^{∗} \end{aligned} n = 7 \begin{aligned}k & 0 & 1 & 2 & 3 & 4 & 5 \\ & D_{λ} & C_{λ} & B_{ρ}^{∗} & D_{λ}^{∗} & C_{λ}^{∗} & B_{ρ} \end{aligned} .

We give equations for the curves B_{ρ} in Appendix B. We choose our coordinate ρ with cusps at the roots of

ρ^{3} − ρ^{2} − 2ρ + 1 = 0 and ρ^{3} − 2ρ^{2} − ρ + 1 = 0.

The automorphisms of P^{1} permuting these cusps are generated by ρ ↦ 1/ρ and ρ ↦ 1 − ρ. We find B_{ρ} ≅ B_{(ρ−1)/ρ} whereas B_{ρ} and B_{1−ρ}^{∗} are related by 7-isogeny.

We may exhibit elements of the Tate-Shafarevich group by comparing two different 5-descents. As we saw in §2.3 we need only apply our first descent theorem for λ = τ^{5} and λ = τf(τ)/g(τ) and compare estimates. In the case τ = 1 the curves in question are

C_{1} ≅ D_{11} ≅ X_{0}(11) and D_{1} ≅ X_{1}(11).

The next two examples duplicate some of the work in [CS].

Example 5.6 Let n = 5 and λ = 1. Then A = ∅, B = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}} and d = 0 where (11) = 𝔭_{1}𝔭_{2}𝔭_{3}𝔭_{4}. Since [A] ≅ ψ ⊕ ψ^{2} is generated by ζ and ϕ = 1 + ζ + ζ^{4} with ζ, ϕ ≢ ±1 (mod 𝔭_{1}) it follows rank(Ξ) = 2. Theorem 3 gives

\begin{matrix}C_{1}(K) ≅ (Z/5Z)^{2} & � (C_{1}/K)(5) = 0 \\ D_{1}(K) ≅ Z/5Z & � (D_{1}/K)(5) = 0. \end{matrix}

Example 5.7 Let n = 5 and λ = 11. Then A = {𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4}}, B = ∅ and d = 0. We have [A] ≅ 1 ⊕ ψ ⊕ ψ ⊕ ψ^{2} ⊕ ψ^{2} ⊕ ψ^{3} and D_{11}[5] ≅ 1 ⊕ ψ. Since C_{1} ≅ D_{11}, the last example and Theorem 3 give

� (C_{11}/K)(5) ≅ ψ ⊕ ψ^{2} ⊕ ψ^{2} ⊕ ψ^{3}.

Noting [K : Q] is prime to 5 it follows � (C_{11}^{∗}/Q)(5) ≅ (Z/5Z)^{2}. According to Cremona’s tables [Cr] this curve C_{11}^{∗} =275B3 is the first curve, by conductor, with � of analytic order 25.

Example 5.8 Let n = 7 and let λ ∈ Q(μ_{7}) ∩ R with η(λ) = λ. The curves C_{λ} and D_{λ} admit complex multiplication by some order in Q(\sqrt{−7}). Since the sets of primes A and B are disjoint we must have A = B = ∅. Similarly from the identity c(λ) + c(η(λ)) = 4 we deduce c(λ) = 2 and d(λ) = 3. Thus

[A] ≅ ψ ⊕ ψ^{2} ⊕ ψ^{4} and I(B, d) ≅ ψ^{4} ⊕ ψ^{5} ⊕ 1.

Now Ξ : [A] × I(B, d) → μ_{7} is G-equivariant and the proof of Theorem 3 reveals ker_{L}(Ξ) ≅ ker_{R}(Ξ) as G-modules. Thus rank(Ξ) = 2 and

rank C_{λ}(K) = rank D_{λ}(K) = 0.

Remark 5.9 The curves in Example 5.8 each admit a pair of models over Q. These models are B_{ρ} and B_{ρ}^{∗} for ρ = 0, 1, ∞ and ρ = −1, \frac{1}{2}, 2, equivalently the curves 49A in Cremona’s tables [Cr]. The quotient of the Klein quartic

X(7) = {a^{3}b + b^{3}c + c^{3}a = 0}

by the automorphism (a : b : c) ↦ (c : a : b) is the elliptic curve X_{0}(49). We refer to [E, §2.1] for explicit formulae. We have just shown that this curve has rank 0 over K = Q(μ_{7}). A straightforward calculation now shows that the only K-rational points on the Klein quartic are the 24 points of inflection.

Remark 5.10 We have no analogue of Example 5.8 in the case n = 5 since the fixed points of η are defined over Q(μ_{5}, \sqrt{−1}) but not over Q(μ_{5}). This may be related to the fact Q(\sqrt{−5}) has Hilbert class field Q(\sqrt{5}, \sqrt{−1}).

The following problem was suggested by John Coates. Our motivation is to find examples to illustrate the theory in [CS]. We continue to write K = Q(μ_{n}) with n = 5 or 7.

Problem 5.11 Find some examples of elliptic curves E/K satisfying

(a) rank E(K) = 0 and � (E/K)(n) = 0.

(b) E admits a model over Q.

(c) The Tamagawa numbers c_{𝔭} = [E(K_{𝔭}) : E_{0}(K_{𝔭})] are prime to n.

(d) E has good ordinary reduction at 𝔩 = (1 − ζ).

(e) E does not admit complex multiplication (over \overline{K}).

It is known that the curve X_{1}(11) satisfies all these conditions for n = 5. We use Theorem 3 to give further examples with E(K)[n] ≠ 0. Accordingly we take E = D_{λ} for some λ ∈ K. Condition (b) tells us that the set Σ(λ) is G-stable and in particular λ ∈ R. Condition (c) tells us that the set A defined by (49) must be empty. Indeed in the case of split multiplicative reduction the Tamagawa number c_{𝔭} is ord_{𝔭}(Δ), so our claim follows by Lemma 1.4 and inspection of the discriminant Δ(D_{λ}) recorded in §1.1.

In the case n = 5 the condition A = ∅ tells us λ ∈ K is a unit. We have either λ = ±1 corresponding to X_{1}(11) or λ = ±ϕ^{j} for some odd integer j. We apply Theorem 3 for |j| ≤ 23 and tabulate the results in Appendix D.

In the case n = 7 the condition A = ∅ tells us λ ∈ K is an exceptional unit, i.e. both λ and λ − 1 are units. In any number field it is known there are only finitely many exceptional units. Nagell [Na, Théorème 8] has determined all the exceptional units in Q(μ_{7}) ∩ R. The solutions satisfying (b) are the roots of

λ^{3} − tλ^{2} + (t − 3)λ + 1 = 0  (53)

for t = 1, 2, −5, 8, −12, 15, −1259, 1262. For a discussion of this problem of computing exceptional units, and further references to the literature, see [Sm] and [Ni]. The roots of (53) for t = 8 are the irrational cusps of X_{1}(7). We apply Theorem 3 for the remaining 7 curves and tabulate the results in Appendix D.

All the curves in Tables 9 and 10 with entry R = 0 satisfy (a), (b) and (c). We check (d) using Lemma 1.6 and check (e) by computing the j-invariant and comparing with the table in [Si2]. Finally we arrive at some solutions to Problem 5.11 which we list by giving their Cremona labels

n = 5 11A3, 150A1, 150A2, 2350N1  n = 7 294B1, 490K1.

In each case reducing modulo a few primes we learn E(K)(n) ≅ Z/nZ. By inspection of the table of §1.2 the reduction map E(K)(n) → Ẽ(O_{K}/𝔩)(n) is an isomorphism. It follows from [CS, Chapter 3] that each curve on our list has rank 0 over Q(μ_{n^{∞}}).

5.3 Computation of Selmer Groups

Let n = 5 or 7. Let λ ∈ K not a cusp of X_{1}(n) and let A, B and d be as defined in §5.1. In this section we prove that the Selmer group S^{(ϕ)}(C_{λ}/K) is as described by Theorem 3. The definition (8) reads

S^{(ϕ)}(C_{λ}/K) = { θ ∈ K^{×}/(K^{×})^{n} | C_{λ,θ}(K_{𝔭}) ≠ ∅ for all 𝔭 }.  (54)

For θ ∈ K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} the exact sequence (7) tells us

C_{λ,θ}(K_{𝔭}) ≠ ∅ ⟺ θ ∈ im (δ : D_{λ}(K_{𝔭}) → K_{𝔭}^{×}/(K_{𝔭}^{×})^{n}).  (55)

Proposition 5.12 below describes the image of δ. If 𝔭 ≠ 𝔩 this description is an immediate consequence of Proposition 3.12 and the observation

β(λ) ≡ 0 (mod 𝔭) ⟺ ord_{𝔭}^{∗}(η(λ)) > 0.  (56)

Proposition 5.12 The map δ = δ_{ϕ} has image

im δ = \{ \begin{matrix}K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} & if 𝔭 ∈ A \\ O_{𝔭}^{×}/(O_{𝔭}^{×})^{n} & if 𝔭 ∉ A ∪ B and 𝔭 ≠ 𝔩 \\ 0 & if 𝔭 ∈ B. \end{matrix}

If 𝔭 ∉ A ∪ B and 𝔭 = 𝔩 then im δ = U_{d+1}(K_{𝔩}^{×})^{n}/(K_{𝔩}^{×})^{n}.

Since we are working over K = Q(μ_{n}) we may identify Z/nZ ≅ μ_{n} and so have connecting maps

\begin{matrix}δ_{ϕ} : D_{λ}(K_{𝔭}) → K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} \\ δϕ̂ : C_{λ}(K_{𝔭}) → K_{𝔭}^{×}/(K_{𝔭}^{×})^{n}. \end{matrix}

Tate local duality tells us im δ_{ϕ} and im δϕ̂ are exact annihilators with respect to the norm residue symbol

(·, ·)_{𝔭} : K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} × K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} → μ_{n}.

We define the “unit” and “not ramified” subgroups

H_{un}^{1}(K_{𝔭}, μ_{n}) = { θ ∈ K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} | ord_{𝔭}(θ) ≡ 0 (mod n) }
H_{nr}^{1}(K_{𝔭}, μ_{n}) = \{ θ ∈ K_{𝔭}^{×}/(K_{𝔭}^{×})^{n} | K_{𝔭}(\sqrt[n]{θ})/K_{𝔭} is unramified \}

and recall from §1.4 that these are exact annihilators with respect to the norm residue symbol. If 𝔭 ≠ 𝔩 then H_{un}^{1}(K_{𝔭}, μ_{n}) = H_{nr}^{1}(K_{𝔭}, μ_{n}) = O_{𝔭}^{×}/(O_{𝔭}^{×})^{n} is a maximal isotropic subspace of K_{𝔭}^{×}/(K_{𝔭}^{×})^{n}. Thus Proposition 5.12 is compatible with Tate local duality in the sense that if we replace λ by η(λ) then im δ is replaced by its exact annihilator. For 𝔭 = 𝔩 we check the compatibility first, as this will shorten our proofs. We begin with a detailed description of the norm residue symbol (·, ·)_{𝔩}.

Lemma 5.13 Let U′_{r} = U_{r}(K_{𝔩}^{×})^{n}/(K_{𝔩}^{×})^{n}. Then

K_{𝔩}^{×}/(K_{𝔩}^{×})^{n} ⊃ U′_{1} ⊃ U′_{2} ⊃ … ⊃ U′_{n} ⊃ U′_{n+1} = 0

and for 1 ≤ r < s ≤ n + 1 we identify U′_{r}/U′_{s} = U_{r}/U_{s}.

(i) H_{un}^{1}(K_{𝔩}, μ_{n}) = U′_{1} and H_{nr}^{1}(K_{𝔩}, μ_{n}) = U′_{n}. (ii) The subgroups U′_{r} and U′_{n+1−r} are exact annihilators. (iii) The norm residue symbol induces a non-degenerate pairing

U_{1}/U_{n} × U_{1}/U_{n} → μ_{n}

and the subspace generated by U_{K} is a maximal isotropic subspace.

Proof. See [CF, Exercise 2] or [W, Exercise 9.3]. For (iii) we note that the natural map U_{K}/U_{K}^{n} → U′_{1}/U′_{n} is injective since K has no unramified abelian extensions. The image is isotropic by Lemmas 1.19 and 1.20 and maximally isotropic by counting. □

By Lemma 1.5(iii) and the table of §5.1

d(λ) + d(η(λ)) = n − 1.  (57)

Thus Proposition 5.12 in the case 𝔭 = 𝔩 is compatible with Tate local duality. It therefore suffices to prove the proposition for, say, c(λ) ≤ c_{max}/2. If c(λ) = 0 then we are already done by Proposition 3.12. The next two lemmas treat the case 0 < c(λ) < c_{max}.

Lemma 5.14 Let λ ∈ K_{𝔩} with 0 < c(λ) < c_{max}. Then c(λ) = ord_{𝔩}(λ − u) where u is any irrational cusp and

ord_{𝔩}(β′(λ)) = ord_{𝔩}(α(λ) − α(u)) = \{ \begin{matrix}ord_{𝔩}(λ − u) & if n = 5 \\ 2 ord(λ − u) & if n = 7. \end{matrix}

Proof. The only subtle part is proving ord_{𝔩}(α(λ) − α(u)) = 2 ord_{𝔩}(λ − u) in the case n = 7. We first compute ord_{𝔩}(α′(u)) = c_{max} and ord_{𝔩}(α′′(u)) = 0 and then consider the Taylor series expansion of α(λ) about u. □

Lemma 5.15 Let λ ∈ K_{𝔩} with 0 < c(λ) < c_{max} and let t = ord_{𝔩}(β′(λ)). Then the map δ : D_{λ}(K_{𝔩}) → K_{𝔩}^{×}/(K_{𝔩}^{×})^{n} has image U′_{t} = U_{t}(K_{𝔩}^{×})^{n}/(K_{𝔩}^{×})^{n}.

Proof. By the proof of Lemma 1.6 our Weierstrass equation (2) for D_{λ} is minimal. We write E = D_{λ} and recall from [Si1, Chapter VII] there is a filtration

E(K_{𝔩}) ⊃ E_{0}(K_{𝔩}) ⊃ E_{1}(K_{𝔩}).

If t < n − 1 then Proposition 3.18 with ν = 1 tells us that the image of δ restricted to E_{1}(K_{𝔩}) is U′_{t+1}. Now E_{0}(K_{𝔩})/E_{1}(K_{𝔩}) ≅ Z/nZ is generated by 1 = (0, 0). So the image of δ restricted to E_{0}(K_{𝔩}) is generated by α(λ) and U′_{t+1}. But Corollary 3.13 and Lemma 5.14 imply α(λ) is a generator for U′_{t}/U′_{t+1}. So δ restricted to E_{0}(K_{𝔩}) has image U′_{t}. The additive reduction tells us the Tamagawa number [E(K_{𝔩}) : E_{0}(K_{𝔩})] is at most 4 and so prime to n. This proves the lemma in the case t < n − 1 and the general case follows by Tate local duality. □

This completes the proof of Proposition 5.12.

Remark 5.16 We briefly indicate how we may prove Proposition 5.12 without using Tate local duality. For the proof of Lemma 5.15 it remains to extend the Hensel lemma calculations of §3.3 to treat the case t ≥ n − 1. If we write F (z) = 1+γ_{1}z+…+γ_{n}z^{n}+… it turns out the correct modification to Lemma 3.17(i) is ord_{𝔩}(γ_{1} − nγ_{n}) = t.

In the case c(λ) = c_{max} our push-out calculations are useless since (2) is no longer a minimal Weierstrass equation. However, replacing λ by η(λ), it suffices to find the image of

δϕ̂ : C_{λ}(K_{𝔩}) → K_{𝔩}^{×}/(K_{𝔩}^{×})^{n}

in the case c(λ) = 0. We take C_{λ} ↪ P^{n−1} with the equations of Proposition 3.7 and let {h(x_{0}, …, x_{n−1}) = 0} be the hyperplane cutting out the divisor n.0. We then compute the image of δϕ̂ via push-out from C_{λ} using the rational function

h(x_{0}, ζ^{−1}x_{1}, …, ζx_{n−1})/h(x_{0}, x_{1}, …, x_{n−1}).

Proposition 5.17 Let λ ∈ K not a cusp of X_{1}(n). Then

S^{(ϕ)}(C_{λ}/K) ≅ \{ θ ∈ [A] | \begin{matrix}θ ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ B \\ and θ^{n−1} ≡ 0, 1 (mod 𝔩^{d+1}) \end{matrix} \}
S^{(ϕ̂)}(D_{λ}/K) ≅ \{ θ ∈ [B] | \begin{matrix}θ ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ A \\ and θ^{n−1} ≡ 0, 1 (mod 𝔩^{n−d}) \end{matrix} \}.

Proof. The description of S^{(ϕ)}(C_{λ}/K) is immediate from the definition (54) and Proposition 5.12. The description of S^{(ϕ̂)}(D_{λ}/K) follows on replacing λ by η(λ). □

5.4 Proof of the First Descent Theorem over Q

We use Proposition 5.17 to give an alternative proof of our first descent theorem over Q. The idea is to relate S^{(ϕ)}(C_{λ}/K) and S^{(ϕ̂)}(D_{λ}/K) via the nth power reciprocity law and then to descend to a statement about Selmer groups over Q. We obtain a proof closer to the sketch given in §2.1.

Let G = Gal(K/Q) and let ψ : G ⟶^{∼} (Z/nZ)^{×} be the cyclotomic character. For A a G-module, written multiplicatively, and χ : G → (Z/nZ)^{×} a character we adopt the notation

A^{χ} = { a ∈ A | σ(a) = a^{χ(σ)} for all σ ∈ G }.

The restriction maps give isomorphisms

\begin{aligned}H^{1}(Q, μ_{n}) & = & (K^{×}/(K^{×})^{n})^{1} \\ π : H^{1}(Q, Z/nZ) & ⟶^{∼} & (K^{×}/(K^{×})^{n})^{ψ} \end{aligned}  (58)

where π depends on a choice of isomorphism Z/nZ ≅ μ_{n}.

Lemma 5.18 Let p ≡ 0, 1 (mod n) and let ξ^{(p)} ∈ H^{1}(Q, Z/nZ) describe F^{(p)}, the degree n cyclic extension of Q unramified outside p.

(i) If p ≡ 1 (mod n) then πξ^{(p)} is a product of primes above p. (ii) If p = n then πξ^{(n)} is a primitive nth root of unity. Moreover in case (i) πξ^{(p)} ∈ U_{n}(K_{𝔩}^{×})^{n} whereas in case (ii) πξ^{(n)} ∉ U_{2}(K_{𝔩}^{×})^{n}.

Proof. The element πξ^{(p)} is a Kummer generator for KF^{(p)}/K and the lemma follows from the fact this extension is unramified outside p. □

Let λ ∈ Q not a cusp of X_{1}(n). The maps (58) induce isomorphisms

\begin{aligned}S^{(ϕ)}(C_{λ}/Q) & ≅ & S^{(ϕ)}(C_{λ}/K)^{1} \\ S^{(ϕ̂)}(D_{λ}/Q) & ≅ & S^{(ϕ)}(C_{η(λ)}/K)^{ψ}. \end{aligned}  (59)

The crucial point here is that [K : Q] is prime to n, so the torsor T/Q has points everywhere locally if and only if T/K has points everywhere locally. For a detailed account of this argument see [Sw].

We write A_{Q} and B_{Q} for the sets of rational primes defined in §2.1 and A_{K}, B_{K} and d_{K} for the quantities defined in §5.1. Then A_{K} is precisely the set of primes above A_{Q}, and B_{K} contains all the primes above B_{Q} \{n}. The condition for n ∈ B_{Q} is equivalent to d_{K} = n − 1. By Proposition 5.17 and Lemma 5.18 we deduce

S^{(ϕ)}(C_{λ}/Q) ≅ {a ∈ [A_{Q}] | a ∈ (Q_{p}^{×})^{n} for all p ∈ B_{Q}}
S^{(ϕ̂)}(D_{λ}/Q) ≅ {b ∈ ⟨B_{Q}⟩ | πb ∈ (K_{𝔭}^{×})^{n} for all 𝔭|p with p ∈ A_{Q}} (60)

where the notation [ ] and ⟨ ⟩ is retained from §3.4. For the rest of this section let us write A = A_{Q} and B = B_{Q}.

We define Ξ : [A] × ⟨B⟩ → μ_{n} via

Ξ(a, b) = ∏_{p∈B} ∏_{𝔭|p}(a, πb)_{𝔭} = ∏_{p∈A} ∏_{𝔭|p}(πb, a)_{𝔭}  (61)

where the two expressions given are equal by the product formula. Indeed if p ∉ A ∪ B then a ∈ H_{un}^{1}(K_{𝔭}, μ_{n}) and πb ∈ H_{nr}^{1}(K_{𝔭}, μ_{n}) so the term (a, πb)_{𝔭} makes no contribution. If n ∉ B then (61) is equivalent to the more appealing definition Ξ(a, b) = (a/πb) = (πb/a) where (·/·) is the power residue symbol.

Lemma 5.19 Let Ξ : [A] × ⟨B⟩ → μ_{n} be the pairing defined above.

(i) If a ∈ [A] and p ∈ B then Ξ(a, ξ^{(p)}) = 1 ⟺ a ∈ (Q_{p}^{×})^{n}. (ii) If p ∈ A and b ∈ ⟨B⟩ then Ξ(p, b) = 1 ⟺ πb ∈ (K_{𝔭}^{×})^{n} for all 𝔭|p.

Proof. The action of Gal(K/Q) tells us the term (a, πb)_{𝔭} is independent of the choice of 𝔭|p.

(i) If p ≠ n and 𝔭 is any prime of K above p then

Ξ(a, ξ^{(p)}) = 1 ⟺ (a, ξ^{(p)})_{𝔭} = 1 ⟺ (a/𝔭) = 1 ⟺ a ∈ (Q_{p}^{×})^{n}.

If p = n then Ξ(a, ξ^{(n)}) = (πξ^{(n)}/a) = (ζ/a) and by Euler’s criterion

Ξ(a, ξ^{(n)}) = 1 ⟺ a^{n−1} ≡ 1 (mod n^{2}) ⟺ a ∈ (Q_{p}^{×})^{n}.

(ii) If 𝔭 is any prime of K above p then

Ξ(p, b) = 1 ⟺ (πb, p)_{𝔭} = 1 ⟺ (πb/𝔭) = 1 ⟺ πb ∈ (K_{𝔭}^{×})^{n}.

□

Finally (60) and Lemma 5.19 give

S^{(ϕ)}(C_{λ}/Q) ≅ {a ∈ [A] | Ξ(a, ξ^{(p)}) = 1 for all p ∈ B} = ker_{L}(Ξ)
S^{(ϕ̂)}(D_{λ}/Q) ≅ {b ∈ ⟨B⟩ | Ξ(p, b) = 1 for all p ∈ A} = ker_{R}(Ξ).

This completes our alternative proof of Theorem 1.

Remark 5.20 The pairing Ξ = Ξ_{Hilb} defined here is related to Ξ = Ξ_{Tate} defined in §3.4 via

−Ξ_{Tate}(a, b) = Ind_{ζ} Ξ_{Hilb}(a, b)  (62)

where Ind_{ζ} is the inverse of the map Z/nZ ≅ μ_{n} used to construct π. This follows from the relationship between the Tate pairing and the Hilbert norm residue symbol. We recall that the restriction map Br(Q_{p}) → Br(K_{𝔭}) satisfies inv res = [K_{𝔭} : Q_{p}] inv. This accounts for the factor [K : Q] ≡ −1 (mod n) in the above formula.

5.5 Proof of the First Descent Theorem over Q(μ_{n})

We now complete the proof of Theorem 3. Proposition 5.17 gives a description of the Selmer groups attached to the isogeny ϕ and its dual. So it only remains to prove the following version of the power reciprocity law.

Proposition 5.21 Let n = 5 or 7 and let K = Q(μ_{n}). Let A, B be disjoint finite sets of primes of K. Let 0 ≤ d ≤ n − 1 be an integer, and suppose if 𝔩 ∈ A then d = 0, and if 𝔩 ∈ B then d = n − 1. Define

[A] = { a ∈ K^{×}/(K^{×})^{n} | ord_{𝔭}(a) ≡ 0 (mod n) for all 𝔭 ∉ A }
I(B, d) = (subgroup of I_{K}/I_{K}^{n} generated by B) ⊕ U_{n−d}/U_{n}.

Then there is a pairing Ξ : [A] × I(B, d) → μ_{n} with

ker_{L}(Ξ) ≅ {a ∈ [A] | a ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ B and a^{n−1} ≡ 0, 1 (mod 𝔩^{d+1})} ker_{R}(Ξ) ≅ {b ∈ [B] | b ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ A and b^{n−1} ≡ 0, 1 (mod 𝔩^{n−d}) }.

Proof. For S a set of primes of K we adopt the notation

[S] = ker (H^{1}(K, μ_{n}) → ∏_{𝔭∉S} H_{un}^{1}(K_{𝔭}, μ_{n}))
= { θ ∈ K^{×}/(K^{×})^{n} | ord_{𝔭}(θ) ≡ 0 (mod n) for all 𝔭 ∉ S }
I(S) = ⊕_{𝔭∈S}H^{1}(K_{𝔭}, μ_{n})/H_{un}^{1}(K_{𝔭}, μ_{n})
= subgroup of I_{K}/I_{K}^{n} generated by S.

Since K has class number 1 there is an exact sequence

0 → U_{K}/U_{K}^{n} → [S] → I(S) → 0.  (63)

5.5. PROOF OF THE FIRST DESCENT THEOREM OVER Q(μ_{n}) 75

For A, B and d as described in the proposition we define

[B, d] = {b ∈ [B] | b^{n−1} ≡ 0, 1 (mod 𝔩^{n−d})}
I(B, d) = \{ \begin{matrix}⊕_{𝔭∈B}H^{1}(K_{𝔭}, μ_{n})/H_{nr}^{1}(K_{𝔭}, μ_{n}) & if 𝔩 ∈ B \\ I(B) ⊕ U_{n−d}/U_{n} & if 𝔩 ∉ B. \end{matrix}

By Lemma 5.13(iii) and applying the snake lemma to the diagram

0 ⟶ U_{K}/U_{K}^{n} ⟶  [B]  ⟶ I(B) ⟶ 0
↓  ↓  ||
0 ⟶ U_{1}/U_{n} ⟶ I(B, n − 1) ⟶ I(B) ⟶ 0

we learn there is an exact sequence

0 → [B] → I(B, n − 1) → (U_{K}/U_{K}^{n})^{∨} → 0.

Since [B, d] ⊂ [B] is the inverse image of I(B, d) ⊂ I(B, n − 1) it follows

0 → [B, d] → I(B, d) → (U_{K}/U_{K}^{n})^{∨}  (64)

is exact. We define pairings

\begin{matrix}Ξ : [A] × I(B, d) → μ_{n}; & (a, b) ↦ ∏_{𝔭}(a, b)_{𝔭} \\ Ξ′ : I(A) × [B, d] → μ_{n}; & (a, b) ↦ ∏_{𝔭}(b, a)_{𝔭} \end{matrix}

where (·, ·)_{𝔭} is the norm residue symbol. The assumptions on A, B and d made in the statement of the proposition are sufficient for these pairings to be well-defined. The exact sequences (63) and (64) now give a diagram

0 ⟶ [B, d] ⟶ I(B, d) ⟶ (U_{K}/U_{K}^{n})^{∨}
↓_{Ξ′} ↓^{Ξ} ||
0 ⟶ I(A)^{∨} ⟶ [A]^{∨} ⟶ (U_{K}/U_{K}^{n})^{∨} ⟶ 0

and the product formula for the norm residue symbol tells us that this diagram is commutative. We compute

ker_{L}(Ξ) = {a ∈ [A] | a ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ B and a^{n−1} ≡ 0, 1 (mod 𝔩^{d+1})}
ker_{R}(Ξ) ≅ ker_{R}(Ξ′)
= {b ∈ [B, d] | b ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ A}
= {b ∈ [B] | b ∈ (K_{𝔭}^{×})^{n} for all 𝔭 ∈ A and b^{n−1} ≡ 0, 1 (mod 𝔩^{n−d})}.

This completes the proof of Proposition 5.21 and so also of Theorem 3. We remark that our proof of Proposition 5.21 holds for n replaced by any odd regular prime. □

Proof of the Second Descent Theorem

6.1 Outline of the Proof

In this chapter we establish our explicit recipe for 5-descent on the elliptic curves E_{τ} over Q whose 5-torsion splits as μ_{5} ⊕ Z/5Z. We begin by showing that Theorem 2, as stated in §2.4, is a consequence of the following more detailed proposition. We write ϕ : E_{τ} → E′_{τ} and ψ̂ : E_{τ} → E′′_{τ} for the isogenies with kernel μ_{5} and Z/5Z respectively.

Proposition 6.1 Let E = E_{τ} for τ ∈ Q with τ ≠ 0.

(i) There are natural inclusions

\begin{aligned}S^{(ψ̂)}(E/Q) & ⊂ & ϕS^{(5)}(E/Q) & ⊂ & S^{(ϕ̂)}(E′/Q) \\ S^{(ϕ)}(E/Q) & ⊂ & ψ̂S^{(5)}(E/Q) & ⊂ & S^{(ψ)}(E′′/Q). \end{aligned}

(ii) Let Ξ_{ϕ} = ([P, Q] [P, R]) and Ξ_{ψ} = (\begin{matrix}[P, R] \\ [Q, R] \end{matrix}). Then

S^{(ψ̂)}(E/Q) ≅ ker_{R}(Ξ_{ψ}),  S^{(ϕ̂)}(E′/Q) ≅ ker_{R}(Ξ_{ϕ})
S^{(ϕ)}(E/Q) ≅ ker_{L}(Ξ_{ϕ}),  S^{(ψ)}(E′′/Q) ≅ ker_{L}(Ξ_{ψ}).

(iii) The image of the map ϕ : S^{(5)}(E/Q) → S^{(ϕ̂)}(E′/Q) is the kernel of Ψ_{R} restricted to S^{(ϕ̂)}(E′/Q) where

Ψ_{R} = (\begin{matrix}[Q, Q] − [Q, Q]^{T} & [Q, R] \\ −[Q, R]^{T} & 0 \end{matrix}) .

78 CHAPTER 6. PROOF OF THE SECOND DESCENT THEOREM

(iv) The image of the map ψ̂: S^{(5)}(E/Q) → S^{(ψ)}(E′′/Q) is the kernel of Ψ_{L} restricted to S^{(ψ)}(E′′/Q) where

Ψ_{L} = (\begin{matrix}0 & [P, Q] \\ −[P, Q]^{T} & [Q, Q] − [Q, Q]^{T} \end{matrix} ) .

We make a few remarks about the proof of Proposition 6.1.

(i) This is a consequence of the assumption that E has split torsion. It is easily proved by examining the first diagram of §6.3. (ii) This is merely a restatement of Theorem 1 for the values λ = τ^{5} and λ = τf(τ)/g(τ). See §2.3 for details. (iii), (iv) We must compute the Cassels-Tate pairing on the Selmer groups S^{(ϕ̂)}(E′/Q) and S^{(ψ)}(E′′/Q). This is the main content of the above proposition, and will occupy us for most of this chapter.

The work of Beaver [Be] gives a proof of Proposition 6.1(iv) in the case ord_{5}(τ) > 0. We shall essentially follow her proof, making the modifications necessary to remove the condition ord_{5}(τ) > 0.

We now explain how Theorem 2 follows from Proposition 6.1. For this we need the following linear algebra lemma.

Lemma 6.2 Let V be a finite dimensional vector space over a field F. Let V = U⊕W and write pr : V → W for the projection map. Let Ψ : V ×V → F be either a symmetric form or a skew-symmetric form, and suppose that Ψ is isotropic on U, i.e. Ψ(u, u′) = 0 for all u, u′ ∈ U. Then

pr(V^{⊥}) = W_{1} ∩ W_{1}^{⊥}

where W_{1} = W ∩ U^{⊥}.

Proof. Let v = u + w ∈ V^{⊥} with u ∈ U and w ∈ W. Then w ∈ W_{1} since Ψ is isotropic on U, and w ∈ W_{1}^{⊥} since u, v ∈ W_{1}^{⊥}. Thus pr(V^{⊥}) ⊂ W_{1} ∩ W_{1}^{⊥}.

Conversely, given w ∈ W_{1} ∩ W_{1}^{⊥} we seek u ∈ U such that u + w ∈ V^{⊥}. But for any choice of u, we have u + w ∈ U^{⊥}, since Ψ is isotropic on U and w ∈ W_{1}. Thus it suffices to find u ∈ U such that u + w ∈ W^{⊥}. We write α : V → W^{∗} for the linear map induced by Ψ. We must show α(w) ∈ α(U). By counting dimensions we have α(U) = (W/W_{1})^{∗}. Finally the assumption w ∈ W_{1}^{⊥} implies α(w) is trivial on W_{1} and we are done. □

Proof of Theorem 2. Taking Galois cohomology of

0 → E[ϕ] → E[5] →^{ϕ} E′[ϕ̂] → 0

and restricting to Selmer groups we obtain an exact sequence

0 → S^{(ϕ)}(E/Q) → S^{(5)}(E/Q) →^{ϕ} S^{(ϕ̂)}(E′/Q).  (65)

We apply Lemma 6.2 with V = [P ∪ Q ∪ R], U = [P], W = [Q ∪ R] and Ψ the pairing of Theorem 2. Proposition 6.1 allows us to identify (65) with the exact sequence

0 ⟶ U ∩ V^{⊥} ⟶ V^{⊥} ⟶^{pr} W_{1}.

Indeed, Proposition 6.1(ii) gives S^{(ϕ)}(E/Q) = U ∩V^{⊥} and S^{(ϕ̂)}(E′/Q) = W_{1}. Then by Proposition 6.1(iii) and Lemma 6.2 we have

ϕS^{(5)}(E/Q) = ker( Ψ | W_{1} × W_{1}) = W_{1} ∩ W_{1}^{⊥} = pr(V^{⊥}).

It follows that S^{(5)}(E/Q) = V^{⊥} as required. □

Remark 6.3 Theorem 2 may also be deduced from the exact sequence

0 → S^{(ψ̂)}(E/Q) → S^{(5)}(E/Q) \underrightarrow{ψ̂} S^{(ψ)}(E′′/Q)  (66)

using Proposition 6.1(iv). In this case we apply Lemma 6.2 with U = [R] and W = [P ∪ Q].

6.2 Definition of the Cassels-Tate Pairing

We briefly recall the definition of the Cassels-Tate pairing. The following treatment is taken from McCallum [Mc]. See also [Mi].

Let K be a number field and let ϕ : C → D be an n-isogeny of elliptic curves over K. For any element (∗) we write (∗)_{v} for the corresponding local element. We consider the following commutative diagram, both in its own right and with K replaced by K_{v} for each place v.

C(K) ⟶^{ϕ} D(K) ⟶^{δ_{ϕ}} H^{1}(K, C[ϕ])
|| ↓ϕ̂ ↓^{ι}
C(K) ⟶^{n} C(K) ⟶^{δ_{n}} H^{1}(K, C[n])
↓^{ϕ}  ||  ↓^{ϕ}
D(K) \underrightarrow{ϕ̂} C(K) ⟶^{δϕ̂} H^{1}(K, D[ϕ̂])

To define the Cassels-Tate pairing S^{(ϕ̂)}(D/K) × S^{(ϕ̂)}(D/K) → Z/nZ we take a, b ∈ S^{(ϕ̂)}(D/K) ⊂ H^{1}(K, D[ϕ̂]) and suppose given a_{1} ∈ H^{1}(K, C[n]) lifting a. At each place v we choose a_{v,1} ∈ im δ_{n} such that ϕ(a_{v,1}) = a_{v}. Then ϕ(a_{v,1}−a_{1,v}) = 0 and so there exists c_{v} ∈ H^{1}(K_{v}, C[ϕ]) with a_{v,1}−a_{1,v} = ι(c_{v}). We now define

⟨a, b⟩ = ∑_{v}(c_{v}, b_{v})_{v}

where (·, ·)_{v} is the Tate pairing (22). It is easy to check that the definition is independent of all choices, and that ⟨a, b⟩ = 0 whenever a is in the image of ϕ : S^{(n)}(C/K) → S^{(ϕ̂)}(D/K). It is well known the Cassels-Tate pairing is alternating, and its kernel is precisely the image just described. This leads to the definition of an alternating pairing on � .

We should point out that from the computational point of view, the difficult part in the above construction is the lifting of a to a_{1}. In general the existence of such a lifting depends on the local triviality of the torsor described by a.

6.3 Split Torsion and Local Pairings

Let E/K to be an elliptic curve whose n-torsion splits as E[n] = ⟨P ⟩ ⊕ ⟨Q⟩. Let ϕ : E → E′ and ψ̂ : E → E′′ be the isogenies defined over K whose kernels are generated by P and Q respectively. We make the identifications

E[ϕ] = ⟨P ⟩ = E[n]/⟨Q⟩ = E′′[ψ]
E[ψ̂] = ⟨Q⟩ = E[n]/⟨P ⟩ = E′[ϕ̂]

and so obtain a commutative diagram with exact diagonals

H^{1}(K, E[ϕ])  =  H^{1}(K, E′′[ψ])
ι_{P} ↘  ↗ ψ̂
H^{1}(K, E[n])
ι_{Q} ↗  ↘ ^{ϕ}
H^{1}(K, E[ψ̂])  =  H^{1}(K, E′[ϕ̂]).

The key point is that ι_{Q} is a splitting of the map ϕ, and so may be used to help compute the Cassels-Tate pairing on S^{(ϕ̂)}(E′/K). Indeed, as McCallum [Mc] explains we may write the pairing as a sum of local pairings

⟨a, b⟩ = ∑_{v}⟨a_{v}, b_{v}⟩_{v}.  (67)

Here we set a_{1,v} = ι_{Q}(a_{v}) and choose a_{v,1} ∈ im δ_{n} with ϕ(a_{v,1}) = a_{v}. Then the local pairing is given by ⟨a_{v}, b_{v}⟩_{v} = (c_{v}, b_{v})_{v} where ι_{P} (c_{v}) = a_{v,1} − a_{1,v} and (·, ·)_{v} is the Tate pairing.

We shall give a criterion for the local pairing ⟨·, ·⟩_{v} to be trivial. At each place v we have commutative diagrams

E(K_{v}) ⟶^{ϕ} E′(K_{v}) ⟶^{δ_{ϕ}} H^{1}(K_{v}, E[ϕ])
↓ψ̂  ↓ϕ̂  ||
E′′(K_{v}) ⟶^{ψ} E(K_{v}) ⟶^{δ_{ψ}} H^{1}(K_{v}, E′′[ψ])

and

E(K_{v}) \underrightarrow{ψ̂} E′′(K_{v}) ⟶^{δψ̂} H^{1}(K_{v}, E[ψ̂])
↓^{ϕ}  ↓^{ψ}  ||
E′(K_{v}) \underrightarrow{ϕ̂} E(K_{v}) ⟶^{δϕ̂} H^{1}(K_{v}, E′[ϕ̂])

It is clear from the diagrams that im δ_{ϕ} ⊂ im δ_{ψ} and im δψ̂ ⊂ im δϕ̂. Tate local duality, specifically Proposition 1.17, tells us

im δ_{ϕ} = im δ_{ψ} ⟺ im δψ̂ = im δϕ̂.

We call v a switching place if these equalities do not hold, and write S for the set of switching places. It is easy to see that S is a subset of the usual set of bad places, i.e. the infinite places, the primes of bad reduction for E and the primes dividing n. Indeed at the good primes it is well known [Si1] that the image of each map δ consists precisely of the unramified cocycles.

Lemma 6.4 If v ∉ S then the local pairing ⟨·, ·⟩_{v} vanishes.

Proof. With the above notation describing the local pairing we have

ι_{P} (c_{v}) = a_{v,1} − a_{1,v}
⟹  c_{v} = ψ̂(a_{v,1})  since ψ̂ι_{P} = 1 and ψ̂ι_{Q} = 0
⟹  c_{v} ∈ im δ_{ψ}  since δ_{ψ} = ψ̂δ_{n}.

We are assuming im δ_{ϕ} = im δ_{ψ}. So we have c_{v} ∈ im δ_{ϕ} and b_{v} ∈ im δϕ̂. Tate local duality now gives ⟨a_{v}, b_{v}⟩_{v} = (c_{v}, b_{v})_{v} = 0. □

We explain the method due to Beaver [Be] that allows us to compute the local pairing in many cases. From now on we assume n is prime. For the next three lemmas we fix a place v of K and suppose P, Q ∈ E(K_{v}). Then E[n] ⊂ E(K_{v}) and H^{1}(K_{v}, E[n]) = Hom(G_{v}, E[n]).

Definition 6.5 For T ∈ E[n] a point of order n we write

δ_{T} : E(K_{v}) → Hom(G_{v}, E[n]/⟨T ⟩)

for the connecting homomorphism corresponding to the dual of the isogeny E → E/⟨T ⟩. Equivalently δ_{T} is the composite

E(K_{v}) ⟶^{δ_{n}} Hom(G_{v}, E[n])⟶ Hom(G_{v}, E[n]/⟨T ⟩).

Lemma 6.6 There is at most one subgroup ⟨T ⟩ ⊂ E[n] with δ_{T} = 0.

Proof. If δ_{T} = 0 then im δ_{n} ⊂ Hom(G_{v}, ⟨T ⟩). But im δ_{n} is maximal isotropic in Hom(G_{v}, E[n]), so by counting we have im δ_{n} = Hom(G_{v}, ⟨T ⟩) and the subgroup ⟨T ⟩ ⊂ E[n] is uniquely determined. □

Remark 6.7 We find there are examples where there is no subgroup ⟨T ⟩ ⊂ E[n] with δ_{T} = 0. However, for E with split multiplicative reduction, Beaver [Be] uses the Tate parametrisation to show that such a subgroup must exist.

Lemma 6.8 Let v ∈ S be a switching place. Suppose P, Q ∈ E(K_{v}) and that there exists T ∈ E[n] with δ_{T} = 0. Then the local pairing ⟨·, ·⟩_{v} may be computed via the formula

⟨a_{v}, b_{v}⟩_{v} = (−ρ(a_{v}), b_{v})_{v}

where ρ : Hom(G_{v}, ⟨Q⟩) → Hom(G_{v}, ⟨P ⟩) is projection along ⟨T ⟩.

Proof. Since v ∈ S we know δ_{P} = δϕ̂ ≠ 0 and δ_{Q} = δ_{ψ} ≠ 0, so the map ρ is well defined. Now ⟨a_{v}, b_{v}⟩_{v} = (c_{v}, b_{v})_{v} with a_{v} ∈ Hom(G_{v}, ⟨Q⟩) and c_{v} ∈ Hom(G_{v}, ⟨P ⟩). But a_{v} and −c_{v} differ by a_{v,1} ∈ im δ_{n} = Hom(G_{v}, ⟨T ⟩) so the stated result is immediate. □

We write

\begin{aligned}π_{P} : Hom(G_{v}, ⟨Q⟩) & ⟶^{∼} & K_{v}^{×}/(K_{v}^{×})^{n} \\ π_{Q} : Hom(G_{v}, ⟨P ⟩) & ⟶^{∼} & K_{v}^{×}/(K_{v}^{×})^{n} \end{aligned}  (68)

for the maps induced by e_{n}(P, ·) and e_{n}(Q, ·). The Tate pairing and norm residue symbol are related as follows.

Lemma 6.9 For x ∈ Hom(G_{v}, ⟨Q⟩) and y ∈ Hom(G_{v}, ⟨P ⟩) we have

e_{n}(P, Q)^{(x,y)_{v}} = (π_{P} (x), π_{Q}(y))_{v}

where (·, ·)_{v} denotes the Tate pairing, respectively the norm residue symbol.

A convenient way to write this result is (x, y)_{v} = Ind_{ζ}(π_{P} (x), π_{Q}(y))_{v} where ζ = e_{n}(P, Q) and Ind_{ζ} : μ_{n} ⟶^{∼} Z/nZ ; ζ ↦ 1. We are now ready to give Beaver’s description of the Cassels-Tate pairing.

Proposition 6.10 Let n be a prime and let E/K be an elliptic curve whose n-torsion splits as E[n] = ⟨P ⟩ ⊕ ⟨Q⟩. Suppose that every switching place v ∈ S satisfies the hypotheses of Lemma 6.8, say with T = i_{v}P + Q. Let ζ = e_{n}(P, Q) and write (·, ·)_{v} for the norm residue symbol.

(i) The Cassels-Tate pairing on S^{(ϕ̂)}(E′/K) is given by

⟨a, b⟩ = ∑_{v∈S} Ind_{ζ}(π_{P} (a_{v}), π_{P} (b_{v}))_{v}^{i_{v}}.

(ii) The Cassels-Tate pairing on S^{(ψ)}(E′′/K) is given by

⟨a, b⟩ = ∑_{v∈S} Ind_{ζ}(π_{Q}(a_{v}), π_{Q}(b_{v}))_{v}^{−1/i_{v}}.

Proof. It suffices to prove (i), since (ii) follows by symmetry P ↔ Q. For a, b ∈ S^{(ϕ̂)}(E′/K) we have

⟨a, b⟩ = ∑_{v∈S}⟨a_{v}, b_{v}⟩_{v}  by Lemma 6.4
= ∑_{v∈S}(−ρ(a_{v}), b_{v})_{v}  by Lemma 6.8
= ∑_{v∈S} Ind_{ζ}(π_{Q}(ρ(a_{v})), π_{P} (b_{v}))_{v}  by Lemma 6.9
= ∑_{v∈S} Ind_{ζ}(π_{P} (a_{v}), π_{P} (b_{v}))_{v}^{i_{v}}

where the last equality follows from ρ(Q) = −i_{v}P. □

6.4 Computation of the Cassels-Tate Pairing

We apply Proposition 6.10 in the case K = Q and n = 5. Since n is odd we may ignore the infinite places. We have E[5] = ⟨P ⟩ ⊕ ⟨Q⟩ and further assume ⟨P ⟩ ≅ μ_{5} and ⟨Q⟩ ≅ Z/5Z as Galois modules. Then E ≅ E_{τ} for some τ ∈ Q with τ ≠ 0 and we recall from §2.3 and §6.1 our short dictionary

\begin{matrix}ϕ : E → E′ & ↔ & ϕ : C_{τ^{5}} → D_{τ^{5}} \\ ψ : E′′ → E & ↔ & ϕ : C_{τf(τ)/g(τ)} → D_{τf(τ)/g(τ)}. \end{matrix}

The images of the connecting maps δ_{ϕ} and δ_{ψ} are given by Proposition 3.19 on taking λ = τ^{5} and λ = τf(τ)/g(τ). By Lemma 2.12 the set of switching primes S is precisely the set of primes Q. In particular 5 is never a switching prime. Our next task is to show that every prime p ∈ Q satisfies the hypotheses of Lemma 6.8 and to compute the switching exponent i_{p}.

Let p be a prime with p ≡ 1 (mod 5) and fix an embedding Q(μ_{5}) ↪ Q_{p}. As explained in §6.3 we seek ⟨T ⟩ ⊂ E[n] with δ_{T} = 0. We find

\begin{aligned}⟨T ⟩ = ⟨P ⟩ & ⟺ & δϕ̂ = 0 \\ & ⟺ & p ∈ P \\ & ⟺ & τ ≡ 0 or ∞ (mod p) \end{aligned}  (69)

and

\begin{aligned}⟨T ⟩ = ⟨Q⟩ & ⟺ & δ_{ψ} = 0 \\ & ⟺ & p ∈ R \\ & ⟺ & τ ≡ ϕ or \overline{ϕ} (mod p). \end{aligned}  (70)

In §4.2 we saw that the action (16) of PSL_{2}(Z/5Z) on X(5) is generated by

(\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) : τ ↦ ζ^{−1}τ and (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) : τ ↦ (ϕτ +1)/(τ −ϕ) where ζ = e_{5}(P, Q).

From either (69) or (70) we deduce

⟨T ⟩ = ⟨i_{p}P + Q⟩ ⟺ τ ≡ ζ^{i_{p}}ϕ or ζ^{i_{p}} \overline{ϕ} (mod p).

This description makes it easy to compute the switching exponent i_{p}. Since we fixed an embedding Q(μ_{5}) ↪ Q_{p}, we have really computed i_{𝔭} where 𝔭|p is the corresponding prime of Q(μ_{5}). Writing ψ : Gal(Q(μ_{5})/Q) ⟶^{∼} (Z/5Z)^{×} for the cyclotomic character, we find

i_{σ𝔭} = ψ(σ)i_{𝔭} for all σ ∈ Gal(Q(μ_{5})/Q).

Proof of Proposition 6.1(iii), (iv). Let K = Q(μ_{5}) and recall our notation (58) for the restriction maps. For each rational prime p we choose 𝔭 a prime of K above p. It is to be understood that all our formulae are independent of these choices. The Tate pairing

(·, ·)_{p} : H^{1}(Q_{p}, μ_{5}) × H^{1}(Q_{p}, Z/5Z) → Z/5Z

and the Hilbert norm residue symbol

(·, ·)_{𝔭} : K_{𝔭}^{×}/(K_{𝔭}^{×})^{5} × K_{𝔭}^{×}/(K_{𝔭}^{×})^{5} → μ_{5}

are related by

(a, b)_{p} = [K_{𝔭} : Q_{p}]^{−1} Ind_{ζ}(a, πb)_{𝔭}.  (71)

We recall from §3.4 our notation F^{(p)} for the degree n cyclic extension of Q unramified outside p. For p ∈ Q ∪ R we define

χ_{p} : (Z/pZ)^{×} → Z/5Z ; a ↦ (a, ξ^{(p)})_{p}

where ξ^{(p)} ∈ H^{1}(Q, Z/5Z) = Hom(G_{Q}, Z/5Z) is some choice of generator for Gal(F^{(p)}/Q). For p ∈ Q we make our choice such that

ord_{𝔭}(πξ^{(p)}) = −1/i_{𝔭}.  (72)

This is possible since im π ⊂ (K^{×}/(K^{×})^{5})^{ψ} and πξ^{(p)} is a product of primes above p. Since ξ^{(p)} ∉ H_{nr}^{1}(Q_{p}, μ_{5}) it is clear that the characters χ_{p} are nontrivial. For p ∈ Q they are given by

χ_{p}(a) = (a, ξ^{(p)})_{p} = Ind_{ζ}(a, πξ^{(p)})_{𝔭} = Ind_{ζ}(a/𝔭)^{−1/i_{𝔭}}.  (73)

Euler’s criterion asserts (a/𝔭) ≡ a^{(p−1)/5} (mod 𝔭). Thus the characters χ_{p} for p ∈ Q are compatible in the sense of Remark 2.16.

By the work of §3.4 the Selmer groups attached to ϕ, ψ and their duals correspond to the left and right kernels of the pairings

\begin{matrix}Ξ_{ϕ} : [P] × ⟨Q ∪ R⟩ → Z/5Z ; & (a, b) ↦ ∑_{p∈Q∪R}(a, b)_{p} \\ Ξ_{ψ} : [P ∪ Q] × ⟨R⟩ → Z/5Z ; & (a, b) ↦ ∑_{p∈R}(a, b)_{p}. \end{matrix}

Proposition 6.10 describes the Cassels-Tate pairing on the Selmer groups S^{(ϕ̂)}(E′/Q) ≅ ker_{R}(Ξ_{ϕ}) and S^{(ψ)}(E′′/Q) ≅ ker_{L}(Ξ_{ψ}) as the restrictions of the pairings

Ψ_{R} : ⟨Q ∪ R⟩ × ⟨Q ∪ R⟩ → Z/5Z ; (b_{1}, b_{2}) ↦ ∑_{p∈Q} Ind_{ζ}(πb_{1}, πb_{2})_{𝔭}^{i_{𝔭}}
Ψ_{L} : [P ∪ Q] × [P ∪ Q] → Z/5Z ; (a_{1}, a_{2}) ↦ ∑_{p∈Q} Ind_{ζ}(a_{1}, a_{2})_{𝔭}^{−1/i_{𝔭}}.

To complete our proof it remains to identify [Q ∪ R] with ⟨Q ∪ R⟩ via p ↦ ξ^{(p)} and to check that the above pairings are represented by the matrices appearing in the statement of the proposition. This is immediate for the pairings Ξ_{ϕ} and Ξ_{ψ}.

86 CHAPTER 6. PROOF OF THE SECOND DESCENT THEOREM For Ψ_{L} we take p ∈ P and q ∈ Q and compute

Ψ_{L}(p, q) = Ind_{ζ}(p, q)_{𝔮}^{−1/i_{𝔮}} by definition
= Ind_{ζ}(p, πξ^{(q)})_{𝔮} by (72)
= χ_{q}(p)  by (73).

For Ψ_{R} we take q ∈ Q and r ∈ R and compute

Ψ_{R}(ξ^{(q)}, ξ^{(r)}) = Ind_{ζ}(πξ^{(q)}, πξ^{(r)})_{𝔮}^{i_{𝔮}}  by definition
= Ind_{ζ}(πξ^{(r)}, q)_{𝔮}  by (72)
= [K_{𝔮} : Q_{q}]^{−1} Ind_{ζ}(πξ^{(r)}, q)_{𝔮} since q ≡ 1 (mod 5)
= [K_{𝔯} : Q_{r}]^{−1} Ind_{ζ}(q, πξ^{(r)})_{𝔯} by the product formula
= χ_{r}(q)  by (71).

Similar calculations for q, q′ ∈ Q with q ≠ q′ give

Ψ_{L}(q, q′) = Ψ_{R}(ξ^{(q)}, ξ^{(q′)}) = χ_{q′}(q) − χ_{q}(q′).

This completes the proof of Proposition 6.1 and so of Theorem 2. □

Application to the Curves of Conductor 11

The elliptic curves over Q with conductor 11 are

\begin{matrix}11A1 & y^{2} + y = x^{3} − x^{2} − 10x − 20 & A_{0} & E_{1} = C_{1} = D_{11} \\ 11A2 & y^{2} + y = x^{3} − x^{2} − 7820x − 263580 & A_{2} & E′′_{1} = C_{11} \\ 11A3 & y^{2} + y = x^{3} − x^{2} & A_{1} & E′_{1} = D_{1}. \end{matrix}

Here the labels 11A1-3 are the Cremona labels [Cr] and A_{0}, A_{1}, A_{2} are the labels used in [CS] chosen such that A_{i} = X_{i}(11) for i = 0, 1. We sometimes write A to denote any one of these curves. For these primeval elliptic curves we ask ourselves the following question.

How does the Mordell-Weil rank of these curves behave as we pass up the tower of fields given by adjoining the 5-power division points?

We write K = Q(μ_{5}) and L = Q(A_{1}[5]). In Example 5.6 we used Theorem 3 to show rank A(K) = 0. In this chapter we show rank A(L) = 0.

7.1 The Field L = Q(A_{1}[5])

let ζ be a primitive 5th root of unity, and let ϕ = 1 + ζ + ζ^{4} be the golden ratio with conjugate \overline{ϕ}. We recall that there is an involution

η : X_{1}(5) → X_{1}(5) ; λ ↦ (ϕ^{5}λ + 1)/(λ − ϕ^{5})

swapping over the rational and irrational cusps. Since A_{1} ≅ C_{η(1)} over K, it follows by Remark 1.8 that η(1) is a Kummer generator for L/K. In other words L = K(ζ, α) with α^{5} = η(1) = (1 + ϕ^{5})/(1 − ϕ^{5}).

Manifestly L/Q in unramified outside 5 and 11. We describe the situation a little more precisely. Let 𝔭_{1}, 𝔭_{2}, 𝔭_{3}, 𝔭_{4} be the primes of K above 11 with 𝔭_{i} = (π_{i}) and π_{i} = 2 + ζ^{i}. Since (η(1)) = 𝔭_{1}^{−1}𝔭_{2}𝔭_{3}𝔭_{4}^{−1} the primes above 11 ramify in L/K, i.e.

(11) = 𝔓_{1}^{5}𝔓_{2}^{5}𝔓_{3}^{5}𝔓_{4}^{5}

with 𝔓_{i}|𝔭_{i}. Let 𝔩 be the prime of K above 5. Since η(1) = \overline{ϕ^{5}}(1 + z) with ord_{𝔩}(z) = ord_{𝔩}(ϕ^{5} − \overline{ϕ^{5}}) = 6 we may define

α_{0} := \overline{ϕ}(1 + \binom{1/5}{1}z + \binom{1/5}{2}z^{2} + …) ∈ K_{𝔩}

such that α_{0}^{5} = η(1). Thus 𝔩 splits in L/K, i.e.

(5) = 𝔏_{0}^{4}𝔏_{1}^{4}𝔏_{2}^{4}𝔏_{3}^{4}𝔏_{4}^{4}

with α ≡ ζ^{i}\overline{ϕ} (mod 𝔏_{i}^{2}).

We compute the units and class group using pari. Thus strictly speaking the following lemma has only be checked modulo the Generalised Riemann Hypothesis (GRH).

Lemma 7.1 Let L = Q(μ_{5}, α) with α^{5} = (1 + ϕ^{5})/(1 − ϕ^{5}).

(i) The unit group of L is generated by ζ, ϕ, (ϕα+1)/(α−ϕ), (α+1)/(α−1) and their Galois conjugates. (ii) The class group of L is isomorphic to Z/5Z as a Gal(L/Q)-module, and is generated by the primes above 11.

Let us write u_{i} = (ϕζ^{i}α + 1)/(ζ^{i}α − ϕ) and v_{i} = (ζ^{i}α + 1)/(ζ^{i}α − 1). We check by hand that these are indeed units. The identity of Lemma 2.10 gives

u_{i}f(u_{i})/g(u_{i}) = η(ε(u_{i})^{5}) = η(α^{5}) = 1

and so the u_{i} have minimal polynomial

xf(x) − g(x) = x^{5} + 2x^{4} + 6x^{3} − 2x^{2} + 4x − 1.

But also

(v_{i} + 1)^{5}/(v_{i} − 1)^{5} = α^{5} = (1 + ϕ^{5})/(1 − ϕ^{5}).

7.2. ESTIMATING RANK A(L) VIA FIRST DESCENT 89

So the v_{i} are the roots of

ϕ^{5}(x^{5} + 10x^{3} + 5x) − (5x^{4} + 10x^{2} + 1) = 0.

Lemma 7.1(i) is equivalent to the statement ϕ, u_{1}, u_{2}, u_{3}, u_{4}, v_{1}, v_{2}, v_{3}, v_{4} is a set of fundamental units. It is an easy exercise to describe the action of Gal(L/Q) on these units.

Let M be the composite of K and Q(μ_{11}) ∩ R. Then L/K and M/K are both Kummer extensions of degree 5, ramified only at the primes above 11. It follows LM/L is unramified at all places. By properties of the Hilbert class field, this confirms that 5 divides the class number of L.

For our descent calculations we must consider the spaces

V_{∅} = { θ ∈ L^{×}/(L^{×})^{5} | ord_{𝔓}(θ) ≡ 0 (mod 5) for all 𝔓 }
= ⟨ζ, ϕ, u_{1}, u_{2}, u_{3}, u_{4}, v_{1}, v_{2}, v_{3}, v_{4}, w⟩

where w = 1+ϕ^{5} ≡ 1−ϕ^{5} mod (L^{×})^{5} is a contribution from the class group, and

V_{11} = { θ ∈ L^{×}/(L^{×})^{5} | ord_{𝔓}(θ) ≡ 0 (mod 5) for all 𝔓 ∤ 11 }
= ⟨ζ, ϕ, u_{1}, u_{2}, u_{3}, u_{4}, v_{1}, v_{2}, v_{3}, v_{4}, w, α, x, x′⟩

where

x = \frac{(ζ^{4}+ζ^{2}−ζ)(α^{2}−α)−1}{(2ϕ−1)(ζα−1)} and x′ = \frac{(ζ^{3}+ζ^{4}−ζ^{2})(1+α)−α^{2}}{(2ϕ−1)α(ζ^{3}α+1)}

have been chosen such that (x) = 𝔓_{1}𝔓_{4}^{−1} and (x′) = 𝔓_{2}𝔓_{3}^{−1}.

7.2 Estimating rank A(L) via First Descent

We begin with an elementary observation.

Lemma 7.2 rank A(L) is a multiple of 4.

Proof. Let ρ be a generator for Gal(L/K). Then the minimal polynomial for ρ acting on A(L) ⊗ Q divides x^{5} − 1. But rank A(K) = 0 and the cyclotomic polynomial x^{4} + x^{3} + x^{2} + x + 1 is irreducible over Q. So ρ has characteristic polynomial a power of x^{4} + x^{3} + x^{2} + x + 1 and the lemma follows. □

Relabelling torsion gives an action of PSL_{2}(Z/5Z) on X(5) ≅ P^{1}. This includes the maps τ ↦ −1/τ and τ ↦ ζτ. So to describe an orbit it suffices to list the values λ = τ^{5} and ignore repeats of the form λ ↔ −1/λ. We compute the orbits corresponding to A_{0} and A_{1}. They are

{1, −ϕ^{15}, π_{1}^{5}, π_{2}^{5}, π_{3}^{5}, π_{4}^{5}}  and  {η(1), u_{0}^{5}, u_{1}^{5}, u_{2}^{5}, u_{3}^{5}, u_{4}^{5}}.

Thus A_{0} is 5-isogenous to A_{1}, A_{2} and the 4 conjugates of A_{3} := D_{π_{1}^{5}}. Similarly A_{1} is 5-isogenous to A_{0} and the 5 conjugates of A_{4} := D_{u_{0}^{5}}. The aim of this section is to prove that the Selmer groups over L, attached to these isogenies and their duals, have the dimensions listed as follows.

\begin{matrix}ϕ : C_{λ} → D_{λ} & λ & η(λ) & c_{i}(λ) & dim_{5} S(ϕ) & dim_{5} S(ϕ̂) \\ A_{0} → A_{1} & 1 & α^{5} & 0 & 8 & 2 \\ A_{0} → A_{2} & −ϕ^{15} & 11 & 6 & 0 & 14 \\ A_{0} → A_{3} & π_{1}^{5} & −ζ^{2}ϕ^{3}π_{2}π_{3}^{−1}π_{4} & 6 & 0 & 12 \\ A_{1} → A_{4} & u_{0}^{5} & u_{1}u_{2}^{−1}u_{3}^{−1}u_{4}α & 0/6 & 0 & 10 \end{matrix}

For λ ∈ L not a cusp of X_{1}(n) we write

A = { 𝔓 prime | ord_{𝔓}(λ) ≠ 0 }
B = { 𝔓 prime | ord_{𝔓}(η(λ)) ≠ 0 }.

We recall that in §1.1 and §5.1 we defined quantities c(λ) and d(λ) describing the behaviour of λ at the prime 𝔩. We now define c_{i}(λ) and d_{i}(λ) to be the corresponding quantities with 𝔩 replaced by 𝔏_{i}.

Proposition 7.3 The map δ : D_{λ}(L_{𝔓}) → L_{𝔓}^{×}/(L_{𝔓}^{×})^{5} has image

im δ = \{ \begin{matrix}L_{𝔓}^{×}/(L_{𝔓}^{×})^{n} & if 𝔓 ∈ A \\ O_{𝔓}^{×}/(O_{𝔓}^{×})^{n} & if 𝔓 ∉ A ∪ B and 𝔓 ∤ 5 \\ 0 & if 𝔓 ∈ B. \end{matrix}

If 𝔭 ∉ A ∪ B and 𝔓 = 𝔏_{i} then im δ = U_{d_{i}+1}(K_{𝔩}^{×})^{n}/(K_{𝔩}^{×})^{n}.

Proof. This is immediate from Propositions 3.12 and 5.12. The key point is that 𝔩 splits in L/K, so there is no need to perform further Hensel lemma calculations at the primes above 5. Indeed the local fields L_{𝔏_{i}} are copies of the field K_{𝔩} we have already studied. □

Our labelling of the 𝔏_{i} is such that

ord_{𝔏_{i}}(α − ϕ) = \{ \begin{matrix}1 & i ≠ 0 \\ 2 & i = 0. \end{matrix}

Since u_{0} = ε(α), an analogue of Lemma 1.5(iii) for the involution ε gives

ord_{𝔏_{i}}(u_{0} − ϕ) = \{ \begin{matrix}1 & i ≠ 0 \\ 0 & i = 0. \end{matrix}

Thus c_{i}(u_{0}^{5}) = 6 if i ≠ 0, and 0 if i = 0. As indicated in the above table, this is the only one of our examples where c_{i}(λ) depends on i. We are now ready to show that the Selmer groups listed have the dimensions claimed. We write S(C_{λ} → D_{λ}/L) instead of S^{(ϕ)}(C_{λ}/L).

(i) S(A_{0} → A_{1}/L) ≅ { θ ∈ V_{∅} | θ ∈ (L_{𝔓_{i}}^{×})^{5} for 1 ≤ i ≤ 4 } We choose characters (O_{L}/𝔓_{i})^{×} → Z/5Z by composing the maps x ↦ x^{2} and 4 ↦ 1. Computing these characters on our basis for V_{∅} we obtain an 11 × 4 matrix.

𝔓_{1} 𝔓_{2} 𝔓_{3} 𝔓_{4}
ζ 1 3 2 4
ϕ 2 3 3 2
u_{1} 2 2 2 2
u_{2} 2 2 2 2
u_{3} 2 2 2 2
u_{4} 2 2 2 2
v_{1} 0 0 0 0
v_{2} 0 0 0 0
v_{3} 0 0 0 0
v_{4} 0 0 0 0
w 1 1 1 1

This matrix has rank 3, so S(A_{0} → A_{1}/L) ≅ (Z/5Z)^{8}.

(ii) S(A_{1} → A_{0}/L) ≅ { θ ∈ V_{11} | θ^{4} ≡ 1 (mod 𝔏_{i}^{5}) for 0 ≤ i ≤ 4 } For each basis element of V_{11} we first substitute α ↦ ζ^{i}α_{0} and then compute the characters χ_{1}, χ_{2}, χ_{3}, χ_{4} defined by (50). We obtain a 14 × 20 matrix.

α ↦ α_{0} α ↦ ζα_{0} α ↦ ζ^{2}α_{0} α ↦ ζ^{3}α_{0} α ↦ ζ^{4}α_{0}
ζ 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0
ϕ 0 3 3 0 0 3 3 0 0 3 3 0 0 3 3 0 0 3 3 0
u_{1} 4 4 2 0 2 1 2 2 3 1 0 4 1 4 1 1 0 0 0 3
u_{2} 2 1 2 2 3 1 0 4 1 4 1 1 0 0 0 3 4 4 2 0
u_{3} 3 1 0 4 1 4 1 1 0 0 0 3 4 4 2 0 2 1 2 2
u_{4} 1 4 1 1 0 0 0 3 4 4 2 0 2 1 2 2 3 1 0 4
v_{1} 3 2 4 2 1 2 3 0 4 2 1 2 2 2 0 1 0 2 2 0
v_{2} 1 2 3 0 4 2 1 2 2 2 0 1 0 2 2 0 3 2 4 2
v_{3} 4 2 1 2 2 2 0 1 0 2 2 0 3 2 4 2 1 2 3 0
v_{4} 2 2 0 1 0 2 2 0 3 2 4 2 1 2 3 0 4 2 1 2
w 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1
α 0 4 4 0 1 4 4 0 2 4 4 0 3 4 4 0 4 4 4 0
x 4 4 0 3 0 1 4 0 1 0 2 3 0 4 0 2 4 1 1 0
x′ 0 1 4 1 4 0 1 4 0 4 4 0 2 4 3 4 2 1 4 0

This matrix has rank 12 so S(A_{1} → A_{0}/L) ≅ (Z/5Z)^{2}. It is spanned by ζ^{4}u_{1}u_{2}^{2}u_{3}^{3}u_{4}^{4} and ϕu_{0}^{3}u_{1}u_{4}α. (iii) By considering an 11 × 16 submatrix of the above 14 × 20 matrix we learn

{ θ ∈ V_{∅} | θ^{4} ≡ 1 (mod 𝔏_{i}^{5}) for 1 ≤ i ≤ 4 }

is spanned by ζ^{4}u_{1}u_{2}^{2}u_{3}^{3}u_{4}^{4}. Since this element is not a 5th power in L_{𝔓} for any of the primes 𝔓|11 we deduce that the Selmer groups

S(A_{0} → A_{2}/L) ≅ \{ θ ∈ V_{∅} | \begin{matrix}θ ∈ (L_{𝔓_{i}}^{×})^{5} for 1 ≤ i ≤ 4 \\ and θ^{4} ≡ 1 (mod 𝔏_{i}^{5}) for 0 ≤ i ≤ 4 \end{matrix} \}
S(A_{0} → A_{3}/L) ≅ \{ θ ∈ V_{∅} | \begin{matrix}θ ∈ (L_{𝔓_{i}}^{×})^{5} for 2 ≤ i ≤ 4 \\ and θ^{4} ≡ 1 (mod 𝔏_{i}^{5}) for 0 ≤ i ≤ 4 \end{matrix} \}
S(A_{1} → A_{4}/L) ≅ { θ ∈ V_{∅} | θ^{4} ≡ 1 (mod 𝔏_{i}^{5}) for 1 ≤ i ≤ 4 }

are all zero. Likewise we check that the Selmer groups

\begin{aligned}S(A_{2} → A_{0}/L) & ≅ & V_{11} \\ S(A_{3} → A_{0}/L) & ≅ & \{ θ ∈ V_{11} | θ ∈ (L_{𝔓_{1}}^{×})^{5} \} \\ S(A_{4} → A_{1}/L) & ≅ & \{ θ ∈ V_{11} | θ^{4} ≡ 1 (mod 𝔏_{0}^{5}) \} \end{aligned}

7.3. ESTIMATING RANK A(L) VIA SECOND DESCENT 93

have the dimensions claimed.

Our best upper bound for rank A(L) obtained by first descent is 8, computed via the 5-isogeny A_{0} → A_{1} and its dual. In the next section we improve on this by using the Cassels-Tate pairing to help perform a full 5-descent. We are in the habit of calling this a second descent.

7.3 Estimating rank A(L) via Second Descent

To apply the results of Chapter 6 we work with the curves A whose 5-torsion is defined over L, namely A_{0} and A_{1}. Proposition 6.10 gives two methods for computing S^{(5)}(A/L) and we carry out both as a check.

(a) We apply Proposition 6.10 with τ = 1 and aim to compute S^{(5)}(A_{0}/L). We have isogenies ϕ : A_{0} → A_{1} and ψ : A_{2} → A_{0} with

im (δ_{ϕ} : A_{1}(L_{𝔓}) → L_{𝔓}^{×}/(L_{𝔓}^{×})^{5}) = \{ \begin{matrix}O_{𝔓}^{×}/(O_{𝔓}^{×})^{5} & if 𝔓∤11 \\ 0 & if 𝔓|11 \end{matrix}

im (δ_{ψ} : A_{0}(L_{𝔓}) → L_{𝔓}^{×}/(L_{𝔓}^{×})^{5}) = \{ \begin{matrix}O_{𝔓}^{×}/(O_{𝔓}^{×})^{5} & if 𝔓∤11 \\ L_{𝔓}^{×}/(L_{𝔓}^{×})^{5} & if 𝔓|11. \end{matrix}

Thus the switching primes are precisely the primes above 11. Since 1 ≡ ζ^{3i}ϕ or ζ^{3i}\overline{ϕ} (mod 𝔭_{i}) the switching exponent at 𝔓_{i} is 3i.

(i) 0 → S^{(ϕ)}(A_{0}/L) → S^{(5)}(A_{0}/L) →^{ϕ} S^{(ϕ̂)}(A_{1}/L)

We already know S^{(ϕ)}(A_{0}/L) ≅ (Z/5Z)^{8} and S^{(ϕ̂)}(A_{1}/L) ≅ (Z/5Z)^{2}. Since S^{(ϕ̂)}(A_{1}/L) contains a contribution from torsion it follows the Cassels-Tate pairing on this Selmer group is trivial. We deduce S^{(5)}(A_{0}/L) ≅ (Z/5Z)^{10}.

(ii) 0 → S^{(ψ̂)}(A_{0}/L) → S^{(5)}(A_{0}/L) \underrightarrow{ψ̂} S^{(ψ)}(A_{2}/L) We already know S^{(ψ̂)}(A_{0}/L) = 0 and S^{(ψ)}(A_{2}/L) ≅ V_{11} ≅ (Z/5Z)^{14}. The Cassels-Tate pairing on S^{(ϕ)}(A_{2}/L) is given by

⟨θ, θ′⟩ = ∑_{i=1}^{4} Ind_{ζ}(θ, θ′)_{𝔓_{i}}^{1/(2i)}

where (·, ·)_{𝔓} is the norm residue symbol. We deduce that the pairing is represented by a matrix of the form

(\begin{matrix}0 & M \\ −M^{T} & ∗ \end{matrix})

where M is an 11 × 3 matrix. By relating to the 11 × 4 matrix recorded above we see M has rank 2 and so the pairing has rank 4. Again we deduce S^{(5)}(A_{0}/L) ≅ (Z/5Z)^{10}.

(b) We apply Proposition 6.10 with τ = α and aim to compute S^{(5)}(A_{1}/L). We have isogenies ϕ : A_{1} → A_{0} and ψ : A_{4} → A_{1} with

im (δ_{ϕ} : A_{0}(L_{𝔓}) → L_{𝔓}^{×}/(L_{𝔓}^{×})^{5}) = \{ \begin{matrix}L_{𝔓}^{×}/(L_{𝔓}^{×})^{5} & if 𝔓|11 \\ O_{𝔓}^{×}/(O_{𝔓}^{×})^{5} & if 𝔓∤5, 11 \\ U_{5}(K_{𝔩}^{×})^{5}/(K_{𝔩}^{×})^{5} & if 𝔓|5 \end{matrix}

im (δ_{ψ} : A_{4}(L_{𝔓}) → L_{𝔓}^{×}/(L_{𝔓}^{×})^{5}) = \{ \begin{matrix}L_{𝔓}^{×}/(L_{𝔓}^{×})^{5} & if 𝔓|11 \\ O_{𝔓}^{×}/(O_{𝔓}^{×})^{5} & if 𝔓∤11, 𝔓≠𝔏_{0} \\ U_{5}(K_{𝔩}^{×})^{5}/(K_{𝔩}^{×})^{5} & if 𝔓 = 𝔏_{0}. \end{matrix}

Thus the switching primes are 𝔏_{1}, 𝔏_{2}, 𝔏_{3}, 𝔏_{4}. Unfortunately the hypotheses of Lemma 6.8 are not satisfied. However all our Selmer groups are contained in V_{11} and so consist of elements prime to 5. Working with the norm residue symbol on U_{1}/U_{5} rather than the whole of K_{𝔩}^{×}/(K_{𝔩}^{×})^{5} it is clear that the formulae of Proposition 6.10 are still valid. Our labelling of the primes 𝔏_{i} is such that 𝔏_{i} has switching exponent i.

(i) 0 → S^{(ϕ)}(A_{1}/L) → S^{(5)}(A_{1}/L) →^{ϕ} S^{(ϕ̂)}(A_{0}/L)

We already know S^{(ϕ)}(A_{1}/L) ≅ (Z/5Z)^{2} and S^{(ϕ̂)}(A_{0}/L) ≅ (Z/5Z)^{8}. The Cassels-Tate pairing on S^{(ϕ̂)}(A_{0}/L) is given by

⟨θ, θ′⟩ = ∑_{i=1}^{4} Ind_{ζ}(θ, θ′)_{𝔏_{i}}^{i}.

We compute S^{(ϕ̂)}(A_{0}/L) using the 11 × 4 matrix, and then use the 14 × 20 matrix, together with the following numerical description [CF, Exercise 2.13] of the norm residue symbol (·, ·)_{𝔩} on U_{1}/U_{5}, to compute the Cassels-Tate pairing. Here γ = 1 − ζ.

1 − γ 1 − γ^{2} 1 − γ^{3} 1 − γ^{4}
1 − γ 1  ζ^{4}  ζ  ζ
1 − γ^{2} ζ  1  ζ^{2}  1
1 − γ^{3} ζ^{4}  ζ^{3}  1  1
1 − γ^{4} ζ^{4}  1  1  1

The Cassels-Tate pairing on S^{(ψ̂)}(A_{0}/L) is found to have rank 6. The kernel of the pairing is ϕS^{(5)}(A_{1}/L) ≅ (Z/5Z)^{2} generated by u_{1}u_{2}^{2}u_{3}^{3}u_{4}^{4} and v_{1}v_{2}^{2}v_{3}^{3}v_{4}^{4}. We deduce S^{(5)}(A_{1}/L) ≅ (Z/5Z)^{4}.

(ii) 0 → S^{(ψ̂)}(A_{1}/L) → S^{(5)}(A_{1}/L) \underrightarrow{ψ̂} S^{(ψ)}(A_{4}/L) We already know S^{(ψ̂)}(A_{1}/L) = 0 and S^{(ψ)}(A_{4}/L) ≅ (Z/5Z)^{10}. The Cassels-Tate pairing on S^{(ψ)}(A_{4}/L) is given by

⟨θ, θ′⟩ = ∑_{i=1}^{4} Ind_{ζ}(θ, θ′)_{𝔏_{i}}^{−1/i}

where (·, ·)_{𝔓} is the norm residue symbol. We compute S^{(ψ)}(A_{4}/L) using the first 4 columns of the 14 × 20 matrix, and then compute the Cassels-Tate pairing using the remaining 16 columns. The kernel of the pairing is ψ̂S^{(5)}(A_{1}/L) ≅ (Z/5Z)^{4} generated by ζ^{4}u_{1}u_{2}^{2}u_{3}^{3}u_{4}^{4}, ϕu_{0}^{3}u_{2}u_{3}, ϕ^{2}α and v_{0}v_{2}^{2}v_{3}^{2}w. Again we deduce S^{(5)}(A_{1}/L) ≅ (Z/5Z)^{4}.

Thus the estimate for rank A(L) given by second descent is 2. But by Lemma 7.2 the rank is a multiple of 4. We deduce rank A(L) = 0 and

\begin{aligned}� (A_{0}/L)[5] & ≅ & (Z/5Z)^{8} \\ � (A_{1}/L)[5] & ≅ & (Z/5Z)^{2}. \end{aligned}

It remains an open problem to determine whether the curves A have a single point of infinite order over the tower of number fields defined by their 5-power division points.

Appendix A

Varieties defined by Pfaffians

The following theorem of Buchsbaum and Eisenbud [BE] applies to projectively Gorenstein varieties over any field K. It is well known that elliptic normal curves satisfy this condition.

Theorem A.1 Let X ↪ P^{r} be projectively Gorenstein of codimension 3 and let I ◁ R = K[x_{0}, …, x_{r}] be the corresponding homogeneous ideal. Then the minimal free resolution for R/I is self-dual and takes the form

0 → R → R^{n} →^{f} R^{n} → R

for some n ≥ 3 odd. The map f determines a skew symmetric form and so may be represented by an n × n skew symmetric matrix M. The ideal I is minimally generated by the (n − 1) × (n − 1) leading Pfaffians of M.

In Chapter 4 we took K an algebraically closed field and n ≥ 5 odd. We saw that any elliptic curve E ↪ P^{n−1} is defined by quadrics, and these quadrics may be written as the 4 × 4 leading Pfaffians of an n × n matrix.

In the case n = 5, the point (a : b) on Y (5) yields a 5 × 5 matrix

M_{5} = (\begin{matrix}0 & −ax_{1} & −bx_{2} & bx_{3} & ax_{4} \\ ax_{1} & 0 & −ax_{3} & −bx_{4} & bx_{0} \\ bx_{2} & ax_{3} & 0 & −ax_{0} & −bx_{1} \\ −bx_{3} & bx_{4} & ax_{0} & 0 & −ax_{2} \\ −ax_{4} & −bx_{0} & bx_{1} & ax_{2} & 0 \end{matrix}) .

It is readily checked that this is the matrix M appearing in Theorem A.1.

In the case n = 7, the point (a : b : c) on Y (7) yields a 7 × 7 matrix

M_{7} = (\begin{matrix}0 & −ax_{1} & −bx_{2} & cx_{3} & −cx_{4} & bx_{5} & ax_{6} \\ ax_{1} & 0 & −ax_{3} & −bx_{4} & cx_{5} & −cx_{6} & bx_{0} \\ bx_{2} & ax_{3} & 0 & −ax_{5} & −bx_{6} & cx_{0} & −cx_{1} \\ −cx_{3} & bx_{4} & ax_{5} & 0 & −ax_{0} & −bx_{1} & cx_{2} \\ cx_{4} & −cx_{5} & bx_{6} & ax_{0} & 0 & −ax_{2} & −bx_{3} \\ −bx_{5} & cx_{6} & −cx_{0} & bx_{1} & ax_{2} & 0 & −ax_{4} \\ −ax_{6} & −bx_{0} & cx_{1} & −cx_{2} & bx_{3} & ax_{4} & 0 \end{matrix}) .

Let Σ ↪ P^{6} be defined by the 6×6 leading Pfaffians of M_{7}. We claim Σ is the secant variety of E. Since the sum of two rank 2 matrices has rank at most 4 it is clear Sec E ⊂ Σ. The next two lemmas show Sec E and Σ both have dimension 3 and degree 14. Since Sec E is irreducible and all components of Σ have codimension at most 3, it follows Sec E = Σ as claimed.

Lemma A.2 Let n ≥ 5 and let E ↪ P^{n−1} be an elliptic normal curve of degree n. Then Sec E ↪ P^{n−1} has dimension 3 and degree n(n − 3)/2.

Proof. We check that the degree is as stated. Projecting away from a general (n − 4)-plane, E is mapped to a plane curve E′ with d nodes. Then E′ has arithmetic genus 1 + d = (n − 1)(n − 2)/2, and Sec E meets the (n − 4)-plane in d = n(n − 3)/2 points. □

Lemma A.3 Let X ↪ P^{r} be the variety described in Theorem A.1 with M a matrix of linear forms. Then X has degree n(n^{2} − 1)/24.

Proof. We may read off the Hilbert polynomial from the minimal resolution, see e.g. [Ha, Lecture 13]. Writing n = 2k + 1,

\begin{aligned}h_{X}(t) & = & \binom{t+2k}{2k} − n\binom{t+k}{2k} + n\binom{t+k−1}{2k} − \binom{t−1}{2k} \\ & = & \frac{n(n^{2}−1)}{24(n−4)!}t^{n−4} + … \end{aligned}

Thus X has degree n(n^{2} − 1)/24. □

Remark A.4 It’s easy to see that if E is a Néron polygon of degree 7 then Σ consists of 14 3-planes.

We write Gr(k, n) for the Grassmannian of k-dimensional subspaces of an n-dimensional vector space V. For k = 2 the Plücker embedding is given by Gr(2, n) ↪ P(∧^{2}V); ⟨u, v⟩ ↦ u ∧ v.

Proposition A.5 Let n ≥ 5 odd. Let E ↪ P^{n−1} be an elliptic normal curve of degree n. Then E is (set-theoretically) a linear section of the Grassmannian Gr(2, n) ↪ P^{N} where N = \binom{n}{2} − 1.

Proof. The image of Gr(2, n) under the Plücker embedding is given by the vanishing of the 4 × 4 leading Pfaffians of a general n × n skew symmetric matrix. Thus our claim is that E is defined by the vanishing of the 4 × 4 leading Pfaffians of an n × n skew symmetric matrix M with entries linear forms on P^{n−1}. This was proved in Chapter 4. □

In the cases n = 5, 7 we may compute the matrix M by computing the minimal resolution for E, respectively Sec E. This approach has the fortunate consequence that Proposition A.5 remains true over a non-algebraically closed field. For example, we may express the equations for T [λ; τ_{0}, …, τ_{n−1}] as defined in §3.1 as the 4 × 4 leading Pfaffians of an n × n matrix M. To find M we could compute minimal resolutions as suggested above, but there is no need since we already know the answer in the case τ_{0} = τ_{1} = … = τ_{n−1} and the general case follows by rescaling our co-ordinates as described in Lemma 3.5.

We would like to use the above observations to write down further examples of smooth curves of genus 1 and degree n in P^{n−1}. This is possible in the case n = 5 since Gr(2, 5) ↪ P^{9} has dimension 6, degree 5 and canonical divisor K = −5H, so a general P^{4} slice is indeed a smooth curve of genus 1. In the case n = 7, Gr(2, 7) ↪ P^{20} has dimension 10, degree 42 and canonical divisor K = −7H. A general P^{6} does not even meet the Grassmannian. A description of the linear subspaces P^{6} that cut out a smooth curve of genus 1 is still sought.

Appendix B

Descent by n-isogeny

Let n = 5 or 7. Let M_{1} be a Galois module of order n and let M_{2} = M_{1}^{∨}. Let χ_{1} and χ_{2} be the corresponding characters G_{Q} → (Z/nZ)^{×} with χ_{1}χ_{2} the cyclotomic character. We consider pairs of n-isogenous elliptic curves ϕ : E → E′ over Q with E[ϕ] ≅ M_{1} and E′[ϕ̂] ≅ M_{2}. For fixed M_{1} and M_{2} these pairs are parametrised by some twist of Y_{1}(n). We believe it should be possible to prove an analogue of Theorem 1 for any pair of Galois modules M_{1} and M_{2} in place of μ_{n} and Z/nZ. We discuss two approaches.

Let K_{1} and K_{2} be the fixed fields of ker χ_{1} and ker χ_{2}. The restriction map gives an isomorphism H^{1}(Q, M_{1}) ⟶^{∼} (K_{2}^{×}/(K_{2}^{×})^{n})^{χ_{2}}. We explain a method for writing down equations over Q for the torsor under E described by θ ∈ (K_{2}^{×}/(K_{2}^{×})^{n})^{χ_{2}}. For simplicity we suppose n = 5 and [K_{2} : Q] = 4 with χ_{2} : σ ↦ 2. Then E ≅ C_{λ} for some λ ∈ K_{2} with σ(λ) = −1/λ. We write ε = 4+2σ +σ^{2} +3σ^{3} for the idempotent in the group ring Z/5Z[Gal(K_{2}/Q)] with σε = 2ε. Then θ = εθ = θ^{4}σ(θ)^{2}σ^{2}(θ)σ^{3}(θ)^{3}. We take V a 5-dimensional vector space over Q and choose x_{0}, …, x_{4} a basis for V ⊗K_{2} with σ(x_{i}) = x_{2i}. Then the curve

T [λ; λ\frac{σ(θ)σ^{3}(θ)}{θσ^{2}(θ)}, θ, \frac{−1}{σ(θ)}, \frac{−1}{σ^{3}(θ)}, σ^{2}(θ)] ⊂ P(V ⊗ K_{2})

is defined over Q. By construction it is the torsor under E described by θ. In the other cases, i.e. for n = 7 and for other values of [K_{2} : Q] dividing n − 1, a similar treatment is easily given.

We give an example with M_{1} the 5-twist of μ_{5}. So here χ_{1} is the inverse of the cyclotomic character and K_{2} = Q(\sqrt{5}).

Example B.1 In §5.2 we saw � (C_{11}^{∗}/Q)[5] ≅ (Z/5Z)^{2}. In particular the 102 APPENDIX B. DESCENT BY n-ISOGENY

curve T = T [11; −11, ϕ, 1, 1, \overline{ϕ}], where ϕ = (1 + \sqrt{5})/2 and \overline{ϕ} = (1 − \sqrt{5})/2, is a counter-example to the Hasse Principle. To obtain a model over Q we make the substitution

\begin{matrix}x_{0} = y_{0}, x_{1} = y_{1} + \sqrt{5}y_{2}, x_{2} = y_{3} + \sqrt{5}y_{4}, \\ x_{3} = y_{3} − \sqrt{5}y_{4}, x_{4} = y_{1} − \sqrt{5}y_{2} \end{matrix}

and obtain equations

\begin{aligned}0 & = & 11y_{0}^{2} − y_{1}^{2} + 5y_{2}^{2} + y_{3}^{2} − 5y_{4}^{2} \\ 0 & = & y_{1}^{2} + 10y_{1}y_{2} + 5y_{2}^{2} + 2y_{0}y_{3} − y_{1}y_{3} − 5y_{2}y_{3} − 5y_{1}y_{4} − 5y_{2}y_{4} \\ 0 & = & y_{1}^{2} + 2y_{1}y_{2} + 5y_{2}^{2} + y_{1}y_{3} + y_{2}y_{3} + 2y_{0}y_{4} + y_{1}y_{4} + 5y_{2}y_{4} \\ 0 & = & 11y_{0}y_{1} + 55y_{0}y_{2} + 2y_{1}y_{3} + 2y_{3}^{2} − 10y_{2}y_{4} + 10y_{4}^{2} \\ 0 & = & 11y_{0}y_{1} + 11y_{0}y_{2} − 2y_{2}y_{3} + 2y_{1}y_{4} − 4y_{3}y_{4}. \end{aligned}

As our example shows, this approach is rather unsatisfactory in that although we obtain equations over Q, these equations tend to be rather unwieldy. This may be viewed as a consequence of the many arbitrary choices made in the above construction.

A related problem is that of determining the Jacobian of a curve of genus 1 and degree n in P^{n−1}. No general procedure is known, but for n ≤ 4 the problem may be resolved by writing down invariants. See [AKMMMP] for a recent account. For n ≥ 5 we suspect that even if the corresponding invariants were found, they would be too large to write down. See [O’N] for the treatment of some special cases.

Our second approach is to work more directly over the fields K_{1} and K_{2}. Assuming we have some way of treating the primes above n, Proposition 3.12 shows how to compute S^{(ϕ)}(E/K_{2}) and S^{(ϕ̂)}(E/K_{1}). Since the field extensions K_{1}/Q and K_{2}/Q are of degree prime to n, it is then straightforward to determine the Selmer groups over Q.

We are left with the problem of resolving local solubility at the primes above n. This we know how to do for the field K = Q(μ_{n}). Unfortunately is not clear how to extend our results to a general number field. However, it is easy enough to make progress for certain specific choices of M_{1} and M_{2}. For example, if the prime above n splits in K_{1}K_{2}/K then Proposition 5.12 suffices to determine local solubility.

We give an example in the case n = 7 with M_{1} the Galois module defined by the inverse of the cyclotomic character ψ : G_{Q} → (Z/7Z)^{×}.

Let B_{ρ} be the elliptic curve with Weierstrass equation

y^{2} = x^{3} − 27c_{4}x − 54c_{6}

where

c_{4} = 7(ρ^{2} − ρ + 1)(ρ^{2} − 3ρ − 3)(3ρ^{2} − 9ρ + 5)(5ρ^{2} − ρ − 1),
c_{6} = 49(ρ^{4} − 6ρ^{3} + 17ρ^{2} − 24ρ + 9)
(3ρ^{4} − 4ρ^{3} − 5ρ^{2} − 2ρ − 1)(9ρ^{4} − 12ρ^{3} − ρ^{2} + 8ρ − 3),

and

Δ(B_{ρ}) = −343(ρ^{3} − ρ^{2} − 2ρ + 1)^{7}(ρ^{3} − 2ρ^{2} − ρ + 1).

We find B_{ρ} ≅ D_{ν(ρ)} over Q(μ_{7}) where ν is the involution of P^{1} exchanging

0, 1, ∞  ↔ roots of ρ^{3} − ρ^{2} − 2ρ + 1 = 0
roots of λ^{3} − 8λ^{2} + 5λ + 1 = 0 ↔ roots of ρ^{3} − 2ρ^{2} − ρ + 1 = 0.

Indeed this is how we constructed the family B_{ρ}. By keeping track of the torsion, we see B_{ρ} is the universal family of elliptic curves containing a labelled copy of the Galois module M_{1}. Thus B_{ρ} ≅ B_{(ρ−1)/ρ} over Q and there is a 7-isogeny ϕ : B_{ρ} → B_{1−ρ}^{∗} defined over Q. Here we recall our notation E^{∗} for the −7-twist of an elliptic curve E over Q.

We give an analogue of Theorem 1 for the isogeny ϕ : B_{ρ} → B_{1−ρ}^{∗}.

Proposition B.2 Let ρ ∈ Q and put

A = { p prime | ρ^{3} − 2ρ^{2} − ρ + 1 ≡ 0 (mod p) }
B = { p prime | ρ^{3} − ρ^{2} − 2ρ + 1 ≡ 0 (mod p) and p ≡ 1 (mod 7) }.

Then there is a pairing Ξ on Z/7Z-vector spaces of dimensions |A| and |B| such that S^{(ϕ)}(B_{ρ}/Q) ≅ ker_{L}(Ξ) and S^{(ϕ̂)}(B_{1−ρ}^{∗}/Q) ≅ ker_{R}(Ξ).

Proof. Apply Theorem 3 with λ = η(ν(ρ)) and consider the action of Gal(K/Q) on the Selmer groups. □

Remark B.3 The proposition gives |A|+|B|−2 rank(Ξ) as an upper bound for the rank. A computation of reduction types shows that the places with root number −1 are (AΔ{7}) ∪ B ∪ {∞} where Δ denotes symmetric difference. As in §2.2 we recover a parity result.

It is hoped that by studying further examples of Galois modules M_{1} and M_{2} we might generalise Theorem 1 to any pair of n-isogenous elliptic curves. Such curves are parametrised by X_{0}(n) ≅ P^{1} for n = 5, 7 and 13. This raises the hope of performing yet higher descents.

Appendix C

Explicit covering maps

Let n = 5 or 7. We recall from §1.2 our notation C_{λ,θ} for certain twists of C_{λ}. The isogeny ϕ : C_{λ} → D_{λ} induces a map ϕ : C_{λ,θ} → D_{λ}. After all C_{λ,θ} is described by a cocycle taking values in μ_{n} ↪ C_{λ} and ϕ quotients out by this action. We obtain a commutative diagram

\begin{matrix}C_{λ,θ} \\ ≅ & ↘ \\ C_{λ} & ⟶ & D_{λ} \end{matrix}

where both degree n maps ϕ are defined over K. If we give equations for C_{λ,θ} via push-out, then the map ϕ : C_{λ,θ} → D_{λ} is trivial to write down. In contrast, if we use the projective method then describing the map ϕ is somewhat more involved. We arrive at some equations by following the proof of [V2, Théorème 9.13]. Case n = 5. Let λ, τ_{0}, …, τ_{4} ∈ K with λ = ∏ τ_{i}. We recall that the curve T = T [λ; τ_{0}, …, τ_{4}] lies on the quadrics

\begin{aligned}A & = & τ_{0}x_{0}^{2} + x_{1}x_{4} − τ_{2}τ_{3}x_{2}x_{3} \\ B & = & τ_{4}x_{4}^{2} + x_{0}x_{3} − τ_{1}τ_{2}x_{1}x_{2} \\ C & = & τ_{2}x_{2}^{2} + x_{1}x_{3} − τ_{0}τ_{4}x_{0}x_{4}. \end{aligned}

The map ϕ : T → D_{λ} is given by (x_{0} : …: x_{4}) ↦ (X/Z, Y/Z) where

X = λτ_{0}τ_{1}x_{0}x_{1}x_{4}, Y = λ^{2}x_{2}x_{4}^{2}, Z = −τ_{0}^{2}τ_{1}x_{0}^{3}.

We justify this assertion by recording the identity

\begin{matrix}−λ^{3}τ_{0}^{3}τ_{1}^{2}x_{0}^{3}x_{4}^{2}((τ_{1}x_{1}^{2} + x_{0}x_{2})A + τ_{2}τ_{3}x_{2}^{2}B + τ_{1}τ_{2}τ_{3}x_{1}x_{2}C) \\ = Y^{2}Z + (1 − λ)XY Z − λY Z^{2} − X^{3} + λX^{2}Z. \end{matrix}

Case n = 7. Let λ, τ_{0}, …, τ_{6} ∈ K with λ^{4}(λ − 1) = ∏ τ_{i}. We recall that the curve T = T [λ; τ_{0}, …, τ_{6}] lies on the quadrics

\begin{aligned}A & = & τ_{0}x_{0}^{2} + x_{1}x_{6} − (1/λ^{2})τ_{2}τ_{3}τ_{4}τ_{5}x_{2}x_{5} \\ B & = & τ_{6}x_{6}^{2} + x_{0}x_{5} − (1/λ^{2})τ_{1}τ_{2}τ_{3}τ_{4}x_{1}x_{4} \\ C & = & (1/λ)τ_{2}τ_{3}τ_{4}x_{2}x_{4} + λx_{1}x_{5} − τ_{0}τ_{6}x_{0}x_{6}. \end{aligned}

The map ϕ : T → D_{λ} is given by (x_{0} : …: x_{6}) ↦ (X/Z, Y/Z) where

X = λ^{2}(λ − 1)τ_{0}τ_{1}x_{0}x_{1}x_{6}, Y = λ^{4}(λ − 1)^{2}x_{2}x_{6}^{2}, Z = −τ_{0}^{2}τ_{1}x_{0}^{3}.

We justify this assertion by recording the identity

\begin{matrix}−λ^{3}(λ − 1)^{3}τ_{0}^{3}τ_{1}^{2}x_{0}^{3}x_{6}^{2}(λ^{3}(τ_{1}x_{1}^{2} + x_{0}x_{2})A + λτ_{2}τ_{3}τ_{4}τ_{5}x_{2}^{2}B + τ_{1}τ_{2}τ_{3}τ_{4}τ_{5}x_{1}x_{2}C) \\ = Y^{2}Z + (1 + λ − λ^{2})XY Z + (λ^{2} − λ^{3})Y Z^{2} − X^{3} − (λ^{2} − λ^{3})X^{2}Z. \end{matrix}

Although not required for the proof of any of our descent theorems, these formulae serve a number of purposes. They enable us to exhibit elements of the Tate-Shafarevich group from an elementary point of view. This we did in the introduction with n = 5, λ = 30 and T = T [30; 1, 1, 2, 3, 5]. The formulae may also be used to turn knowledge of Q-points on the torsors C_{λ,θ} into knowledge of generators for D_{λ}(Q)/ϕC_{λ}(Q). As explained in [Cr, §3.6] it is often a good idea to search for rational points on the torsors covering an elliptic curve, rather than on the curve itself. However, in our case it seems likely the lack of a convenient algorithm searching for points on T [λ; τ_{0}, …, τ_{n−1}] will more than outweigh the advantage that the points sought have smaller height.

Appendix D

Tables

The construction of tables of examples has played an important role in the development of our descent theorems. We now illustrate Theorems 1–3 by reproducing some of these tables. Many of the examples listed are accessible to calculation by hand. However we have constructed the tables electronically, not least in the hope of keeping them free from errors.

Where possible we identify the curves listed with those in Cremona’s tables [Cr]. We recall that Cremona lists the curves by conductor, with a letter to distinguish isogeny classes with the same conductor, and a number to distinguish curves within each isogeny class. In the construction of our tables we also use Cremona’s programs mwrank, which performs a 2-descent, and findinf, which searches for points of infinite order. The L-values listed are those computed by pari. In the interests of clear presentation, if not of rigour, we have recorded the L-value as 0 whenever the computed value vanishes to 20 decimal places.

The reader’s attention is drawn to Remark 1.2 to explain why, for instance, the entries λ = ±1 in Table 1 are identical. The tables have been constructed so as not to contain repeats of this form.

Table 1 (n = 5) and Table 2 (n = 7)

We apply Theorem 1 for λ ∈ Z with |λ| ≤ 50. For each pair of nisogenous curves C_{λ} and D_{λ} we list the conductor N, the Cremona labels #, the sets of primes A and B, the rank of the pairing Ξ, the dimensions of the Selmer groups S^{(ϕ)}(C_{λ}/Q) and S^{(ϕ̂)}(D_{λ}/Q) and our estimate r_{ϕ} for the rank obtained by descent via n-isogeny. Finally we compare with r = rank D_{λ}(Q) as computed by mwrank and L(D_{λ}, 1) as computed by pari. For n = 5 and λ = −47, −46, −31, −19, 38, mwrank fails to completely determine rank D_{λ}(Q) since there are elements of order 2 in the Tate-Shafarevich group. In these cases our descent via 5-isogeny suffices to determine the rank. For n = 7 many of the curves considered are too large for us to run mwrank and we omit the column r accordingly.

Table 3 (n = 5) and Table 4 (n = 7)

We apply Theorem 1 for λ ∈ Q with C_{λ} and D_{λ} curves of conductor N ≤ 5300. The list of curves considered is taken from Cremona’s tables by searching for curves with rational torsion of order 5, 7 or 10. The quantities tabulated are as described for Tables 1 and 2, except that r = rank D_{λ}(Q) is taken directly from Cremona’s tables, and we omit the L-value.

Table 5 (n = 5)

We apply Theorem 2 for τ ∈ Z with |τ| ≤ 15. For each example we list the sets of primes P, Q and R, the rank of the pairing Ψ and the estimates r_{ϕ}, r_{ψ} and r_{5} for rank E_{τ}(Q). The estimates r_{ϕ} and r_{ψ} are the Selmer ranks (30) obtained by applying Theorem 1 with λ = τ^{5} respectively λ = τf(τ)/g(τ). The estimate r_{5} ≤ min{r_{ϕ}, r_{ψ}} is the Selmer rank (31) obtained by applying Theorem 2. Finally we record L(E_{τ}, 1) as computed by pari. Nearly all the curves in this table are too large for us to run mwrank.

The remaining tables give details of descent calculations over K = Q(μ_{n}). For E an elliptic curve over Q we recall our notation E^{∗} for the 5- respectively −7-twist of E.

Table 6 (n = 5) and Table 7 (n = 7)

We apply Theorem 3 for λ ∈ Z with −15 ≤ λ ≤ 20. The quantities tabulated are the conductor N and Cremona labels # for C_{λ} and D_{λ}, the conductor N^{∗} and Cremona labels #^{∗} for C_{λ}^{∗} and D_{λ}^{∗}, the sets of rational primes A↓ and B↓ below A and B, the numbers of primes a = |A| and b = |B|, the cusp number c = c(λ), the rank of the pairing Ξ, the dimensions of the Selmer groups S^{(ϕ)}(C_{λ}/K) and S^{(ϕ̂)}(D_{λ}/K) and finally our estimates r, r^{∗} and R for rank D_{λ}(Q), rank D_{λ}^{∗}(Q) and rank D_{λ}(K) obtained by descent via n-isogeny. The estimate R is the Selmer rank (52) computed using Theorem 3. The estimates r and r^{∗} are computed by considering the action of Gal(K/Q) on the Selmer groups S^{(ϕ)}(C_{λ}/K) and S^{(ϕ̂)}(D_{λ}/K). Notice that care must be taken here to allow for the Galois twist involved when we identify S^{(ϕ̂)}(D_{λ}/K) as a subgroup of H^{1}(K, μ_{n}) = K^{×}/(K^{×})^{n}. Clearly we have r + r^{∗} ≤ R, and the same inequality holds for the true ranks. An entry in bold indicates an estimate we know to be equal to the true rank, e.g. by running findinf to exhibit sufficiently many points of infinite order. An entry in italics indicates an estimate we know to be larger than the true rank, e.g. by running mwrank or by computing an L-value. For reasons of space we omit the columns a and b in Table 6 and the columns N^{∗} and #^{∗} in Table 7.

Table 8 (n = 7)

We apply Theorem 3 for some values λ = η(ν(ρ)) with ρ ∈ Q. Here ν is the involution of P^{1} described in Appendix B. This choice is made so that over K we may identify ϕ : B_{ρ} → B_{1−ρ}^{∗} with ϕ : C_{λ} → D_{λ}. The quantities tabulated are the conductor N for B_{ρ} and B_{ρ}^{∗}, the sets of rational primes A↓ and B↓ below A and B, the numbers of primes a = |A| and b = |B|, the cusp number c = c(λ), the rank of the pairing Ξ, the dimension of the Selmer groups S^{(ϕ)}(C_{λ}/K) and S^{(ϕ̂)}(D_{λ}/K) and finally our estimates r, r^{∗} and R for rank B_{ρ}(Q), rank B_{ρ}^{∗}(Q) and rank B_{ρ}(K) obtained by descent via 7-isogeny. The entries in bold and italics have the same meanings as described for Tables 6 and 7.

Table 9 (n = 5) and Table 10 (n = 7)

We apply Theorem 3 for some values λ ∈ K chosen such that A = ∅ and D_{λ} admits a model E over Q. For each example we list λ, η(λ), the conductor N and Cremona label # for E, the set of rational primes B↓ below B, the number of primes b = |B|, the cusp number c = c(λ), the rank of the pairing Ξ, the dimension of the Selmer groups S^{(ϕ)}(C_{λ}/K) and S^{(ϕ̂)}(D_{λ}/K) and finally our estimates r, r^{∗} and R for rank E(Q), rank E^{∗}(Q) and rank E(K) obtained by descent via n-isogeny. The entries in bold and italics have the same meaning as described for Tables 6 and 7.

In the case n = 5 we write ϕ for the golden ratio, and take λ = ϕ^{j} for j an odd integer. In each example we have taken E = C_{η(λ)}.

In the case n = 7 we take λ = u_{t} a root of

λ^{3} − tλ^{2} + (t − 3)λ + 1 = 0

for some t ∈ Z. The only values t ∈ Z for which this polynomial splits over K are t = 1, 2, −5, 8, −12, 15, −1259, 1262. For the entries above the line we have taken E = C_{−2}, C_{5} and C_{22/3}. For the entries below the line we have taken E = B_{1}, B_{2}, B_{3} and B_{9/4}.

Table 1 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ  N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r L(D_{λ}, 1)
−50 30490  {2, 5}  ∅  0 2 0 1 1 0
−49 20573  {7}  ∅  0 1 0 0 0 0.30766099
−48 16986  {2, 3}  ∅  0 2 0 1 1 0
−47 128075  {47}  ∅  0 1 0 0  0.63400946
−46 120566  {2, 23} {2621} 1 1 0 0  3.21920195
−45 37785  {3, 5} {11} 1 1 0 0 0 1.63505070
−44 53218  {2, 11} {41} 1 1 0 0 0 1.66142256
−43 99803  {43} {11, 211} 0 1 2 2 2 0
−42 93450 {2, 3, 7} ∅ 0 3 0 2 0 4.29287969
−41 87371  {41} {2131} 0 1 1 1 1 0
−40 20390  {2, 5}  ∅  0 2 0 1 1 0
−39 76011  {3, 13}  ∅  0 2 0 1 1 0
−38 70718  {2, 19} {1861} 1 1 0 0 0 0.92118046
−37 65675  {37} {71} 0 1 1 1 1 0
−36 10146  {2, 3}  ∅  0 2 0 1 1 0
−35 56315  {5, 7}  ∅  0 2 0 1 1 0
−34 51986  {2, 17} {11} 1 1 0 0 0 0.99514042
−33 47883  {3, 11} {1451} 1 1 0 0 0 1.01579851
−32  550 K2, 3 {2} {5, 11} 1 0 1 0 0 2.07491286
−31 40331  {31} {1301} 0 1 1 1  0
−30 36870  {2, 3, 5} ∅  0 3 0 2 0 5.42048094
−29 33611  {29} {61} 0 1 1 1 1 0
−28 15274  {2, 7} {1091} 1 1 0 0 0 2.27161017
−27 3075 L2, 1 {3}  {41} 0 1 1 1 1 0
−26  806 F2, 1 {2, 13} {31} 1 1 0 0 0 2.38703680
−25 4495 D2, 1 {5}  {31} 0 1 1 1 1 0
−24 5034 E2, 1 {2, 3} ∅ 0 2 0 1 1 0
−23 17963  {23} {11, 71} 0 1 2 2 2 0
−22 15950 {2, 11} ∅ 0 2 0 1 1 0
−21 14091  {3, 7} {11, 61} 1 1 1 1 1 0
−20 6190  {2, 5}  ∅  0 2 0 1 1 0
−19 10811  {19}  ∅  0 1 0 0  1.17090283
−18 3126 C2, 1 {2, 3} {521} 1 1 0 0 0 3.02909724
−17 8075  {17}  ∅  0 1 0 0 0 0.31396220

Table 1 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N  #  A B Ξ ϕ ϕ̂ r_{ϕ} r L(D_{λ}, 1)
−16 862 E2, 1 {2} {431} 0 1 1 1 1 0
−15 5835  {3, 5} ∅ 0 2 0 1 1 0
−14 4886 F2, 1 {2, 7} ∅ 0 2 0 1 1 0
−13 4043 A2, 1 {13} {311} 0 1 1 1 1 0
−12 1650 S2, 1 {2, 3} {11} 1 1 0 0 0 3.87816291
−11 2651 C2, 1 {11} {241} 0 1 1 1 1 0
−10 2090 N2, 1 {2, 5} {11} 1 1 0 0 0 2.15518557
−9 537 E2, 1 {3} ∅ 0 1 0 0 0 0.91494535
−8 302 A2, 1 {2} {151} 0 1 1 1 1 0
−7 175 A1, 2 {7} {5} 0 1 1 1 1 0
−6 606 F2, 1 {2, 3} {101} 1 1 0 0 0 2.84676247
−5 395 C2, 1 {5} ∅ 0 1 0 0 0 0.62499546
−4 118 B2, 1 {2} ∅ 0 1 0 0 0 1.39538063
−3 123 A2, 1 {3} {41} 0 1 1 1 1 0
−2 50 B3, 1 {2} ∅ 0 1 0 0 0 0.95681123
±1 11 A1, 3 ∅ {11} 0 0 1 0 0 0.25384186
2 38 B2, 1 {2} ∅ 0 1 0 0 0 0.81924563
3 75 C2, 1 {3} ∅ 0 1 0 0 0 0.62723493
4 58 B2, 1 {2} ∅ 0 1 0 0 0 1.03320751
5 155 A2, 1 {5} {31} 0 1 1 1 1 0
6 186 B2, 1 {2, 3} {31} 1 1 0 0 0 1.95289767
7 203 A2, 1 {7} ∅ 0 1 0 0 0 0.35054172
8 50 B4, 2 {2} ∅ 0 1 0 0 0 0.95681123
9 57 C2, 1 {3} ∅ 0 1 0 0 0 0.58650959
10 110 A2, 1 {2, 5} {11} 1 1 0 0 0 1.35954181
11 11 A2, 1 {11} ∅ 0 1 0 0 0 0.25384186
12 66 C3, 1 {2, 3} {11} 1 1 0 0 0 1.19161482
13 325 E2, 1 {13} ∅ 0 1 0 0 0 0.44965292
14 574 J2, 1 {2, 7} {41} 1 1 0 0 0 2.12965065
15 885 D2, 1 {3, 5} ∅ 0 2 0 1 1 0
16 158 C2, 1 {2} ∅ 0 1 0 0 0 1.54436244
17 1717 C2, 1 {17} {101} 0 1 1 1 1 0
18 150 A2, 4 {2, 3} {5} 1 1 0 0 0 1.76925925
19 2869 B2, 1 {19} {151} 0 1 1 1 1 0

Table 1 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ  N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r L(D_{λ}, 1)
20 1790 D2, 1 {2, 5}  ∅  0 2 0 1 1 0
21 4389 K2, 1 {3, 7} {11} 1 1 0 0 0 1.57719020
22 5302  {2, 11} {241} 0 2 1 2 2 0
23 6325  {23} {11} 0 1 1 1 1 0
24 1866 I2, 1 {2, 3} {311} 1 1 0 0 0 4.28015800
25 1745 E2, 1 {5}  ∅  0 1 0 0 0 0.55338675
26 10114  {2, 13} ∅  0 2 0 1 1 0
27 1293 E2, 1 {3} {431} 0 1 1 1 1 0
28 6650 {2, 7} ∅ 0 2 0 1 1 0
29 15109  {29} {521} 0 1 1 1 1 0
30 17070  {2, 3, 5} ∅  0 3 0 2 0 6.02423606
31 19189  {31}  ∅  0 1 0 0 0 0.23502226
32 1342 C2, 1 {2} {11, 61} 1 0 1 0 0 1.14697681
33 23925 {3, 11} ∅ 0 2 0 1 1 0
34 26554  {2, 17} {11, 71} 1 1 1 1 1 0
35 29365 {5, 7} ∅ 0 2 0 1 1 0
36 5394  {2, 3} {31} 1 1 0 0 0 4.19212658
37 1147 B2, 1 {37} {31} 0 1 1 1 1 0
38 38950 {2, 19} {41} 1 1 0 0 4.02140423
39 42549  {3, 13} {1091} 0 2 1 2 2 0
40 11590  {2, 5} {61} 1 1 0 0 0 2.89912975
41 50389 {41} ∅ 0 1 0 0 0 0.18962514
42 54642  {2, 3, 7} {1301} 1 2 0 1 1 0
43 11825  {43} {5, 11} 0 1 2 2 2 0
44 31922  {2, 11} {1451} 1 1 0 0 0 1.79541292
45 22935  {3, 5} {11} 1 1 0 0 0 1.76439797
46 74014  {2, 23} ∅  0 2 0 1 1 0
47 79477  {47}  ∅  0 1 0 0 0 0.17058134
48 10650  {2, 3} {71} 1 1 0 0 0 3.35622748
49 13027  {7} {1861} 0 1 1 1 1 0
50 19490  {2, 5}  ∅  0 2 0 1 1 0

Table 2 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over Q

λ  N  A  B  Ξ ϕ ϕ̂ r_{ϕ} L(D_{λ}, 1)
−50 74076990 {2, 3, 5, 17} {11173} 1 3 0 2 2.05696503
−49 9597070 {2, 5, 7} {71} 1 2 0 1 0
−48 5429046 {2, 3, 7} {129263} 1 2 0 1 0
−47 34327578 {2, 3, 47} {2969} 1 2 0 1 0
−46 247533866 {2, 23, 47} {114493} 1 2 0 1 0
−45 74208810 {2, 3, 5, 23} ∅ 0 4 0 3 0
−44 33294030 {2, 3, 5, 11} {7, 29, 71} 3 1 0 0 5.11304620
−43 89409298 {2, 11, 43} ∅ 0 3 0 2 0
−42 159666654 {2, 3, 7, 43} {211} 1 3 0 2 5.53178972
−41 142190706 {2, 3, 7, 41} {71, 1163} 2 2 0 1 0
−40 31569590 {2, 5, 41} {5923} 1 2 0 1 0
−39 27955590 {2, 3, 5, 13} {43, 1667} 2 2 0 1 0
−38 98720466 {2, 3, 13, 19} {29, 2297} 2 2 0 1 0
−37 86875334 {2, 19, 37}  {7}  1 2 0 1 0
−36 12699066 {2, 3, 37}  ∅  0 3 0 2 9.21681632
−35 11098290 {2, 3, 5, 7} {1289} 1 3 0 2 7.51425902
−34 57977990 {2, 5, 7, 17} ∅ 0 4 0 3 0
−33 50280186 {2, 3, 11, 17} {1093} 1 3 0 2 2.07210713
−32 2713854 {2, 3, 11} ∅ 0 3 0 2 6.23082208
−31 2333246  {2, 31} {37633} 1 1 0 0 0.93821430
−30 31944570 {2, 3, 5, 31} {7, 701} 2 2 0 1 0
−29 27197070 {2, 3, 5, 29} {43}  1 3 0 2 23.0964755
−28 11515378 {2, 7, 29} {113} 1 2 0 1 0
−27 1077258 {2, 3, 7}  ∅  0 3 0 2 2.47367425
−26 1802814 {2, 3, 13}  {29}  1 2 0 1 0
−25 2697370 {2, 5, 13} {20749} 1 2 0 1 0
−24 556530 {2, 3, 5} ∅ 0 3 0 2 11.9897036
−23 2278794 {2, 3, 23} {7, 337} 2 1 0 0 6.42266497
−22 7402274 {2, 11, 23}  ∅  0 3 0 2 2.30008612
−21 5956566 {2, 3, 7, 11}  ∅  0 4 0 3 0
−20 2372790 {2, 3, 5, 7} {11299} 1 3 0 2 9.37961008
−19 1869790 {2, 5, 19} {757} 1 2 0 1 0
−18  970482 {2, 3, 19} {8513} 1 2 0 1 0
−17  745518 {2, 3, 17} {7309} 1 2 0 1 0

114 APPENDIX D. TABLES

Table 2 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over Q

λ  N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r L(D_{λ}, 1)
−16 211582  {2, 17} {7, 127} 1 1 1 1  0
−15 157470  {2, 3, 5} {29} 1 2 0 1  0
−14 920010  {2, 3, 5, 7} {337} 1 3 0 2  8.20530033
−13 657566  {2, 7, 13} {3613} 1 2 0 1  0
−12 229242  {2, 3, 13} ∅  0 3 0 2  2.96566337
−11 155298  {2, 3, 11} ∅  0 3 0 2  3.37997297
−10 4730 K2, 1 {2, 5, 11} {43} 1 2 0 1 1 0
−9 42630  {2, 3, 5} {7, 29} 2 1 0 0 0 4.54417718
−8 6378  {2, 3}  ∅  0 2 0 1 1 0
−7 10766  {2, 7}  ∅  0 2 0 1 1 0
−6 22386  {2, 3, 7}  ∅  0 3 0 2 0 4.02654416
−5 10470  {2, 3, 5}  ∅  0 3 0 2 0 5.14484219
−4 2110 H2, 1 {2, 5} {211} 1 1 0 0 0 1.96226711
−3  678 D2, 1 {2, 3} {113} 1 1 0 0 0 2.79409146
−2  294 B1, 2 {2, 3}  {7} 1 1 0 0 0 2.21206490
−1, 2  26 B2, 1  {2}  ∅  0 1 0 0 0 0.62096535
3  174 B2, 1 {2, 3} {29} 1 1 0 0 0 1.85980396
4  258 F2, 1 {2, 3} {43} 1 1 0 0 0 2.13616829
5  490 K1, 2 {2, 5}  {7} 1 1 0 0 0 1.40933201
6 1230 K2, 1 {2, 3, 5}  ∅  0 3 0 2 0 3.53252671
7  546 F2, 1 {2, 3, 7}  ∅  0 3 0 2 0 2.67156776
8  574 I2, 1  {2, 7}  ∅  0 2 0 1 1 0
9  762 G2, 1 {2, 3} {127} 1 1 0 0 0 2.91591947
10 7530  {2, 3, 5}  ∅  0 3 0 2 0 5.63627952
11 46090  {2, 5, 11} ∅  0 3 0 2 0 2.37770891
12 42042  {2, 3, 11} {7} 1 2 0 1 1 0
13 71058  {2, 3, 13} {911} 1 2 0 1 1 0
14 226954  {2, 7, 13} {29, 43} 2 1 0 0  1.54832039
15 346710  {2, 3, 5, 7} {127} 1 3 0 2  9.58816728
16 63870  {2, 3, 5} {2129} 1 2 0 1  0
17 91358  {2, 17}  ∅  0 2 0 1  0
18 339762  {2, 3, 17} ∅  0 3 0 2  7.92763280
19 463638  {2, 3, 19} {7} 1 2 0 1  0
20 71630  {2, 5, 19} {29} 1 2 0 1  0

Table 2 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over Q

λ  N  A  B  Ξ ϕ ϕ̂ r_{ϕ} L(D_{λ}, 1)
21 1226190 {2, 3, 5, 7} {5839} 1 3 0 2 10.5517170
22 3181794 {2, 3, 7, 11} {71} 1 3 0 2 4.85751653
23 4073806 {2, 11, 23}  ∅  0 3 0 2 2.56499226
24 1288506 {2, 3, 23}  ∅  0 3 0 2 7.13460156
25  322530 {2, 3, 5} {827} 1 2 0 1 0
26 1598870 {2, 5, 13} {7} 1 2 0 1 0
27 1090986 {2, 3, 13} {71, 197} 2 1 0 0 1.44673411
28  664482 {2, 3, 7}  ∅  0 3 0 2 2.71225343
29 7229642 {2, 7, 29}  ∅  0 3 0 2 0.84937377
30 17357370 {2, 3, 5, 29} {71, 281} 2 2 0 1 0
31 20700870 {2, 3, 5, 31}  ∅  0 4 0 3 0
32 1533694  {2, 31}  {29} 1 1 0 0 1.01837396
33 1807806 {2, 3, 11} {7, 43} 2 1 0 0 6.74846858
34 33914694 {2, 3, 11, 17} ∅  0 4 0 3 0
35 39568690 {2, 5, 7, 17}  ∅  0 4 0 3 0
36 7658490 {2, 3, 5, 7}  ∅  0 4 0 3 0
37 8854914 {2, 3, 37} {39887} 1 2 0 1 0
38 61176466 {2, 19, 37} {3347} 1 2 0 1 0
39 70168254 {2, 3, 13, 19} {113} 1 3 0 2 1.75363708
40 20046390 {2, 3, 5, 13} {7}  1 3 0 2 20.1140714
41 22828390 {2, 5, 41}  ∅  0 3 0 2 2.74954086
42 103642014 {2, 3, 7, 41}  ∅  0 4 0 3 0
43 117265386 {2, 3, 7, 43} {29} 1 3 0 2 1.47312805
44 66141482 {2, 11, 43} ∅ 0 3 0 2 0.40395817
45 24799830 {2, 3, 5, 11} {337} 1 3 0 2 5.43281068
46 55640910 {2, 3, 5, 23} {6203} 1 3 0 2 2.61176176
47 186768694 {2, 23, 47} {7, 43} 2 1 0 0 0.71807052
48 26057082 {2, 3, 47} {92401} 1 2 0 1 0
49 4144854 {2, 3, 7}  {29} 1 2 0 1 0
50 7367570 {2, 5, 7}  ∅  0 3 0 2 0.64283549

Table 3 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N #  A  B Ξ ϕ ϕ̂ r_{ϕ} r
1 11 A1, 3  ∅  {11} 0 0 1 0 0
11 11 A2, 1 {11}  ∅  0 1 0 0 0
2 38 B2, 1 {2}  ∅  0 1 0 0 0
−2 50 B3, 1 {2}  ∅  0 1 0 0 0
8 50 B4, 2 {2}  ∅  0 1 0 0 0
9 57 C2, 1 {3}  ∅  0 1 0 0 0
4 58 B2, 1 {2}  ∅  0 1 0 0 0
12 66 C3, 1 {2, 3} {11} 1 1 0 0 0
9/2 66 C4, 2 {2, 3} {11} 1 1 0 0 0
3 75 C2, 1 {3} ∅ 0 1 0 0 0
10 110 A2, 1 {2, 5} {11} 1 1 0 0 0
−4 118 B2, 1 {2} ∅ 0 1 0 0 0
−3 123 A2, 1 {3} {41} 0 1 1 1 1
4/3 150 A1, 3 {2, 3} {5} 1 1 0 0 0
18 150 A2, 4 {2, 3} {5} 1 1 0 0 0
5 155 A2, 1 {5} {31} 0 1 1 1 1
16 158 C2, 1 {2} ∅ 0 1 0 0 0
−7 175 A1, 2 {7} {5} 0 1 1 1 1
6 186 B2, 1 {2, 3} {31} 1 1 0 0 0
7 203 A2, 1 {7} ∅ 0 1 0 0 0
32/3 246 B2, 1 {2, 3} {41} 1 1 0 0 0
13/2 286 D2, 1 {2, 13} {11} 1 1 0 0 0
−8 302 A2, 1 {2} {151} 0 1 1 1 1
13 325 E2, 1 {13} ∅ 0 1 0 0 0
3/2 366 B2, 1 {2, 3} {61} 1 1 0 0 0
−5 395 C2, 1 {5} ∅ 0 1 0 0 0
−3/2 426 A2, 1 {2, 3} {71} 1 1 0 0 0
−9 537 E2, 1 {3} ∅ 0 1 0 0 0
11/2 550 K1, 2 {2, 11} {5} 1 1 0 0 0
−32 550 K2, 3 {2} {5, 11} 1 0 1 0 0
48/5 570 L3, 1 {2, 3, 5} ∅  0 3 0 2 0
100/9 570 L4, 2 {2, 3, 5} ∅  0 3 0 2 0
14 574 J2, 1 {2, 7} {41} 1 1 0 0 0
−6 606 F2, 1 {2, 3} {101} 1 1 0 0 0

Table 3 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r
7/5 665 D2, 1 {5, 7}  ∅  0 2 0 1 1
25/2 710 D2, 1 {2, 5} {71} 1 1 0 0 0
27/2 786 M2, 1 {2, 3} {131} 1 1 0 0 0
−26 806 F2, 1 {2, 13} {31} 1 1 0 0 0
−4/3 834 G2, 1 {2, 3}  ∅  0 2 0 1 1
−16 862 E2, 1  {2}  {431} 0 1 1 1 1
45/4 870 I3, 1 {2, 3, 5}  ∅  0 3 0 2 0
50/3 870 I4, 2 {2, 3, 5}  ∅  0 3 0 2 0
23/2 874 E2, 1 {2, 23}  ∅  0 2 0 1 1
15 885 D2, 1 {3, 5}  ∅  0 2 0 1 1
5/2 890 G2, 1 {2, 5}  ∅  0 2 0 1 1
21/2 1050 O2, 1 {2, 3, 7}  ∅  0 3 0 2 0
37 1147 B2, 1  {37}  {31} 0 1 1 1 1
−15/7 1155 N2, 1 {3, 5, 7} {11} 1 2 0 1 1
8/3 1254 K2, 1 {2, 3} {11} 1 1 0 0 0
27 1293 E2, 1 {3} {431} 0 1 1 1 1
−5/2 1310 C2, 1 {2, 5} {131} 1 1 0 0 0
32 1342 C2, 1  {2} {11, 61} 1 0 1 0 0
122/11 1342 C3, 2 {2, 11, 61} ∅  0 3 0 2 0
−17/3 1479 F2, 1 {3, 17}  ∅  0 2 0 1 1
7/2 1526 E2, 1  {2, 7}  ∅  0 2 0 1 1
32/13 1586 D2, 1 {2, 13} {61} 1 1 0 0 0
−9/2 1650 R2, 1 {2, 3} {11} 1 1 0 0 0
−12 1650 S2, 1  {2, 3} {11} 1 1 0 0 0
16/3 1686 C2, 1 {2, 3} {281} 1 1 0 0 0
17 1717 C2, 1  {17} {101} 0 1 1 1 1
25 1745 E2, 1  {5}  ∅  0 1 0 0 0
20 1790 D2, 1 {2, 5}  ∅  0 2 0 1 1
24 1866 I2, 1  {2, 3} {311} 1 1 0 0 0
−85/8 1870 H2, 1 {2, 5, 17} {11} 1 2 0 1 1
−8/3 1914 O2, 1 {2, 3} {11} 1 1 0 0 0
−54 1914 P2, 1  {2, 3} {11} 1 1 0 0 0
−68/3 1938 J2, 1 {2, 3, 17} ∅  0 3 0 2 0
144/13 1950 Y2, 1 {2, 3, 13} ∅  0 3 0 2 0

Table 3 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N  #  A  B Ξ ϕ ϕ̂ r_{ϕ} r
9/4 1986 G2, 1  {2, 3} {331} 1 1 0 0 0
−10 2090 N2, 1  {2, 5} {11} 1 1 0 0 0
5/4 2110 E2, 1  {2, 5} {211} 1 1 0 0 0
56/5 2170 Q2, 1 {2, 5, 7} {31} 1 2 0 1 1
29/3 2175 J1, 2  {3, 29} {5} 1 1 0 0 0
5/3 2235 F2, 1  {3, 5} ∅ 0 2 0 1 1
−5/4 2290 D2, 1  {2, 5} ∅ 0 2 0 1 1
19/2 2318 E2, 1 {2, 19} {61} 1 1 0 0 0
47/4 2350 N1, 2 {2, 47} {5} 1 1 0 0 0
54/5 2370 M2, 1 {2, 3, 5} ∅ 0 3 0 2 0
34/3 2550 EE2, 1 {2, 3, 17} ∅ 0 3 0 2 0
−11 2651 C2, 1  {11} {241} 0 1 1 1 1
−5/3 2715 C2, 1  {3, 5} {181} 1 1 0 0 0
−9/4 2766 I2, 1  {2, 3} {461} 1 1 0 0 0
−7/2 2786 D2, 1  {2, 7} ∅ 0 2 0 1 1
27/4 2850 W2, 1 {2, 3} ∅ 0 2 0 1 1
19 2869 B2, 1  {19} {151} 0 1 1 1 1
17/2 3026 D2, 1 {2, 17} ∅ 0 2 0 1 1
−27 3075 L2, 1  {3} {41} 0 1 1 1 1
−18 3126 C2, 1  {2, 3} {521} 1 1 0 0 0
25/3 3135 H2, 1  {3, 5} {11} 1 1 0 0 0
49/4 3206 E2, 1  {2, 7} ∅ 0 2 0 1 1
15/2 3270 H2, 1 {2, 3, 5} ∅ 0 3 0 2 0
101/9 3333 G2, 1 {3, 101} {11} 1 1 0 0 0
92/7 3542 R2, 1 {2, 7, 23} {11} 1 2 0 1 1
81/8 3786 G2, 1  {2, 3} {631} 1 1 0 0 0
173/8 3806 K2, 1 {2, 173} {11} 1 1 0 0 0
7/4 3850 T2, 1  {2, 7} {11} 1 1 0 0 0
8/5 4010 E2, 1  {2, 5} {401} 1 1 0 0 0
7/3 4011 D2, 1  {3, 7} {191} 1 1 0 0 0
−13 4043 A2, 1  {13} {311} 0 1 1 1 1
21 4389 K2, 1  {3, 7} {11} 1 1 0 0 0
89/8 4450 K2, 1 {2, 89} ∅ 0 2 0 1 1
−25 4495 D2, 1  {5} {31} 0 1 1 1 1

Table 3 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r
−16/3 4650 LL2, 1 {2, 3} {31} 1 1 0 0 0
9/8 4650 PP2, 1 {2, 3} {31} 1 1 0 0 0
−56 4774 J2, 1  {2, 7} {11, 31} 1 1 1 1 1
−7/4 4774 K2, 1  {2, 7} {11, 31} 1 1 1 1 1
−8/5 4790 C2, 1  {2, 5}  ∅  0 2 0 1 1
−9/8 4854 C2, 1  {2, 3}  ∅  0 2 0 1 1
−14 4886 F2, 1  {2, 7}  ∅  0 2 0 1 1
25/4 4910 G2, 1  {2, 5} {491} 1 1 0 0 0
−24 5034 E2, 1  {2, 3}  ∅  0 2 0 1 1
43/4 5074 D2, 1 {2, 43}  ∅  0 2 0 1 1

Table 4 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over Q

λ N  #  A  B  Ξ ϕ ϕ̂ r_{ϕ} r
2 26 B2, 1  {2}  ∅  0 1 0 0 0
3 174 B2, 1  {2, 3} {29} 1 1 0 0 0
4 258 F2, 1  {2, 3} {43} 1 1 0 0 0
−2 294 B1, 2  {2, 3} {7} 1 1 0 0 0
5 490 K1, 2  {2, 5} {7} 1 1 0 0 0
7 546 F2, 1 {2, 3, 7} ∅  0 3 0 2 0
8 574 I2, 1 {2, 7} ∅ 0 2 0 1 1
−3 678 D2, 1  {2, 3} {113} 1 1 0 0 0
9 762 G2, 1  {2, 3} {127} 1 1 0 0 0
−9/2 858 K2, 1 {2, 3, 11} ∅  0 3 0 2 0
6 1230 K2, 1 {2, 3, 5} ∅  0 3 0 2 0
−4 2110 H2, 1  {2, 5} {211} 1 1 0 0 0
−10 4730 K2, 1 {2, 5, 11} {43} 1 2 0 1 1
5/2 5010 H2, 1 {2, 3, 5} ∅  0 3 0 2 0

Table 5 (n = 5) Descent via [5] : E_{τ} → E_{τ} over Q

τ  P  Q  R Ψ r_{ϕ} r_{ψ} r_{5} L(E_{τ}, 1)
−15 {3, 5} {11, 3761, 58321} ∅  4 0 4 0
−14 {2, 7} {41, 1091, 30941} {11} 4 1 3 1 0
−13 {13} {11, 3061, 22621} {181} 4 2 2 0
−12 {2, 3} {11, 41, 3221} {5, 31} 4 2 2 2 0
−11 {11} {41, 71, 251, 271} {131} 4 3 3 1 0
−10 {2, 5} {11, 31, 61, 401} ∅ 4 1 5 1 0
−9 {3} {11, 31, 151, 761} ∅  4 2 4 0 0.09302147
−8 {2} {11, 491, 2801} {71} 2 2 2 2 0
−7 {7} {311, 661} {5, 11} 4 2 0 0 1.15762623
−6 {2, 3} {11, 31, 61, 71} {41} 4 2 4 2 0
−5 {5}  {11, 31, 991}  ∅  2 1 3 1 0
−4 {2}  {11, 461}  ∅  2 0 2 0 1.08064637
−3 {3}  {31, 181}  {11} 2 1 1 1 0
−2 {2}  {11}  {5} 2 0 0 0 2.07491286
±1 ∅  {11}  ∅  0 0 0 0 0.25384186
2 {2}  {11, 61}  ∅  2 0 2 0 1.14697681
3 {3}  {11, 41}  {5} 2 1 1 1 0
4 {2}  {181, 521}  {11} 2 1 1 1 0
5 {5} {11, 101, 461}  ∅  2 1 3 1 0
6 {2, 3} {11, 191, 991}  ∅  4 0 4 0 0.28811770
7 {7} {11, 31, 61, 331} {41} 4 3 3 1 0
8 {2}  {661, 1181} {5, 11} 4 2 0 0 0.47598299
9 {3} {11, 491, 9091} {71} 2 2 2 2 2.32573719
10 {2, 5} {11, 761, 13421}  ∅  4 0 4 0 4.14475951
11 {11} {31, 401, 19141}  ∅  2 1 3 1 0
12 {2, 3} {11, 71, 251, 2411} {131} 6 2 4 0
13 {13} {11, 41, 71, 101} {5, 31} 4 4 2 2 0
14 {2, 7} {11, 31, 1531, 3061} {181} 4 2 4 2
15 {3, 5} {41, 1091, 61681} {11} 4 1 3 1 0

Table 6 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over K = Q(μ_{5})

λ N  #  N^{∗}  #^{∗} A↓ B↓ c Ξ ϕ ϕ̂ r r^{∗} R
−15 5835  29175  {3, 5} {389} 0 1 3 1 1 1 2
−14 4886 F2, 1 122150  {2, 7} {349} 0 0 4 2 1 1 4
−13 4043 A2, 1 101075  {13} {311} 0 2 1 2 1 0 1
−12 1650 S2, 1  1650 D1, 2 {2, 3} {11} 4 3 1 4 0 1 3
−11 2651 C2, 1 66275  {11} {241} 0 2 4 2 1 1 4
−10 2090 N2, 1 10450  {2, 5} {11, 19} 0 3 1 3 0 0 2
−9 537 E2, 1 13425  {3} {179} 0 1 2 1 0 1 1
−8 302 A2, 1 7550  {2} {151} 0 1 2 3 1 0 3
−7 175 A1, 2 175 C1, 2 {7} ∅ 6 2 1 2 1 0 1
−6 606 F2, 1 15150  {2, 3} {101} 0 2 2 2 0 0 2
−5 395 C2, 1 1975 D2, 1 {5} {79} 0 1 2 1 0 1 1
−4 118 B2, 1 2950 A2, 1 {2} {59} 0 1 2 1 0 1 1
−3 123 A2, 1 3075 D2, 1 {3} {41} 0 2 1 2 1 0 1
−2 50 B3, 1 50 A1, 3 {2} ∅ 4 2 1 1 0 0 0
±1 11 A1, 3  275 B2, 1 ∅ {11} 0 2 0 2 0 0 0
2 38 B2, 1  950 A2, 1 {2} {19} 0 1 2 1 0 1 1
3 75 C2, 1 75 A1, 2 {3} ∅ 4 2 1 1 0 0 0
4 58 B2, 1 1450 A2, 1 {2} {29} 0 1 2 1 0 1 1
5 155 A2, 1  775 C2, 1 {5} {31} 0 2 1 2 1 0 1
6 186 B2, 1 4650 E2, 1 {2, 3} {31} 0 3 1 1 0 0 0
7 203 A2, 1 5075 C2, 1 {7} {29} 0 1 2 1 0 1 1
8 50 B4, 2 50 A2, 4 {2} ∅ 4 2 1 1 0 0 0
9 57 C2, 1 1425 B2, 1 {3} {19} 0 1 2 1 0 1 1
10 110 A2, 1  550 B2, 1 {2, 5} {11} 0 3 1 1 0 0 0
11 11 A2, 1 275 B3, 2 {11} ∅ 0 0 6 0 0 2 4
12 66 C3, 1 1650 B3, 1 {2, 3} {11} 0 3 1 1 0 0 0
13 325 E2, 1 325 D1, 2 {13} ∅ 4 2 1 1 0 0 0
14 574 J2, 1 14350  {2, 7} {41} 0 3 1 1 0 0 0
15 885 D2, 1 4425 C2, 1 {3, 5} {59} 0 1 3 1 1 1 2
16 158 C2, 1 3950 A2, 1 {2} {79} 0 1 2 1 0 1 1
17 1717 C2, 1 42925  {17} {101} 0 1 2 3 1 0 3
18 150 A2, 4 150 B2, 4 {2, 3} ∅ 6 3 1 1 0 0 0
19 2869 B2, 1 71725  {19} {151} 0 1 3 3 1 1 4
20 1790 D2, 1 8950  {2, 5} {179} 0 1 3 1 1 1 2

122 APPENDIX D. TABLES

Table 7 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over K = Q(μ_{7})

λ N # A↓ B↓ a b c Ξ ϕ ϕ̂ r r^{∗} R
−15 157470  {2, 3, 5} {29, 181} 4 9 0 5 2 4 1 1 4
−14 920010  {2, 3, 5, 7} {13, 337} 5 9 0 5 3 4 2 1 5
−13 657566  {2, 7, 13} {3613} 6 6 0 4 5 2 1 2 5
−12 229242  {2, 3, 13} {2939} 6 3 0 1 8 2 2 0 8
−11 155298  {2, 3, 11} {13, 181} 5 6 0 3 5 3 2 0 6
−10 4730 K2, 1 {2, 5, 11} {43} 5 6 0 5 3 1 1 1 2
−9 42630  {2, 3, 5} {29} 4 6 4 6 1 6 0 1 5
−8 6378  {2, 3} {1063} 3 3 0 2 4 1 1 0 3
−7 10766  {2, 7} {769} 3 3 0 2 4 1 1 0 3
−6 22386  {2, 3, 7} {13, 41} 4 6 0 2 5 4 2 1 7
−5 10470  {2, 3, 5} {349} 4 3 0 2 5 1 2 0 4
−4 2110 H2, 1 {2, 5} {211} 3 6 0 5 1 1 0 0 0
−3  678 D2, 1 {2, 3} {113} 3 6 0 5 1 1 0 0 0
−2  294 B1, 2 {2, 3}  ∅  3 0 4 5 1 1 0 0 0
−1, 2  26 B2, 1  {2}  {13} 2 3 0 2 3 1 0 0 2
3  174 B2, 1 {2, 3}  {29} 3 6 0 5 1 1 0 0 0
4  258 F2, 1 {2, 3}  {43} 3 6 0 5 1 1 0 0 0
5  490 K1, 2 {2, 5}  ∅  3 0 4 5 1 1 0 0 0
6 1230 K2, 1 {2, 3, 5} {41} 4 3 0 2 5 1 2 0 4
7  546 F2, 1 {2, 3, 7} {13} 4 3 0 2 5 1 2 0 4
8 574 I2, 1 {2, 7} {41} 3 3 0 2 4 1 1 0 3
9  762 G2, 1 {2, 3} {127} 3 6 0 5 1 1 0 0 0
10 7530  {2, 3, 5} {251} 4 3 0 2 5 1 2 0 4
11 46090  {2, 5, 11} {419} 5 3 0 2 6 1 2 1 5
12 42042  {2, 3, 11} {13} 5 3 4 6 2 3 1 0 3
13 71058  {2, 3, 13} {911} 6 6 0 5 4 1 1 0 3
14 226954  {2, 7, 13} {29, 43} 6 12 0 8 1 4 0 1 3
15 346710  {2, 3, 5, 7} {13, 127} 5 9 0 5 3 4 2 1 5
16 63870  {2, 3, 5} {2129} 4 6 0 4 3 2 1 0 3
17 91358  {2, 17} {2687} 3 3 0 2 4 1 1 0 3
18 339762  {2, 3, 17} {3331} 4 3 0 1 6 2 2 0 6
19 463638  {2, 3, 19} {83} 4 3 4 5 2 4 1 1 4
20 71630  {2, 5, 19} {13, 29} 4 9 0 5 2 4 1 1 4

Table 8 (n = 7) Descent via ϕ : B_{ρ} → B_{1−ρ}^{∗} over K = Q(μ_{7})

ρ  N  A↓  B↓  a b c Ξ ϕ ϕ̂ r r^{∗} R
1  49  ∅  ∅  0 0 2 2 1 1 0 0 0
2  49  ∅  ∅  0 0 2 2 1 1 0 0 0
3  637  ∅  {13} 0 3 0 1 2 2 1 1 2
4  58261 {29}  {41} 6 3 2 4 5 2 1 0 5
5  45227 {71}  {13} 6 3 4 7 2 2 1 1 2
6  88543 {139} {13} 3 3 2 2 4 4 1 1 6
7  3290791 {239} {281} 6 6 2 7 2 2 2 0 2
8  7998809 {13, 29} {433} 9 3 2 5 7 1 2 0 6
9 17283721 {13, 43} {631} 9 6 2 8 4 1 1 0 3
10  4878097 {113} {881} 6 3 0 2 7 1 2 0 6
11 62863619 {13, 83} {29, 41} 6 9 2 5 4 7 1 2 9
12 15614683 {1429} {223} 6 3 4 7 2 2 1 1 2
13 181277509 {1847} {2003} 3 6 2 5 1 4 0 1 3
14 288934331 {2339} {2521} 6 6 2 7 2 2 0 2 2
15 445176319 {41, 71} {3121} 9 3 2 4 8 2 2 2 8
16 666121729 {43, 83} {13, 293} 9 6 2 5 7 4 2 1 9
17 138799703 {617} {4591} 6 3 0 2 7 1 2 0 6
18 1385670559 {5167} {13, 421} 6 9 2 8 1 4 0 1 3
5/2  27391 {13}  {43} 3 6 2 4 2 5 2 1 5
7/3  61789 {13}  {97} 3 3 2 2 4 4 1 1 6
9/4  8869  ∅  {181} 0 3 2 2 1 4 0 1 3
11/5  61103 {29}  {43} 6 6 4 8 1 4 0 1 3
13/6  1883021 {83} {463} 3 6 2 4 2 5 2 1 5
15/7  5507159 {167} {673} 3 6 2 5 1 4 0 1 3
17/8  1882433 {41} {937} 3 3 0 1 5 2 2 1 5
19/9 27743261 {449} {13, 97} 6 6 2 5 4 4 1 1 6
21/10 53312441 {659} {13, 127} 6 9 2 7 2 5 0 1 5
23/11 95564651 {13, 71} {2113} 9 3 2 4 8 2 2 0 8
25/12 23158037 {29, 43} {379} 12 6 4 10 5 2 3 0 5
27/13 262858001 {1637} {29, 113} 3 12 2 5 1 10 1 2 9
29/14 410478341 {2099} {13, 307} 3 6 2 2 4 7 1 2 9
31/15 88688873 {13, 29} {4801} 9 3 0 2 10 1 3 0 9
33/16 913434431 {13, 251} {29, 197} 6 12 2 8 1 7 0 2 6
35/17 1312079909 {41, 97} {6733} 6 3 2 2 7 4 2 1 9

124 APPENDIX D. TABLES Table 9 (n = 5) Descent via ϕ : C_{λ} → D_{λ} over K = Q(μ_{5})

λ η(λ) N # B↓ b c Ξ ϕ ϕ̂ r r^{∗} R
ϕ^{−23} 377/34 320450  {2, 13, 17, 29} 5 2 1 1 5 3 1 4
ϕ^{−21} 521/47 612175  {47, 521} 5 0 1 1 4 0 1 3
ϕ^{−19} 144/13 1950 Y2 {2, 3, 13} 3 2 1 1 3 2 0 2
ϕ^{−17} 199/18 29850  {2, 3, 199} 4 0 1 1 3 1 1 2
ϕ^{−15}  11  11 A2  {11}  4 6 2 0 6 0 2 4
ϕ^{−13} 76/7  6650  {2, 7, 19} 4 0 1 1 3 1 1 2
ϕ^{−11} 21/2  1050 O2  {2, 3, 7}  3 2 1 1 3 2 0 2
ϕ^{−9}  29/3  2175 J1  {3, 29}  3 0 1 1 2 0 1 1
ϕ^{−7}  8  50 B4  {2}  1 2 1 1 1 0 0 0
ϕ^{−5}  11/2  550 K1  {2, 11}  5 0 2 0 3 0 1 1
ϕ^{−3}  3  75 C2  {3}  1 2 1 1 1 0 0 0
ϕ^{−1}  4/3  150 A1  {2, 3}  2 0 1 1 1 0 0 0
ϕ  −2  50 B3  {2}  1 2 1 1 1 0 0 0
ϕ^{3}  −7  175 A1  {7}  1 0 0 2 1 1 0 1
ϕ^{5}  0, ∞
ϕ^{7}  18  150 A2  {2, 3}  2 0 1 1 1 0 0 0
ϕ^{9}  13  325 E2  {13}  1 2 1 1 1 0 0 0
ϕ^{11}  47/4  2350 N1  {2, 47}  2 0 1 1 1 0 0 0
ϕ^{13}  34/3  2550 EE2 {2, 3, 17} 3 2 1 1 3 2 0 2
ϕ^{15} 123/11 33825  {3, 11, 41} 9 0 2 0 7 1 2 5
ϕ^{17} 89/8 4450 K2 {2, 89} 3 2 1 1 3 1 1 2
ϕ^{19} 322/29 233450  {2, 7, 23, 29} 5 0 1 1 4 2 1 3
ϕ^{21} 233/21 122325  {3, 7, 233} 3 2 1 1 3 2 0 2
ϕ^{23} 843/76 800850  {2, 3, 19, 281} 8 0 1 1 7 2 2 6

Table 10 (n = 7) Descent via ϕ : C_{λ} → D_{λ} over K = Q(μ_{7})

λ  η(λ)  N  #  B↓  b c Ξ ϕ ϕ̂ r r^{∗} R
u_{2}  −2  294 B1  {2, 3}  3 0 2 1 1 0 0 0
u_{−12}  5  490 K1  {2, 5}  3 0 2 1 1 0 0 0
u_{1262} 22/3  61446  {2, 3, 11, 19} 6 0 2 1 4 2 1 3
u_{1}  u_{1}  49 A1  ∅  0 2 2 1 1 0 0 0
u_{15}  u_{15}  49 A2  ∅  0 2 2 1 1 0 0 0
u_{−5} u_{55/13}  637 A2  {13}  3 0 1 2 2 1 1 2
u_{−1259} u_{1441/181} 8869  {181}  3 2 2 1 4 0 1 3

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Electronic References

C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier pari/gp A semi-general computer algebra package, implementing almost all the algorithms presented in [Coh].

http://www.parigp-home.de

J.E. Cremona mwrank A program for 2-descent on elliptic curves over Q. findinf Searches for rational points on elliptic curves over Q.

http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs

J.E. Cremona Modular elliptic curve data for conductors up to 5300.

http://www.maths.nottingham.ac.uk/personal/jec/ftp/data

Wolfram Research Inc. mathematica A general computer algebra package.

http://www.wolfram.com