Chapter 5

Modular curves and Kuga-Sato varieties

5.1 Generalities

5.1.1 Level structures on elliptic curves

For this section, E will denote a fixed elliptic curve over a base scheme S. We use the following notations:

• [×n]: E → E is the endomorphism “multiplication by n” of E;

• E[n] is the kernel of [×n]; it is a finite flat group scheme over S, and if n is invertible on S (which will in practice always be the case) is a finite étale group scheme over S;

• E[tors] denotes the direct limit of the group schemes E[n] with respect to inclusions.

We will only make use of E[tors] when S is a ℚ-scheme, in which case it is actually a scheme: in fact, if E[n]^{∗} denotes the complement E[n] − ∪_{m<n}E[m] (the closed subscheme of “points of exact order n” on E) then (if S is a ℚscheme) E[n] is the disjoint union of the E[m]^{∗}, for m running over all positive divisors of n, and E[tors] is the disjoint union of E[n]^{∗}, n ≥ 1. It is therefore an ind-étale group scheme over S. There is a natural morphism E[tors] → E which is the direct limit of the inclusions E[n] ⊂ E. Of course this morphism it not a closed immersion (since otherwise E[tors] would be of finite type over S!) and E[tors] is not a closed subscheme of E.

Definition 5.1.1.1. Let n be a positive integer. A level n structure on E/S is an isomorphism

α_{n}: (ℤ/n)_{S}^{2} ⟶^{∼} E[n]

Last revision 4 December 1996 5:2

5:1 between the constant group scheme (ℤ/n)^{2} over S and the kernel of multiplication by n on E.

Equivalently, let us write as usual E(S) for the set of sections of E → S. Assume that S is connected. Then a level n structure on E/S is the same thing as an injective homomorphism of abelian groups

α_{n}: (ℤ/n)^{2} ↪ E(S).

So one can define level n structures without reference to group schemes. However, using group schemes gives greater flexibility, and we will use it constantly in what follows.

Observe that there can be no level n structure on E/S unless n is invertible on S. For if s ∈ S is a point of characteristic p, then the fibre E_{s} has, by virtue of the level structure, exactly n^{2} k(s)-rational points of order dividing n, which is only possible if n ≢ 0 (mod p).

Let m and n be positive integers with m|n. If α_{m} and α_{n} are level m and level n structures on E/S respectively, such that the diagram

(ℤ/n)_{S}^{2} ⟶_{α_{n}} E[n] ↪ E
reduction mod m↓  ↓  ↓^{[×n/m]}
(ℤ/m)_{S}^{2} ⟶_{α_{m}} E[m] ↪ E

(5.1.1.1)

commutes, then we say that α_{m} and α_{n} are compatible. Given a level n structure α_{n} and a positive divisor m of n, there exists exactly one level m structure which is compatible with α_{n}.

By passage to the inverse limit we obtain the following notion:

Definition 5.1.1.2. Assume that S is a ℚ-scheme. A complete level structure on E/S is a family of level n structures α_{n}, one for each n ≥ 1, such that whenever m|n the α_{m} and α_{n} are compatible.

Assume S has chacteristic zero. For positive integers m, n with m|n, multiplication by n/m induces a finite morphism

[×\frac{n}{m}]: E[n] → E[m]

and we may consider the inverse limit of the group schemes E[n] with respect to these morphisms. As they are finite morphisms, the inverse limit

T̂(E/S) =^{def} lim_{⟵}_{n} E[n]

exists (cf. 5.2.4 below) - it is a profinite group scheme over S.

Definition 5.1.1.2 (bis). A complete level structure on E/S is an isomorphism

α̂: ℤ̂_{S}^{2} ⟶^{∼} T̂(E/S)

of profinite group schemes over S.

We have canonical isomorphisms (e.g. of sheaves on S for the flat topology)

T̂(E/S) ⊗ ℚ/ℤ = lim_{⟶}_{n} T̂(E/S) ⊗ \frac{1}{n}ℤ/ℤ

= lim_{⟶}_{n} E[n]

= E[tors]

and therefore a third equivalent definition is:

Definition 5.1.1.2. A complete level structure on E/S is an isomorphism

α_{∞}: (ℚ/ℤ)_{S}^{2} ⟶^{∼} E[tors]

One also defines

V_{f}(E/S) =^{def} T̂(E/S) ⊗_{ℤ} ℚ

= lim_{⟶}_{n}((T̂(E/S))_{n}, ×n/m)

which is an inductive limit of profinite group schemes. It is also an inverse limit of (ind-finite!) group schemes and finite morphisms

V_{f}(E/S) = lim_{⟵}_{n}((E[tors])_{n}, [×n/m])

and is therefore a true group scheme.

5.1.2 Pairings

For any elliptic curve E/S and any n ≥ 1 the e_{n}-pairing is a bilinear nondegenerate alternating form

e_{n}: E[n] ×_{S} E[n] → μ_{n/S}.

If α_{n} is a level n structure on E/S, then it determines an n-th root of unity on S

det(α_{n}) =^{def} e_{n}(α_{n}\binom{1}{0}, α_{n}\binom{0}{1}) ∈ μ_{n}(S)

which is a primitive n-th root of unity (since e_{n} is nondegenerate), called the determinant of α_{n}. It therefore gives rise to a morphism

S → Spec ℤ[ζ_{n}]
det(α_{n}) ← ζ_{n}

For the properties of e_{n} see [K-M, §2.8.5]. If m|n then the diagram

\begin{matrix}E[n] ×_{S} E[n] & ⟶_{e_{n}} & μ_{n} \\ ^{([× \frac{n}{m}],[× \frac{n}{m}])}↓ & & ↓^{× \frac{n}{m}} \\ E[m] ×_{S} E[m] & ⟶_{e_{m}} & μ_{m} \end{matrix}  (5.1.2.1)

commutes [K-M, p.505], and therefore one has a nondegenerate form

ê: T̂(E/S) ×_{S} T̂(E/S) → ℤ̂(1) = lim_{⟵}_{n} μ_{n}

such that ê ⊗ ℤ/nℤ = e_{n}. Therefore a complete level structure α̂ on E/S gives rise to a generator det(α̂) of ℤ̂(1)(S), and thus a morphism

S → Spec ℤ[ζ_{∞}] =^{def} Spec ⋃ ℤ[ζ_{n}].

5.1.3 Group actions

Let E/S be an elliptic curve and α_{n} a level n structure on E/S. If g ∈ GL_{2}(ℤ/nℤ) then α_{n} ∘ g is another level n structure on E/S, and if S is connected, every level n structure on E/S is of the form α ∘ g for a unique g. More generally, without assuming the existence of any level n structure on E/S, we can form the scheme

P_{n} =^{def} Isom((ℤ/n)_{S}^{2}, E[n]).

For any S′/S, P_{n}(S′) is the set of level n structures on E ×_{S} S′/S′. Although in general the representability of Isom can be rather subtle, in this case it is rather straightforward. The functor which associates to S′/S the set of group scheme homomorphisms (ℤ/n)_{S}^{2} → E[n] is represented by E[n] ×_{S} E[n] (since to give such a homomorphism is equivalent to giving the images of \binom{1}{0} and \binom{0}{1} in E[n]). The scheme P_{n} is the open and closed subscheme of E[n] ×_{S} E[n] which represents homomorphisms which are isomorphisms. So in particular, P_{n} is finite and étale over S. A detailed discussion of representability of these sort of things can be found in [K-M, §1.6].

Composition with g ∈ GL_{2}(ℤ/nℤ) defines a right action of the finite group GL_{2}(ℤ/nℤ) on P_{n}, or (what amounts to the same thing) an action of the constant group scheme GL_{2}(ℤ/nℤ)_{S} on P_{n}. Over a geometric point s of S, there exist #GL_{2}(ℤ/nℤ) distinct level n structures on E_{s}, permuted transitively by GL_{2}(ℤ/nℤ). This implies that P_{n}/S is a torsor under GL_{2}(ℤ/nℤ), for the étale topology of S. In particular, the quotient of P_{n} by GL_{2}(ℤ/nℤ) exists and is presisely S.

Slightly more generally, let m|n, and fix a level m structure α_{m}: (ℤ/m)_{S}^{2} ⟶^{∼} E[m]. It determines a section of P_{m} over S. There is an obvious morphism P_{n} → P_{m}, and we can form the P_{n}(α_{m}) ⊂ P_{n}, the fibre over this section. For any S′/S, P_{n}(α_{m})(S′) is the set of level n structures on E ×_{S} S′/S′ compatible with α_{m}.

Proposition 5.1.3.1. P_{n}(α_{m})/S is a K_{n,m}-torsor, where

K_{n,m} =^{def} ker[GL_{2}(ℤ/nℤ) ⟶^{reduction mod m} GL_{2}(ℤ/mℤ)].

5.2 Modular curves and Kuga-Sato varieties

5.2.1 Moduli of elliptic curves with level structure

We are mainly interested in the modular curves over ℚ attached to principal congruence subgroups. Unless stated otherwise, we will always assume that the level n is an integer greater than or equal to 3. The basic representability result is then:

Theorem 5.2.1.1. There exists a smooth, affine curve M_{n} over ℚ and an elliptic curve f_{n}: X_{n} → M_{n}, together with a level n structure

α_{n}^{univ} : (ℤ/n)_{M_{n}}^{2} ⟶^{∼} X_{n}[n],

which represents the functor

“isomorphism classes of elliptic curves with level n structure in characteristic zero”.

Explicitly, let (E/S, α_{n}) be an elliptic curve with level n structure in characteristic zero. Then there is a unique morphism ϕ: S → M_{n}, and a unique S-isomorphism of elliptic curves ψ: E → X_{n} ×_{M_{n},ϕ} S such that the following diagram commutes:

(ℤ/n)_{S}^{2}  ⟶^{id×ϕ}  (ℤ/n)_{M_{n}}^{2}
↓^{α_{n}}  ↓^{α_{n}^{univ}}
E[n]  X_{n}[n]
∩  ∩
E  ⟶_{ψ}^{∼}  X_{n} ×_{M_{n},ϕ} S ⟶^{pr_{1}} X_{n}
↘  ↙pr_{2}  ↓^{f_{n}}
S  ⟶^{ϕ} M_{n}

(5.2.1.1)

In particular, the S-valued points of M_{n} are in 1-1 correspondence with isomorphism classes of such pairs (E/S, α_{n}).

There is a straightforward generalisation to schemes in which n is invertible:

Theorem 5.2.1.2. There exists a smooth, affine curve M_{n} over ℤ[1/n] and an elliptic curve X_{n}/M_{n}, together with a level n structure

α_{n}^{univ} : (ℤ/n)_{M_{n}}^{2} ⟶^{∼} X_{n}[n],

which represents the functor

“isomorphism classes of elliptic curves with level n structure”.

We will not describe the construction of M_{n}; the reader will find detailed accounts in [D-R] and [K-M] of much more general results. As stated here, the theorem is due to Igusa [reference], and a succinct sketch of his proof of the existence of M_{n} can be found in the Introduction to [D-R].

The curve M_{n} is not geometrically connected. The Weil pairing on X_{n}[n] defines a canonical morphism M_{n} → Spec ℚ(ζ_{n}), with connected geometric fibre. However, one can find a smooth curve M′_{n} over ℚ such that M_{n} ⟶^{∼} M′_{n}⊗ℚ(ζ_{n}). There is a modular description of M′_{n}, as the moduli space for elliptic curves E/S plus an isomorphism μ_{n} ×ℤ/n ⟶^{∼} E[n] of determinant one [D-R, V.4.4]. But for our purposes it is better to work with M_{n}.

It is useful to have a modular description of the points of X_{n}. Consider the functor on ℚ-schemes S:

S → {isomorphism classes of triples (E/S, α_{n}, P) where (E/S, α_{n}) is an elliptic curve with level n structure and P ∈ E(S) is a section}

Proposition 5.2.1.3. The above functor is represented by X_{n}, and the universal triple (corresponding to the tautological point id_{X_{n}} ∈ X_{n}(X_{n})) is

(X_{n} ×_{M_{n}} X_{n} ⟶^{pr_{1}} X_{n}, α_{n}^{univ} × id_{X_{n}}, Δ_{X_{n}/M_{n}})

where Δ_{X_{n}/M_{n}}: X_{n} → X_{n} ×_{M_{n}} X_{n} is the diagonal morphism.

Let us provisionally denote the functor in question X′. We first define a morphism from (the functor represented by) X_{n} to X′: let x ∈ X_{n}(S) be a point of X_{n}, and f_{n}(x) ∈ M_{n}(S) its image in M_{n}. We associate to x the triple (E/S, α_{n}, u) where:

E = X_{n} ×_{M_{n},f_{n}(x)} S = f_{n}(x)^{∗}X_{n};
α_{n} = f_{n}(x)^{∗}(α_{n}^{univ});
P = (x, id_{S}): S → X_{n} ×_{M_{n},f_{n}(x)} S

In particular, if S = X_{n} and x = id_{X_{n}}, then to x is associated the triple described in the proposition. In the other direction, if (E/S, α_{n}, P) ∈ X′(S) then by Theorem 5.2.1.1 we have a unique classifying map ϕ: S → M_{n} and isomorphism ψ: E ⟶^{∼} X_{n} ×_{M_{n},ϕ} S. We may then associate to this triple the point x ∈ X_{n}(S) defined as the composite

S ⟶^{u} E ⟶^{ψ} X_{n} ×_{M_{n},ϕ} S ⟶^{pr_{1}} X_{n}.

We leave it to the reader to verify that these are indeed morphisms of functors which are mutual inverses.

Definition 5.2.1.4. Let k be a non-negative integer, and n ≥ 3. The k-th Kuga-Sato variety of level n is the fibre product

X_{n}^{k} =^{def} \underbrace{X_{n} ×_{M_{n}} ⋯ ×_{M_{n}} X_{n}}_{k copies}

Evidently X_{n}^{0} = M_{n} and X_{n}^{1} = X_{n}. The Kuga-Sato variety X_{n}^{k} is smooth and quasi-projective over ℚ, of dimension k + 1.

In analogy with the modular description of X_{n}, there exists a modular interpretation of X_{n}^{k}: its S-valued points are in 1-1 correspondence with (k+2)-tuples (E/S, α_{n}, P_{1}, …, P_{k}) where (E/S, α_{n}) is an elliptic curve with level n structure and P_{i} ∈ E(S) are k sections.

5.2.2 Change of level

Suppose that m, n ≥ 3 and m|n. For any S, we can associate to each pair (E/S, α_{n}) ∈ M_{n}(S) the pair (E/S, [×\frac{n}{m}] ∘ α_{n}) ∈ M_{m}(S). This defines a morphism

ρ_{n,m}: M_{n} → M_{m}.

Explicitly, ρ_{n,m} is the classifying map for the elliptic curve with level m structure (X_{n}/M_{n}, [×\frac{n}{m}] ∘ α_{n}^{univ}).

There is more than one natural way to find a morphism X_{n} → X_{m} which covers ρ_{n,m}. The most obvious construction is the following: consider the morphism of functors

X_{n}(S) → X_{m}(S)
(E/S, α_{n}, u) ↦ (E/S, [×\frac{n}{m}] ∘ α_{n}, u)

This induces a morphism which we will denote σ_{n,m}^{naive} : X_{n} → X_{m}. Another, and in practice more useful, choice is the morphism σ_{n,m}: X_{n} → X_{m}, where

σ_{n,m}(S): (E/S, α_{n}, u) ↦ (E/S, [×\frac{n}{m}] ∘ α_{n}, (\frac{n}{m})u).

These morphisms fit into a commutative diagram

(ℤ/n)_{/M_{n}}^{2} reduction mod m ⟶ (ℤ/m)_{/M_{n}}^{2} ⟶^{id×ρ_{n,m}} (ℤ/m)_{/M_{m}}^{2}

α_{n}^{univ}↓^{≀}  ↓^{≀}  ↓^{α_{m}^{univ}}
X_{n}[n]  ⟶  X_{n}[m]  X_{m}[m]
∩  ∩  ∩
X_{n}  ⟶^{[× \frac{n}{m}]}  X_{n}  ⟶^{σ_{n,m}^{naive}}  X_{m}
σ_{n,m}
f_{n} ↘  ↓^{f_{n}}  ↓^{f_{m}}
M_{n}  ⟶_{ρ_{n,m}}  M_{m}

(5.2.2.1)

in which the bottom right-hand square is Cartesian, and σ_{n,m} is the composite

σ_{n,m} = σ_{n,m}^{naive} ∘ [×\frac{n}{m}]: X_{n} → X_{m}.

The morphisms ρ_{n,m}, σ_{n,m}^{naive}, σ_{n,m} are all finite and étale. For the first two this follows from Proposition 5.1.3.1 above and for σ_{n,m} it follows because [×\frac{n}{m}] is finite étale as well.

5.2.3 Group actions

We first define some group schemes over ℤ which will be used in the sequel.

Definition 5.2.3.1. • G denotes the group scheme GL_{2} over ℤ.

• G^{(1)} denotes the semidirect product 𝔾_{a}^{2} ×G, where G acts on 𝔾_{a}^{2} by left multiplication: so for any ring R

G^{(1)}(R) = \{(a, γ) | a = \binom{a_{1}}{a_{2}} ∈ R^{2}, γ ∈ G(R)\}

with multiplication

(a, γ)(a′, γ′) = (a + γa′, γγ′)

• G^{(k)} is the semidirect product 𝔾_{a}^{2k} ×G, with

G^{(k)}(R) = {(a_{1}, …, a_{k}, γ) | a_{1}, …, a_{k} ∈ R^{2}, γ ∈ G(R)}

Next we define actions (on the right) of G(ℤ/nℤ) on M_{n} and G^{(1)}(ℤ/nℤ) on X_{n} which are compatible with the projection G^{(1)}(ℤ/nℤ) → G(ℤ/nℤ). In terms of the functors represented, the actions are as follows:

g ∈ G(ℤ/nℤ), (E/S, α_{n}) ∈ M_{n}(S)

(E/S, α_{n})g =^{def} (E/S, α_{n} ∘ g);

(a, g) ∈ G^{(1)}(ℤ/nℤ), (E/S, α_{n}, P) ∈ X_{n}(S)

(E/S, α_{n}, P)(a, g) =^{def} (E/S, α_{n} ∘ g, P + α_{n}(a)).

We check at once:

((E/S, α_{n}, P)(a, g))(a′, g′) = (E/S, α_{n} ∘ g, P + α_{n}(a))(a′, g′)
= (E/S, α_{n} ∘ g ∘ g′, P + α_{n}(a) + α_{n} ∘ g(a′))
= (E/S, α_{n}, P)(a + ga′, gg′)

so that we do indeed have an action of the semidirect product.

Proposition 5.2.3.2. Let m, n ≥ 3 with m|n. Let g ∈ G(ℤ/nℤ), a ∈ (ℤ/n)^{2}. Then the following diagrams are commutative:

\begin{matrix}M_{n} & ⟶^{ρ_{n,m}} & M_{m} \\ ↓^{g} & & ↓^{gmodm} \\ M_{n} & ⟶^{ρ_{n,m}} & M_{m} \end{matrix} X_{n} ⟶^{σ_{n,m}} X_{m}

↓^{(a,g)}  ↓^{(amodm,gmodm)}
X_{n} ⟶^{σ_{n,m}} X_{m}

If a = (n/m)b for some b ∈ (ℤ/m)^{2}, then the following diagram commutes:

X_{n} ⟶^{σ_{n,m}^{naive}} X_{m}
↓^{(a,g)}  ↓^{(b,gmodm)}
X_{n} ⟶^{σ_{n,m}^{naive}} X_{m}

Proposition 5.2.3.3. (i) ρ_{n,m}: M_{n} → M_{m} identifies M_{m} with the quotient of M_{n} by K_{n,m}.

(ii) σ_{n,m}^{naive} : X_{n} → X_{m} identifies X_{m} with the quotient X_{n}/K_{n,m}.

(iii) σ_{n,m}: X_{n} → X_{m} identifies X_{m} with the quotient of X_{n} by

K_{n,m}^{(1)} =^{def} ker [G^{(1)}(ℤ/nℤ) → G^{(1)}(ℤ/mℤ)] .

(i), (ii) Let x = (E/S, α_{m}) ∈ M_{m}(S). Then ρ_{n,m}^{−1}(x) is the set of level n structures on E/S compatible with α_{m}, which by Proposition 5.1.3.1 above is the set of points of a K_{n,m}-torsor. Hence M_{m} = M_{n}/K_{n,m}, and the same argument applies to σ_{m,n}^{naive}.

(iii) Since σ_{n,m} = σ_{n,m}^{naive}∘[×n/m], and [×n/m] identifies X_{n} with its quotient by the subgroup scheme (mℤ/nℤ)^{2} ⟶^{∼} X_{n}[n/m], this part follows from (ii).

5.2.4 Generalities concerning inverse limits

We need to review some elementary facts about projective limits in the category of schemes. The basic reference for this material is [EGA IV,8.1]. Although the results hold in greater generality we will consider only index sets I which are partially ordered and directed: this means precisely that we have a binary relation ≤ on I satisfying the axioms:

• if i, j, k ∈ I and i ≤ j, j ≤ k, then i ≤ k;

• if i, j ∈ I with i ≤ j and j ≤ i then i = j;

• if i, j ∈ I then there exists k ∈ I with i ≤ k and j ≤ k.

Such an ordered set I may be regarded as a category in the usual way (with Hom(i, j) a singleton if i ≤ j and empty otherwise), and a projective system in a category C is then a contravariant functor I^{opp} → C. Explicitly, it is a collection of objects X_{i}, one for each i ∈ I, and morphisms λ_{ij}: X_{j} → X_{i}, one for each pair (i, j) with i ≤ j, such that:

• λ_{ii} = id_{X_{i}} for every i ∈ I; and

• λ_{ij} ∘ λ_{jk} = λ_{ik} whenever i ≤ j ≤ k.

Recall that an inverse limit of such a projective system is an object X of C which represents the functor

Y → lim_{⟵}_{i} Hom_{C}(Y, X_{i}).

Now let S be a fixed scheme, and (X_{i}, λ_{ij}) a projective system of S-schemes and S-morphisms. Assume:

There exists i_{0} ∈ I such that for every i ≥ i_{0} the morphism λ_{i_{0}i}: X_{i} → X_{i_{0}} is affine.

(5.2.4.1)

Then X_{i} = Spec A_{i} for some direct system of O_{X_{i_{0}}} -algebras A_{i}, i ≥ i_{0}. From [EGA IV, 8.2.3, 8.2.5, 8.2.9] we have:

Proposition 5.2.4.1. (i) X =^{def} Spec(lim_{⟶}_{i≥i_{0}} A_{i}) is the inverse limit of (X_{i}) in

the category of S-schemes (and also in the category of all schemes).

(ii) The underlying topological space sp(X) of X is the inverse limit (in the category of topological spaces) of (sp(X_{i})).

(iii) If Y_{i_{0}} → X_{i_{0}} is any morphism, write

Y_{i} =^{def} Y_{i_{0}} ×_{X_{i_{0}}} X_{i}, i ≥ i_{0}.

Then the inverse limit of (Y_{i}) exists and equals Y_{i_{0}} ×_{X_{i_{0}}} X.

By the definition of inverse limit, we have

Hom_{S}(T, lim_{⟵}_{i}(X_{i})) = lim_{⟵}_{i} Hom_{S}(T, X_{i})

and in general not much can be said about Hom_{S}(lim_{⟵}_{i}(X_{i}), T), unless we make further hypotheses.

Proposition 5.2.4.2 ([EGA IV.8.13.2]). Let (X_{i})_{i∈I}, (Y_{j})_{j∈J} be projective systems of S-schemes satisfying (5.2.4). There is a natural homomorphism

lim_{⟵}_{j}(lim_{⟶}_{i} Hom_{S}(X_{i}, Y_{j})) → lim_{⟵}_{j} Hom_{S}(X, Y_{j}) = Hom_{S}(X, Y)

which is an isomorphism if each X_{i} is quasi-compact and quasi-separated, and each Y_{j} is locally of finite presentation over S.

5.2.5 Interlude: elliptic curves up to isogeny

Throughout this section, S will be a fixed quasi-compact scheme over ℚ. Let (Ell/S) be the category of elliptic curves over S and S-homomorphisms.

Definition 5.2.5.1. The category (Ell-Isog/S) of elliptic curves up to isogeny over S is the category whose objects are E ⊗ℚ, where E is an object of (Ell/S), and morphisms

Hom_{S}(E ⊗ ℚ, E′ ⊗ ℚ) = Hom_{S}(E, E′) ⊗ ℚ

The inclusion Hom_{S}(E, E′) ⊂ Hom_{S}(E, E′) ⊗ ℚ defines a functor from (Ell/S) to (Ell-Isog/S). If f: E → E′ is any isogeny of elliptic curves over S, then since S is quasi-compact there is an integer n ≥ 1 and an isogeny f′: E′ → E such that f′ ∘ f = [×n]_{E}, f ∘ f′ = [×n]_{E′}. Hence the functor transforms isogenies to isomorphisms, and conversely any homomorphism whose image in (Ell-Isog/S) is an isomorphism is already an isogeny. In other words, (Ell-Isog/S) is obtained from (Ell/S) by formally inverting all isogenies.

There is a concrete realisation of (Ell-Isog/S). We first need a rather general construction. Let C be an additive category, and X an object of C. For every n ∈ ℤ we have an endomorphism [×n]_{X} of X. Consider the set I of positive integers ordered by divisibility, and let (X_{n}, [×n/m]) be the following projective system indexes by I:

• X_{n} = X for all n ≥ 1;

• if m|n the corresponding morphism X_{n} → X_{m} is [×n/m]_{X}.

We will denote such an inverse system ((X)_{n}, [×n/m]_{X}).

Definition 5.2.5.2. If the limit exists, we write

X̃ = lim_{⟵}_{n}((X)_{n}, [×n/m]_{X}).

The association X → X̃ is clearly functorial, so we have a homomorphism

Hom_{C}(X, Y) → Hom_{C}(X̃, Ỹ).  (5.2.5.1)

whenever X̃ and Ỹ both exist. Let d ≥ 1 be an integer. Then the morphisms for n ≥ 1

(X)_{n} ⟶^{id_{X}} (X)_{n}d

define a morphism of projective systems

((X)_{n}, [×n/m]_{X}) → ((X)_{n}, [×n/m]_{X})

whose composite with [×d] induces the identity on X̃. Therefore the map (5.2.5) extends to a homomorphism

Hom_{C}(X, Y) ⊗ ℚ → Hom_{C}(X̃, Ỹ).  (5.2.5.2)

Under certain circumstances this is an isomorphism. For example, let C be the category of group schemes over S, and let X, Y be elliptic curves over S (or more generally abelian schemes). Then since multiplication by n ≥ 1 is finite (and therefore affine), X̃, Ỹ exist by Proposition 5.2.4.1. Then by Proposition 5.2.4.2,

Hom_{S}(X̃, Ỹ) = lim_{⟵}_{n′}(lim_{⟶}_{n} Hom_{S}(X, Y))

where the inverse (resp. direct) limit on the right hand side is taken with respect to composition with [×n/m]_{X} (resp. [×n′/m′]_{Y}). Now for any abelian group G, the direct limit of the system ((G)_{n}, [×n/m]_{G}) is G⊗ℚ, hence the transition morphisms in the inverse system are all isomorphisms, and thus (5.2.5) is an isomorphism.

Definition 5.2.5.3. The category (pro-Ell/S) of pro-elliptic curves over S is the category of group schemes over S isomorphic to Ẽ, for some elliptic curve E/S, and group scheme homomorphisms.

We have a functor E → Ẽ from (Ell/S) to (pro-Ell/S). By the lemma, this gives an equivalence of between the categories (pro-Ell/S) and (Ell-Isog/S).

Certain constructions can be carried out very nicely using the concept of pro-elliptic curve. For example, let E/S be an (usual) elliptic curve. The projection onto the (n = 1)-component of the inverse system ((E)_{n}, [×n/m]) is a morphism of group schemes proj_{1} : Ẽ → E. By Proposition 5.2.4.1(iii), we get immediately:

Proposition 5.2.5.4. (i) T̂(E/S) is canonically isomorphic to proj_{1}^{−1}(e) ⊂ Ẽ. (ii) V_{f}(E/S) is canonically isomorphic to the fibre product E[tors]×_{E,proj_{1}} Ẽ,

and also to { \atop E[tors]} (the result of applying the ˜ -construction to E[tors]).

(iii) There is a canonical isomorphism

Ẽ/T̂(E/S) ⟶^{∼} E

which induces on “torsion” subgroups the isomorphism V_{f}(E/S)/T̂(E/S) = E[tors].

In particular, V_{f}(E/S) depends functorially only on the pro-elliptic curve Ẽ, so we will often write it as Ẽ[tors] to indicate this. One can think of it as the “pro-torsion subscheme” of Ẽ (although it is not a subscheme, only a direct limit of subschemes).

Definition 5.2.5.5. Let Ẽ be a pro-elliptic curve over S. A complete level structure on Ẽ is an isomorphism

α_{f}: 𝔸_{f}^{2}_{/S} ⟶^{∼} Ẽ[tors]

Proposition 5.2.5.6. There is a bijection

\left\{{isomorphism classes of pairs (E/S, α_{∞}), where E/S is an ellip- \atop tic curve and α_{∞} is a complete level structure on E}\right\}
↕
\left\{{isomorphism classes of pairs (Y/S, α_{f}), where Y/S is a pro- \atop elliptic curve and α_{f} is a complete level structure on Y}\right\}

given by (E/S, α_{∞}) → (Y, α_{f}), where Y = Ẽ/S and

α_{f} = α̃_{∞}: 𝔸_{f}^{2} = { \atop (ℚ/ℤ)^{2}} ⟶^{∼} { \atop E[tors]}

We have to show that if (Y/S, α_{f}) is a pro-elliptic curve with level structure, then Y = Ẽ for some elliptic curve E/S and α_{f} = α̃_{∞} for some complete level structure α_{∞} on E. We may assume that S is connected. Then α_{f}(ℤ̂_{S}^{2}) ⊂ V_{f}(Y/S) is a constant closed subgroup scheme. We want to show that the quotient Y/α_{f}(ℤ̂_{S}^{2}) is an elliptic curve. In fact, write Y = F̃ for some elliptic curve F/S. Then the subgroups α_{f}(ℤ̂_{S}^{2}) and T̂(F/S) ⊂ V_{f}(Y/S) are commensurable, so as Y/T̂(F/S) exists and equals F, the quotient E = Y/α_{f}(ℤ̂_{S}^{2}) exists and is isogenous to F. It is clear that α_{f} induces an isomorphism α_{∞}: (ℚ/ℤ)_{S}^{2} ⟶^{∼} E[tors], and this sets up the required bijection.

5.2.6 The inverse limit as an inverse limit

We have already defined (Definition 5.2.2) for each pair m|n with m, n ≥ 3 the morphisms ρ_{n,m}: M_{n} → M_{m}, σ_{n,m}^{naive}, σ_{n,m}: X_{n} → X_{m}. We may therefore form the inverse limits over integers ≥ 3 ordered by divisibility:

M =^{def} lim_{⟵}_{n}((M_{n})_{n}, ρ_{n,m})

X^{naive} =^{def} lim_{⟵}_{n}((X_{n})_{n}, σ_{n,m}^{naive})

X =^{def} lim_{⟵}_{n}((X_{n})_{n}, σ_{n,m})

Both X^{naive} and X are schemes over M. Since the square

X_{n} ⟶^{σ_{n,m}^{naive}} X_{m}
f_{n}↓  ^{f_{m}}↓
M_{n} ⟶^{ρ_{n,m}} M_{m}

is Cartesian (5.2.2) we have isomorphisms for every n ≥ 3

X^{naive} ⟶^{∼} X_{n} ×_{M_{n}} M

and in particular X^{naive} is an elliptic curve over M. Also, since σ_{n,m} = [×n/m]∘ σ_{n,m}^{naive}, we have a canonical morphism X → X^{naive} which on the components at level n is [×n]. Therefore this induces an isomorphism

X = lim_{⟵}_{n}((X)_{n}, [×n/m]) = { \atop X^{naive}};

therefore X is a pro-elliptic curve over M.

We clearly have a modular description of the inverse limits, by taking the inverse limits of the functors of points of M_{n} and X_{n}:

M(S) = {isomorphism classes of pairs (E/S, α_{∞}) where E/S is an elliptic curve and α_{∞} is a complete level structure on E/S}

X^{naive}(S) = {isomorphism classes of triples (E/S, α_{∞}, u) where (E/S, α_{∞}) ∈ M(S) and u ∈ E(S) is a section}

X(S) = {isomorphism classes of triples (E/S, α_{∞}, ũ) where (E/S, α_{∞}) ∈ M(S) and ũ ∈ Ẽ(S) is a section}

From this we deduce actions of G(ℤ̂) on M, (ℚ/ℤ)^{2} ×G(ℤ̂) on X^{naive}, and G^{(1)}(ℤ̂) = ℤ̂^{2} ×G(ℤ̂) on X, as follows: let g ∈ G(ℤ̂), a ∈ ℤ̂^{2}, b ∈ (ℚ/ℤ)^{2}. Then:

• (E/S, α_{∞})g = (E/S, α_{∞} ∘ g) ∈ M(S);

• (E/S, α_{∞}, u)(b, g) = (E/S, α_{∞} ∘ g, u + α_{∞}(b)) ∈ X^{naive}(S);

• (E/S, α_{∞}, ũ)(a, g) = (E/S, α_{∞} ∘ g, u + α̂(a)) ∈ X(S);

Here α̂: ℤ̂_{S}^{2} ⟶^{∼} T̂(E/S) and α_{∞} = α̂ ⊗ ℚ/ℤ: (ℚ/ℤ)_{S}^{2} ⟶^{∼} E[tors] are the two alternative descriptions (Definitions 5.1.1.2) of the complete level structure on E/S.

Consider the subgroup ℤ̂^{2} ⊂ G^{(1)}(ℤ̂); it acts on X by translation by sections belonging to the subgroup T̂(X^{naive}/M) ⊂ V_{f}(X^{naive}/M) = X[tors]. Therefore the projection X → X^{naive} identifies X^{naive} with the quotient of X by ℤ̂^{2}.

5.2.7 The inverse limit not as an inverse limit

We can give a nicer description of X and M using pro-elliptic curves. Combining the description of the inverse limit in the previous section and the bijection of Proposition 5.2.5.6 gives the following:

Proposition 5.2.7.1. (i) The scheme M represents the functor on quasi-compact S/ℚ

S → {isomorphism classes of pairs (Y/S, α_{f}) where Y/S is a pro-elliptic curve and α_{f} is a com- plete level structure on Y/S}

(ii) The scheme X represents the functor on quasi-compact S/ℚ

S → {isomorphism classes of triples (Y/S, α_{f}, u) where (Y/S, α_{f}) ∈ M(S) and u ∈ Y (S) is a section}

Using this description we can define actions of the finite adelic groups G_{f} = G(𝔸_{f}) and G_{f}^{(1)} = G^{(1)}(𝔸_{f}) on M and X: if g ∈ G_{f} and a ∈ 𝔸_{f}^{2}, then

(Y/S, α_{f})g = (Y/S, α_{f} ∘ g) ∈ M(S);
(Y/S, α_{f}, u)(a, g) = (Y/S, α_{f} ∘ g, u + α(a)) ∈ X(S);

These actions are compatible with the actions of G(ℤ̂), G^{(1)}(ℤ̂) defined in the last section. The normaliser of ℤ̂^{2} ⊂ G_{f}^{(1)} is 𝔸_{f}^{2} ×G(ℤ̂), and so acting on the quotient X/ℤ̂^{2} = X^{naive} is the quotient of this normaliser by ℤ̂^{2}. In other words, we obtain an action of (𝔸_{f}^{2} ×G(ℤ̂))/ℤ̂^{2} = (ℚ/ℤ)^{2} ×G(ℤ̂) on X^{naive}, which is the same as that defined in the previous section.

It is because of the action of G_{f} that X is a much better object than X^{naive}.

5.3 Complex points

In this section we will write down explicit descriptions of the complex points on the modular varieties and the various maps between them. Unfortunately there is more than one way to do this. It has become traditional in the literature on Shimura varieties to represent a modular variety as the quotient of a discrete group acting on the left of some complex manifold, and we will conform to that convention here. There is still more than one way to write down the parameterisation; we will give two, and indicate how to pass from one to the other.

In [Be*] and [Be**] the opposite convention is used—the complex parameterisation of modular varieties given there has the discrete group acting on the right. This means that unfortunately all the formulae we give will later differ from the ones there, and the translation from one to the other is an exercise which, although essentially trivial, admits a high probability of error. We give below some hints in this direction.

5.3.1 Complex points at finite level

Throughout we will use the notation H^{±} = ℂ − ℝ for the union of the upper and lower half-planes, and H for the upper half-plane when needed. We always take the left action of G(ℝ) on H^{±} given by linear fractional transformations:

(\begin{matrix}a & b \\ c & d \end{matrix}) : τ ↦ \frac{aτ + b}{cτ + d}

and we define

j(γ, τ) =^{def} det(γ)(cτ + d)^{−1} for all τ ∈ H^{±} and γ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ G(ℝ)

which satisfies the relations

j(γγ′, τ) = j(γ, γ′(τ))j(γ′, τ), j(γ, τ) (1 −τ) = (1 −γ(τ)) γ

Next we define a left action of G^{(1)}(ℝ) = ℝ^{2} ×G(ℝ) on ℂ × H^{±} by

(\binom{v_{1}}{v_{2}}, γ): (u, τ) ↦ (j(γ, τ)u + v_{1} − v_{2}γ(τ), γ(τ)).

The reader will check that this is indeed an action of the semidirect product. By combining this with the action by left multiplication, we obtain actions on the left of G(ℤ) on H^{±} × G(ℤ/nℤ) and of G^{(1)}(ℤ) on ℂ × H^{±} × G^{(1)}(ℤ/nℤ). Now we define some complex manifolds and morphisms between them:

M_{n}^{an} = G(ℤ)\H^{±} × G(ℤ/nℤ)  (5.3.1.1)
X_{n}^{an} = G^{(1)}(ℤ)\ℂ × H^{±} × G^{(1)}(ℤ/nℤ)
f_{n}^{an} : X_{n}^{an} → M_{n}^{an}, f_{n}^{an}(u, τ, a, g) = (τ, g)
ρ_{n,m}^{an} : M_{n}^{an} → M_{m}^{an}, ρ_{n,m}^{an}(τ, g) = (τ, g (mod m))
σ_{n,m}^{an} : X_{n}^{an} → X_{m}^{an}, σ_{n,m}^{an}(u, τ, a, g) = (u, τ, a (mod m), g (mod m))
σ_{n,m}^{naive,an} : X_{n}^{an} → X_{m}^{an}, σ_{n,m}^{an}(u, τ, a, g) = (mu/n, τ, a (mod m), g (mod m))
det^{an} : M_{n}^{an} → (ℤ/nℤ)^{∗}, det^{an}(τ, g) = ±det(g) if ± Im(τ) > 0.

It is easy to check that these maps are well-defined, and in particular that every point on the fibre f_{n}^{an−1}(τ, g) ⊂ X_{n}^{an} has a representative of the form (u, τ, a, g). The fibres of the map det^{an} are readily seen to be copies of the “classical” modular curve

Γ(n)\H, Γ(n) =^{def} {γ ∈ SL_{2}(ℤ) | γ ≡ 1 (mod n)}

In particular, the connected components of X^{an} and M^{an} are indexed by (ℤ/nℤ)^{∗}. If (τ, g) ∈ M_{n}^{an} we define a map λ_{τ,g}: ℂ → f_{n}^{an−1}(τ, g) by λ_{τ,g}(u) = (u, τ, 0, g).

Lemma 5.3.1.1. λ_{τ,g} is surjective, and identifies f_{n}^{an−1}(τ, g) with the quotient ℂ/(nℤ + nτℤ).

We define a group operation X_{n}^{an} ×_{M_{n}^{an}} X_{n}^{an} → X_{n}^{an} by

(u, τ, a, g) ⊕ (u′, τ, a′, g) = (u + u′, τ, a + a′, g)

which makes X_{n}^{an} a group manifold over M_{n}^{an}, whose fibres are the complex tori ℂ/(nℤ + nτℤ). We also define for each a ∈ (ℤ/n)^{2} a section a^{∗} of f_{n}^{an} by

a^{∗}: (τ, g) ↦ (0, τ, ga, g).

For each (τ, g) ∈ M_{n}^{an}, the sections a^{∗} determine an isomorphism of (ℤ/n)^{2} with the points of order dividing n of the fibre f_{n}^{an−1}(τ, g). Thus we obtain an isomorphism

α_{n}^{univ,an} : (ℤ/n)^{2} × M_{n}^{an} ⟶^{∼} ker ([×n]: X_{n}^{an} → X_{n}^{an})

by α_{n}^{univ,an}(a, (τ, g)) = a^{∗}(τ, g). We also have right actions on G(ℤ/nℤ) and G^{(1)}(ℤ/nℤ) on M_{n}^{an}, X_{n}^{an} respectively, given by right multiplication. The isomorphism α_{n}^{univ,an} is then G(ℤ/nℤ)-equivariant, if we define the action of G(ℤ/nZ) on (ℤ/n)^{2} × M_{n}^{an} as being the product of the action just defined on M_{n}^{an} and the right action g: a ↦ ag^{−1} on (ℤ/n)^{2}.

Lemma 5.3.1.2. Let (τ, g) ∈ M_{n}^{an}, and consider the level n structure α_{n}^{univ,an} on f_{n}^{an−1}(τ, g). In terms of the isomorphism of Lemma 5.3.1.1 it is given by

a ↦ −b_{1} + b_{2}τ ∈ ℂ/nℤ + nτℤ, b = ga ∈ (ℤ/nℤ)^{2}.

Theorem 5.3.1.3. There are canonical isomorphisms M_{n}(ℂ) ⟶^{∼} M_{n}^{an} and X_{n}(ℂ) ⟶^{∼} X_{n}^{an}, which are equivariant with respect to the actions of G(ℤ/nℤ), G^{(1)}(ℤ/nℤ), and which transform f_{n}, ρ_{n,m}, α_{n}^{univ}, … into f_{n}^{an}, ρ_{n,m}^{an}, α_{n}^{univ,an}, … and such that the diagram

M_{n}(ℂ)  ⟶^{∼} M_{n}^{an}
↓^{det}  ↓_{det}^{an}
(Spec ℚ(ζ_{n}))(ℂ) ⟵^{∼} (ℤ/nℤ)^{∗}
∪  ∪
e^{−2πin/a}  ←  a

is commutative.

We will defer proving this until the next section, when we will check the analogous assertion at infinite level (from which the present result follows at once by passage to the quotient).

5.3.2 Complex points at infinite level

We define now:

M^{an} = G(ℚ)\H^{±} × G(𝔸_{f})
X^{an} = G^{(1)}(ℚ)\ℂ × H^{±} × G^{(1)}(𝔸_{f})
f^{an} : X^{an} → M^{an}, f^{an}(u, τ, a, g) = (τ, g).

For the moment we regard these simply as topological spaces, with obvious right actions of G_{f} and G_{f}^{(1)} respectively. Noticing that if H is either G or G^{(1)}, then

H(𝔸_{f}) = H(ℚ)H(ℤ̂) and H(ℚ) ∩ H(ℤ̂) = H(ℤ),

we see that the natural maps

G(ℤ)\H^{±} × G(ℤ̂) → M^{an}
G^{(1)}(ℤ)\ℂ × H^{±} × G^{(1)}(ℤ̂) → X^{an}

are isomorphisms. Hence

M_{n}^{an} = M^{an}/K_{n}, X_{n}^{an} = X^{an}/K_{n}^{(1)}

and

M^{an} = lim_{⟵}_{n} M_{n}^{an}, X^{an} = lim_{⟵}_{n} X_{n}^{an}

where the inverse limits are with respect to the transition morphisms ρ_{n,m}^{an}, σ_{n,m}^{an}. Therefore M^{an} and X^{an} are pro-analytic spaces. The action of the subgroup G(ℤ̂) ⊂ G_{f} on M^{an} is the inverse limit of the action of the finite groups G(ℤ/nℤ) on the M_{n}^{an}’s, and likewise for X^{an}.

Theorem 5.3.2.1. There exists a commutative diagram

X(ℂ) ⟶^{Ξ} X^{an}
f↓  ^{f^{an}}↓
M(ℂ) ⟶^{Θ} M^{an}

where Ξ and Θ are isomorphisms which are equivariant with respect to the actions of G_{f}^{(1)} and G_{f}, and which transform the universal level structure α_{f}^{univ} on X/M into the level structure on X^{an}/M^{an}

α_{f}^{an} : 𝔸_{f}^{2} × M^{an} → X^{an}
(a, (τ, g)) ↦ (0, τ, ga, g)

The proof of this comparison is not difficult but we will perform it carefully to ensure that the normalisations are all correct. We will first construct a parameterisation of a covering of the moduli problem X. First we make some general remarks about pro-elliptic curves over ℂ.

Let Y be a pro-elliptic curve over ℂ; the space H^{0}(Y, Ω_{Y}) is 1-dimensional [**]. If we choose a presentation Y = lim_{⟵} E_{n}, where E_{n} are elliptic curves, then

H^{0}(Y, Ω_{Y}) is canonically isomorphic to each of the spaces H^{0}(E_{n}, Ω_{E_{n}}). Its dual Lie(Y) is the tangent space at the origin to Y, considered as a complex Lie group, which in turn is canonically isomorphic to the tangent space of each of the E_{n}’s at the origin. Therefore there is an exponential map

exp: H^{0}(Y, Ω_{Y})^{∗} → Y (ℂ)

which is the inverse limit of the exponential maps of the E_{n}’s, and which depends functorially only on Y. Its image is dense, since its composition with any of the projections Y → E_{n} is surjective.

We also have an inclusion V_{f}(Y) ↪ Y (ℂ). The sum of these two maps is therefore a canonical homomorphism

exp_{𝔸}: Lie(Y) × V_{f}(Y) → Y (ℂ)

of topological groups, whose source is a free 𝔸_{ℚ}-module of rank 2.

If W is a free 𝔸_{ℚ}-module of rank m, recall that a lattice in W is a ℚ-subspace L ⊂ W (necessarily of dimension m) such that the canonical map L⊗_{ℚ} 𝔸_{ℚ} → W is an isomorphism. (Equivalently, it is a ℚ-subspace of W which is discrete and cocompact.)

Lemma 5.3.2.2. exp_{𝔸} is surjective, and its kernel is a lattice in Lie(Y) × V_{f}(Y).

We now set up the moduli problem. Consider a quintuple (Y, ω, δ, α_{f}, P), where:

• Y is a pro-elliptic curve over ℂ;

• ω ∈ H^{0}(Y, Ω_{Y}) is a non-zero differential form;

• δ: Λ =^{def} ker(exp_{𝔸}) ⟶^{∼} ℚ^{2} is an isomorphism;

• α_{f}: 𝔸_{f}^{2} ⟶^{∼} V_{f}(Y) is a complete level structure on Y; and

• P ∈ Y (ℂ) is a point.

The set of isomorphism classes of such quintuples is clearly a covering of X(ℂ), the map being simply “forget ω and δ”. This data gives rise to a commutative diagram, which we will first write down and then carefully explain:

𝔸_{ℚ}^{2}
↗  ∪  ↖
ℝ^{2}  ℚ^{2}  𝔸_{f}^{2}
(ω_{1} ω_{2})↓^{≀}  ^{δ}↑^{≀}  ^{g}↑^{≀}
δ_{ℝ} ≀  ℂ  Λ  𝔸_{f}^{2}  ^{≀ δ_{f}}
∫ ω↓^{≀}  ∩  ^{α_{f}} ↓^{≀}
\overline{}Lie(Y)  Λ ⊗_{ℚ} 𝔸_{ℚ}  V_{f}(Y) \overline{}
↘  ∥  ↙
Lie(Y) × V_{f}(Y)

The middle vertical arrows come from the choice of δ and the maps just decribed. Here δ_{𝔸} = δ ⊗_{ℚ} 𝔸 is an isomorphism since Λ is a lattice. So the real and finite adelic components of δ_{𝔸} are isomorphisms

δ_{ℝ}: Lie(Y) ⟶^{∼} ℝ^{2}, δ_{f}: V_{f}(Y) ⟶^{∼} 𝔸_{f}^{2}

We define g = δ_{f} ∘α_{f} ∈ G_{f} to make the right hand part of the diagram commute.

We already have an isomorphism ∫ ω: Lie(Y) ⟶^{∼} ℂ which is the transpose of the map ℂ ⟶^{∼} H^{0}(Y, Ω_{Y}), 1 ↦ ω. Therefore there exist ω_{1}, ω_{2} ∈ ℂ^{∗} with ω_{2}/ω_{1} ∉ ℝ making the left hand part of the diagram commute; explicitly, ω_{i} is the “period”

ω_{i} = ∫_{δ_{ℝ}^{−1}(e_{i})} ω

where e_{i} ∈ ℚ^{2} are the standard basis elements. We thus obtain a parameterisation of Y:

ϕ: Y (ℂ) ⟶^{∼} \frac{Lie(Y) × V_{f}(Y)}{Λ} ⟶^{∼} (∫ ω, α_{f}^{−1})\frac{ℂ × 𝔸_{f}^{2}}{L}

for some lattice L ⊂ ℂ × 𝔸_{f}^{2}. The image of the basis {δ^{−1}(e_{i})} of Λ is the basis {(ω_{i}, g^{−1}e_{i})} of L.

Let (P_{ℝ}, P_{f}) ∈ Lie(Y) × V_{f}(Y) be any point whose image under exp_{𝔸} is P ∈ Y (ℂ). To the quintuple (Y, ω, δ, α_{f}, P) we associate

(z, ω_{1}, ω_{2}, a, g) ∈ ℂ × ℂ^{∗} × ℂ^{∗} × 𝔸_{f}^{2} ×G_{f}

where ω_{i}, g are as above, z = ∫_{0}^{P_{ℝ}} ω = (ω_{1} ω_{2}) (δ_{ℝ}(P_{ℝ})), and a = δ_{f}(P_{f}) = gα_{f}^{−1}(P_{f}). This association is well-defined apart from the choice of lifting (P_{ℝ}, P_{f}) of P, so we have a well-defined point of the quotient

Z^{an} =^{def} ℚ^{2}\ℂ × (ℂ^{∗})^{2} × 𝔸_{f}^{2} ×G_{f} − {ω_{2}/ω_{1} ∈ ℝ}

where the action of ℚ^{2} is

t: (z, ω_{1}, ω_{2}, a, g) ↦ (z + ∑ t_{i}ω_{i}, ω_{1}, ω_{2}, a + t, g).

There is a bijection between Z^{an} and the set of isomorphism classes of quintuples (Y, ω, δ, α_{f}, P). On both sets we have an action of G_{f}^{(1)} on the right—on Z^{an} it is given by right multiplication, and on the moduli space by

(Y, ω, δ, α_{f}, P)(b, γ) = (Y, ω, δ, α′_{f}, P′), α′_{f} = α_{f}γ, P′ = P + α_{f}(b).

Since δ_{f} ∘ α_{f} = g and a = δ_{f}(P_{f}) we obtain from (Y, ω, δ, α′_{f}, P′) the point (z, ω_{1}, ω_{2}, a′, g′) with a′ = δ_{f}(P_{f} + α_{f}(b)) = a + gb, g′ = gγ, so the bijection is G_{f}^{(1)}-equivariant.

We have two further group actions to consider in order to pass to the quotient X^{an}. Firstly, if γ ∈ G(ℚ) we can replace δ by δ′ = γδ. Then (z, ω_{1}, ω_{2}, a, g) is replaced by (z′, ω′_{1}, ω′_{2}, a′, g′) where

(ω′_{1} ω′_{2}) = (ω_{1} ω_{2}) γ^{−1} = \frac{1}{detγ} (\begin{matrix}dω_{1} − cω_{2} \\ −bω_{1} + aω_{2} \end{matrix}) if γ = (\begin{matrix}a & b \\ c & d \end{matrix})

z′ = z, a′ = γa, g′ = γg

This is clear since g = δ_{f} ∘ α_{f} and (ω_{1} ω_{2}) ∘ δ_{ℝ} = ∫ ω = (ω′_{1} ω′_{2}) ∘ δ′_{ℝ}.

Next, if λ ∈ ℂ^{∗} we may replace ω by ω′ = λω, which transforms our point into (z′, ω′_{1}, ω′_{2}, a′, g′) where

z′ = λz, ω′_{i} = λω_{i}, a′ = a, g′ = g.

Consider the mapping ψ: Z^{an} → X^{an} given by

ψ: (z, ω_{1}, ω_{2}, a, g) ↦ (u, τ, a, g) with u = \frac{z}{ω_{1}} , τ = −\frac{ω_{2}}{ω_{1}} .

It is well-defined, since if z′ = z + ∑ t_{i}ω_{i}, a′ = a + t then

ψ(z′, ω_{1}, ω_{2}, a′, g) = (u′, τ′, a′, g)

with

u′ = \frac{z + ∑ t_{i}ω_{i}}{ω_{1}} = u + t_{1} − t_{2}τ.

So the action of ℚ^{2} is compatible with the one used to define the quotient X^{an}. The fibres of ψ are obviously preserved by the action of ℂ^{∗}. The same is true of the action of G(ℚ), since (in an obvious notation)

τ′ = −\frac{ω′_{2}}{ω′_{1}} = \frac{−aω_{2} + bω_{1}}{−cω_{2} + dω_{1}} = γ(τ)

u′ = \frac{z′}{ω′_{1}} = \frac{uω_{1}}{(−cω_{2} + dω_{1})/detγ} = j(γ, τ)u

agreeing with the action of G(ℚ) used to define X^{an}. Finally it is easy to check that the fibres of ψ are principal homogeneous spaces under ℂ^{∗} × G(ℚ), so ψ gives an isomorphism ℂ^{∗} × G(ℚ)\Z^{an} ⟶^{∼} X^{an}, which is evidently G_{f}^{(1)}- equivariant. Combining this with the earlier isomorphism, we get the desired G_{f}^{(1)}-equivariant isomorphism Ξ: X(ℂ) ⟶^{∼} X^{an}.

All the properties in the theorem are now more or less automatic, apart from the assertion concerning det^{an}. This is easiest checked at finite level: let (τ, g) ∈ M_{n}^{an}. By Lemma 5.3.1.1 above, the fibre of X_{n}^{an} at this point is the elliptic curve ℂ/nℤ + nτℤ, and the level n structure (ℤ/n)^{2} → ℂ/nℤ + nτℤ is given by

a ↦ (−1 τ) ga

since if b_{1}, b_{2} ∈ ℤ then

λ_{τ,g}: b_{1} + b_{2}τ ↦ (b_{1} + b_{2}τ, τ, 0, g) ≡ (0, τ, \binom{−b_{1}}{b_{2}}, g) ∈ X_{n}^{an}[n].

According to the normalisation of the e_{n}-pairing given in [K-M, §2.8, especially (2.8.5.2)] we have

e_{n}(1, τ) = e^{±2πi/n} if τ ∈ H^{±}

and so the level n structure given by (τ, g) has determinant exp(∓2πidet(g)/n), as required.

5.4 Other normalisations

We will explain two other ways in which to normalise the action of the groups, and give a dictionary between them and the present convention.

5.4.1 Another left action

Let ^{∗}G^{(1)} be the semidirect product taken in the other direction: regard points of 𝔾_{a}^{2} as row vectors, and then

∗G^{(1)} = G× 𝔾_{a}^{2}

with multiplication

(g, v)(g′, v′) = (gg′, vg′ + v′)

We have an isomorphism

∗μ: G^{(1)} ⟶^{∼} ^{∗}G^{(1)}
(v, g) ↦ (^{t}g^{−1},^{t}(g^{−1}v)).

The finite group ^{∗}G^{(1)}(ℤ/nℤ) acts on X_{n} on the right by

g: (E/S, α_{n}, P) ↦ (E/S, α_{n} ∘^{t}g^{−1}, P)
v: (E/S, α_{n}, P) ↦ (E/S, α_{n}, P + α_{n}(^{t}v))

We have a left action of ^{∗}G^{(1)}(ℚ) on H^{±} × ℂ by

(γ, v): (τ, u) ↦ (γ(τ), \frac{u + v_{1}τ + v_{2}}{cτ + d}) where γ = (\begin{matrix}a & b \\ c & d \end{matrix})

The map

ℂ × H^{±} → H^{±} × ℂ
(u, τ) ↦ (\frac{−1}{τ}, \frac{−u}{τ})

transforms the action of x ∈ G^{(1)}(ℚ) into that of ^{∗}μ(x) ∈ ^{∗}G^{(1)}(ℚ), and this gives a different parameterisation of the complex points of X:

X(ℂ) = ^{∗}G^{(1)}(ℚ)\H^{±} × ℂ ×^{∗}G_{f}^{(1)}

with the level structure

M(ℂ) × 𝔸_{f}^{2} → X(ℂ)
((τ, g), a) ↦ (τ, 0, g,^{t}a)

5.4.2 A right action

There is another way to set up the moduli problems for elliptic curves, which is used in [De-Rap] and [De-FMR]. There, a level n structure on E/S is defined to be an isomorphism β_{n}: E[n] ⟶^{∼} (ℤ/n)_{S}^{2}. Let us denote the modular varieties defined in this way as M^{opp}, X^{opp}, etc. Thus for example X_{n}^{opp}(S) is the set of isomorphism classes of triples

(E/S, β_{n}: E[n] ⟶^{∼} (ℤ/n)_{S}^{2}, P ∈ E(S)).

Let ^{†}G^{(1)} be the following semidirect product of G and 𝔾_{a}^{2} (as before, viewed as column vectors):

†G^{(1)} = G× 𝔾_{a}^{2}

with multiplication

(g, v)(g′, v′) = (gg′, g′^{−1}v + v′)

Then ^{†}G^{(1)}(ℤ/nℤ) acts on X_{n}^{opp} on the left by

(g, v): (E/S, β_{n}, P) ↦ (E/S, g ∘ β_{n}, P + β_{n}^{−1}(v)).

Let ^{†}μ: G^{(1)} ⟶^{∼} ^{†}G^{(1)} be the isomorphism

†μ: (v, g) ↦ (g, −g^{−1}(v)).

Then the morphism X_{n} → X_{n}^{opp} which is given on points by

(E/S, α_{n}, P) ↦ (E/S, α_{n}^{−1}, P)

transforms the action of x ∈ G^{(1)}(ℤ/nℤ) on M_{n} into that of^{†}μ(x)^{−1} ∈ ^{†}G^{(1)}(ℤ/nℤ) on M_{n}^{opp}. (It obviously transforms right multiplication by y ∈ G_{f}^{(1)} into left multiplication by ^{†}μ(y)^{−1}.)

The level structure on X^{opp} is given by

M^{opp} × 𝔸_{f}^{2} ⟵^{∼} X^{opp}[tors] ⊂  X^{opp}
∪  ∪
((τ, g), a) ⟶ (τ, 0, g, −a)

We have a right action of ^{†}G^{(1)}(ℚ) on H^{±} × ℂ, given as

(γ, v): (τ, u) ↦ (\frac{aτ + c}{bτ + d}, \frac{u + (τ 1) γv}{bτ + d} ) where γ = (\begin{matrix}a & b \\ c & d \end{matrix})

and the complex points of X^{opp} are

X^{opp}(ℂ) = H^{±} × ℂ ×^{†}G_{f}^{(1)} mod ^{†}G^{(1)}(ℚ).

The isomorphism X ⟶^{∼} X^{opp} is given by

(τ, u, x) ↦ (\frac{−1}{τ}, \frac{−u}{τ},^{†}μ(x)^{−1})

as one checks that this map transforms left multiplication by x ∈ G^{(1)}(ℚ) on ℂ × H^{±} × G_{f}^{(1)} into right multiplication by ^{†}μ(x)^{−1} on H^{±} × ℂ ×^{†}G_{f}^{(1)}.

This convention is also used (with some modifications) in [Be]. It has certain advantages over the convention we have adopted—from the point of view of [De-FMR] it is particularly natural—but unfortunately is at variance with almost the entire literature on Shimura varieties.

5.5 Cusps and compactification

5.5.1 The Tate curve over ℂ

In this section (which is a parody of the rather profound treatment of the Tate curve over ℤ[[q]] in [De-Ra, VII.1]) we describe the construction of certain degenerating elliptic curves over ℂ.

We fix in the sequel an integer n ≥ 1; later we will assume that n ≥ 3. We let q_{n} be a complex variable, and D = {q_{n} ∈ ℂ | |q_{n}| < 1} be the unit disc, and D^{∗} = D − {0} the punctured disk. We will define a complex manifold Tate_{n}, a proper holomorphic ϕ: Tate_{n} → D, and an embedding over D

α_{n}^{can} : D × (ℤ/n)^{2} ↪ Tate_{n}

with the following properties:

1. If q_{n} ≠ 0, then ϕ^{−1}(q_{n}) ⟶^{∼} ℂ^{∗}/⟨q_{n}^{n}⟩;

2. In terms of this isomorphism,

α_{n}^{can} : (q_{n}, e_{1}) ↦ e^{−2πi/n} ∈ ℂ^{∗}/⟨q_{n}^{n}⟩
(q_{n}, e_{2}) ↦ q_{n} ∈ ℂ^{∗}/⟨q_{n}^{n}⟩;

3. π^{−1}(0) is canonically isomorphic to the standard Néron polygon with n sides:

C_{n} = ℙ^{1}(ℂ) × (ℤ/n)/{(0, i) ∼ (∞, i + 1)}

and α_{n}^{can}(0, e_{1}) ↦ (e^{−2πi/n}, 0), α_{n}^{can}(0, e_{2}) ↦ (1, 1) ∈ C_{n}.

We call Tate_{n} the n-sided Tate curve over D. To construct it, define for each i ∈ ℤ/n the manifold

U_{i} = {(x, y, q_{n}) ∈ ℂ^{2} × D | xy = q_{n}}.

We will write the coordinates on U_{i} as (x_{i}, y_{i+1}, q_{n}). Then we define an equivalence relation on ⊔_{i}U_{i} generated by the following pre-equivalence relation: if (x_{i}, y_{i+1}, q_{n}) ∈ U_{i} with x_{i} ≠ 0, then

(x_{i}, y_{i+1}, q_{n}) ∼ (x_{i−1}, y_{i}, q_{n}) = (q_{n}x_{i}, 1/x_{i}, q_{n}) ∈ U_{i−1}.

This identifies {x_{i} ≠ 0} ⊂ U_{i} with {y_{i} ≠ 0} ⊂ U_{i−1}. Then define Tate_{n} to be the quotient:

Tate_{n} =^{def} ⨆_{i∈ℤ/n} U_{i}/ ∼ .

The morphism ϕ is simply given by (x_{i}, y_{i+1}, q_{n}) ↦ q_{n} ∈ D on each U_{i}. The map

ℂ^{∗} × D × ℤ/n → ⊔U_{i}
(t, q_{n}, i) ↦ (t, q_{n}/t, q_{n}) ∈ U_{i}

defines by passage to the quotient a map

ℂ^{∗} × D × ℤ/n → Tate_{n}  (5.5.1.1)

which is a morphism over D. Over D^{∗} it gives an isomorphism

ℤ\ℂ^{∗} × D^{∗} × ℤ/n ⟶^{∼} Tate_{n} −ϕ^{−1}(0),

where the action of ℤ on the left hand side is given by

m: (t, q_{n}, i) ↦ (tq_{n}^{−m}, q_{n}, i + m).

We next define α_{n}^{can} by

α_{n}^{can} : (q_{n}, a) ↦ (e^{−2πia_{1}/n}, q_{n}e^{2πia_{1}/n}) ∈ U_{a_{2}}.

We leave to the reader to check properties (1) and (2). For (3), we just write down the necessary maps: we have a morphism

ℙ × ℤ/n → Tate_{n}
(t, i) ↦ (t, 0, 0) ∈ U_{i} if t ≠ ∞
(0, 1/t, 0) ∈ U_{i−1} if t ≠ 0.

We write Tate_{n}^{reg} ⊂ Tate_{n} for the open set on which ϕ is smooth; clearly it is the complement of the n points (0, 0, 0) ∈ U_{i}. The morphism (5.5.1) identifies Tate_{n}^{reg} with the quotient of ℂ^{∗} × D × ℤ/n by the equivalence relation

(t, q_{n}, i) ∼ (tq_{n}^{−m}, q_{n}, i + m) if m ∈ ℤ and q_{n} ≠ 0.

In particular, Tate_{n}^{reg} is an analytic group over D, and α_{n}^{can} is a homomorphism of anaylic groups over D. Moreover one checks at once that the group law extends to an action

Tate_{n}^{reg} ×_{D} Tate_{n} → Tate_{n}

and that in particular α_{n}^{can} gives an action of (ℤ/n)^{2} on Tate_{n}.

A final observation is that there is a unique invariant relative holomorphic 1-form ω^{can} ∈ H^{0}(Tate_{n}^{reg}, Ω_{Tate_{n}^{reg} /D}^{1}) whose pullback via (5.5.1) is dt/t.

Proposition 5.5.1.1. Tate_{n} is a projective analytic space over D.

Let Z_{n} ⊂ Tate_{n} be the image of α_{n}^{can}. Then Z_{n} is a divisor and the invertible sheaf O_{Tate_{n}}(Z_{n}) is ample relative to ϕ, since the fibres of ϕ are reduced of dimension 1, and Z_{n} meets each of their irreducible components.

In fact if n ≥ 2 then an explicit embedding Tate_{n} ↪ ℙ^{n^{2}−1}(ℂ) × D can be given by ϑ-functions. The most classical example is n = 2. Consider the following minor modifications of Jacobi’s ϑ-functions:

ϑ_{1}(t, q_{2}) = ϑ_{2}(−t, q_{2}) = (t − t^{−1}) ∏_{m≥1}(1 − q_{2}^{2m}t)(1 − q_{2}^{2m}t^{−1})
ϑ_{4}(t, q_{2}) = ϑ_{3}(−t, q_{2}) = ∏_{m≥1}(1 − q_{2}^{2m−1}t)(1 − q_{2}^{2m−1}t^{−1})

and let

F_{1} = 1, F_{2} = \frac{ϑ_{3}ϑ_{4}}{ϑ_{1}ϑ_{2}} , F_{3} = \frac{ϑ_{2}ϑ_{4}}{ϑ_{1}ϑ_{3}} , F_{4} = \frac{ϑ_{2}ϑ_{3}}{ϑ_{1}ϑ_{4}} .

Then one has classical relations [some nice ref.]:

ϑ_{2}(1, q_{2})^{2}F_{2}F_{3} − ϑ_{4}(1, q_{2})^{2}F_{3}F_{4} + ϑ_{3}(1, q_{2})^{2}F_{2}^{2} = 0
ϑ_{3}(1, q_{2})^{2}F_{2}F_{3} − ϑ_{4}(1, q_{2})^{2}F_{2}F_{4} + q_{2}ϑ_{2}(1, q_{2})^{2}F_{1}^{2} = 0

which for q_{2} ≠ 0 define a nonsingular quartic in ℙ^{3}(ℂ). For q_{2} = 0 we obtain

F_{2}(F_{3} − F_{4}) = 0 = 4F_{2}F_{3} − F_{3}F_{4} + F_{4}^{2}

wich is a Néron 2-gon in ℙ^{3}(ℂ). It is a pleasant exercise to verify that the functions F_{i} are holomorphic functions on Tate_{2} − Im(α_{2}^{can}) which form a basis for H^{0}(Tate_{2}, O(Z_{2})).

5.5.2 Compactification over ℂ

We will write down rather explicitly the compactification X_{n}(ℂ) ↪ \overline{X}_{n}(ℂ) in terms of the Tate curve. If q_{n} ∈ D^{∗} then the fibre Tate_{n}(q_{n}) of the Tate curve, together with the level n structure α_{n}^{can}, determines a complex point of M_{n}, so we obtain classifying maps

ϕ_{Tate,n}: D^{∗} → M_{n}(ℂ), ψ_{Tate,n}: Tate_{n} ×_{D}D^{∗} → X_{n}(ℂ)

which are given by the following diagram.

Tate_{n} ×_{D}D^{∗}
≀↑^{(5.5.1)}
ℤ\ℂ^{∗} × D^{∗} × ℤ/nℤ  D^{∗}
≀↑  ^{≀}↑
N_{ℤ}^{(1)}\ℂ × H × N_{ℤ/nℤ}^{(1)} ⟶ N_{ℤ}\H × N_{ℤ/nℤ}
↓  ↓
X_{n}^{an} = G_{ℤ}^{(1)}\ℂ × H^{±} × G_{ℤ/nℤ}^{(1)} ⟶ G_{ℤ}\H × G_{ℤ/nℤ} = M_{n}^{an}

Here we have used the following groups over ℤ:

N =^{def} \{(\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})\} ⊂ GL_{2}; N^{(1)} = 𝔾_{a}^{2}× N ⊂ G^{(1)}.

The unlabelled isomorphism in the left-hand column is given by

(u, τ, a, (\begin{matrix}1 & b \\ 0 & 1 \end{matrix})) ∈ ℂ × H × (ℤ/nℤ)^{2}× N_{ℤ/nℤ}
↓
(e^{2πi(u−a_{1}+ba_{2})/n}, e^{2πi(τ−b)/n}, a_{2}) ∈ ℂ^{∗} × D^{∗} × ℤ/nℤ

and the bottom square in the diagram is Cartesian. The map ϕ_{Tate,n} extends to a holomorphic map D → \overline{M}_{n}(ℂ) which is a local isomorphism at 0; define ∞_{ℂ} to be ϕ_{Tate,n}(0).

We can combine this map with the action of G_{ℤ/nℤ} to get an equivariant map

D × G_{ℤ/nℤ} → \overline{M}_{n}(ℂ)
(q_{n}, g) ↦ ϕ_{Tate,n}(q_{n})g

If q_{n} ≠ 0 then (for any choice of the logarithm)

ϕ_{Tate,n}(q_{n})g = (\frac{n}{2πi} log q_{n}, g) ∈ G_{ℤ}\H^{±} × G_{ℤ/nℤ}

and if b ∈ ℤ/nℤ then

ϕ_{Tate,n}(q_{n}) · ± \binom{1 b}{1} g = (\frac{n}{2πi} log q_{n}, ± \binom{1 b}{1} g)

∼ (\frac{n}{2πi} log q_{n} − b, g)

= ϕ_{Tate,n}(q_{n}e^{−2πin/b})g.

Then M_{n}^{∞}(ℂ) is the orbit of ∞_{ℂ} under G_{ℤ/nℤ}, with stabiliser ±N_{ℤ/nℤ}, hence

M_{n}^{∞}(ℂ) ⟶^{∼} ±N_{ℤ/nℤ}\G_{ℤ/nℤ}.

From the theory of elliptic surfaces it follows that ψ_{Tate,n} extends to an isomorphism over D

Tate_{n} ⟶^{∼} \overline{X}_{n}(ℂ) ×_{\overline{M}_{n}(ℂ),ϕ_{Tate,n}} D.

(Alternatively, we could define \overline{X}_{n}(ℂ) by gluing copies of the Tate curve to X_{n}(ℂ), and then showing that the resulting surface was actually an algebraic surface over ℚ—which is not hard to do directly.) We can extend ψ_{Tate,n} to a G_{ℤ/nℤ}-equivariant map

Tate_{n} ×G_{ℤ/nℤ} → \overline{X}_{n}(ℂ)
(x, g) ↦ ψ_{Tate,n}(x)g

which can also be explicitly computed away from {q_{n} = 0} by

ℤ\(ℂ^{∗} × D^{∗} × ℤ/nℤ) × G_{ℤ/nℤ} → G^{(1)}\ℂ × H^{±} × G_{ℤ/nℤ}^{(1)}
(t, q_{n}, c, g) ↦ (\frac{n}{2πi} log t, \frac{n}{2πi} log q_{n}, \binom{0}{c}, g)

Next we compute for b ∈ ℤ/nℤ

ψ_{Tate,n}(t, q_{n}, c) · ± \binom{1 b}{1} g = (\frac{n}{2πi} log t, \frac{n}{2πi} log q_{n}, \binom{0}{c}, ± \binom{1 b}{1} g)

∼ (±\frac{n}{2πi} log t + bc, \frac{n}{2πi} log q_{n} − b, \binom{0}{±c}, g)

= ψ_{Tate,n}(t^{±1}e^{2πibc/n}, q_{n}e^{−2πin/b}, ±c)g.

The G_{ℤ}^{(1)}-equivalence between the first and second right-hand expressions is given by

(\binom{±bc}{0}, ± \binom{1 −b}{1}) ∈ G_{ℤ}^{(1)}.

So we have the following parameterisation of the smooth part of the fibre of \overline{X}_{n} over M_{n}^{∞}:

\overline{X}_{n}^{reg}(ℂ) ×_{\overline{M}_{n}} M_{n}^{∞}(ℂ) = ±N_{ℤ/nℤ}\ℂ^{∗} × ℤ/nℤ × G_{ℤ/nℤ}

where the action of ±N_{ℤ/nℤ} is

± \binom{1 b}{1} : (t, c, g) ↦ (t^{±1}e^{2πibc/n}, c, ± \binom{1 b}{1} g).

In particular, the connected component of the identity X̂_{n} ×M_{n}^{∞} (which is noncanonically isomorphic to 𝔾_{m} ×M_{n}^{∞}) has a G_{ℤ/nℤ}-equivariant parameterisation as

{±1}\ℂ^{∗} × (N_{ℤ/nℤ}\G_{ℤ/nℤ})

where −1 acts as (t, N_{ℤ/nℤ}g) ↦ (t^{−1}, N_{ℤ/nℤ}(−g)).

5.5.3 Variation: compactification at infinite level

Consider m, n with m|n. It is easy to see that the Cartesian square in diagram (5.2.2) extends to a commutative square

\overline{X}_{n} ⟶^{σ̄_{n,m}^{naive}} \overline{X}_{m}
↓f̄_{n}  ↓f̄_{m}
\overline{M}_{n} ⟶^{ρ̄_{n,m}} \overline{M}_{m}

which however is not Cartesian if n > m; for the fibres of f̄_{n} (resp. f̄_{m}) over cusps are n-gons (resp. m-gons). In fact, the finite morphism ρ̄_{n,m} is ramified at each cusp, which ramification degree n/m, so the fibre product

\overline{X}_{m} ×_{\overline{M}_{m}} \overline{M}_{n}

has singularities analytically of the form xy = q_{n}^{n/m}. The morphism from \overline{X}_{n} to the fibre product is the minimal resolution of these singularities (which is a sequence of n/m − 1 simple blowups at each singularity).

So the morphisms σ̄_{n,m}^{naive} are not affine, and in fact the inverse limit of (\overline{X}_{n}, σ̄_{n,m}^{naive}) does not exist in the category of schemes. Hoewver for σ_{n,m} the situation is somewhat better.

Proposition 5.5.3.1. σ_{n,m} extends to a finite morphism σ̄_{m,n}: \overline{X}_{n} → \overline{X}_{m}.

We have to check that the closure of the graph of σ_{n,m}: X_{n} → X_{m} in \overline{X}_{n} × \overline{X}_{m} is finite over \overline{X}_{m} and maps isomorphically to \overline{X}_{n}, and this can be checked after extending scalars to ℂ and working in the analytic category. Moreover the question is local in the neighbourhood of the cusps. So we are reduced to verifying that the morphism

Tate_{n} ×_{D}D^{∗}  ⟶  Tate_{m} ×_{D}D^{∗}
∥  ∥
ℤ\ℂ^{∗} × D^{∗} × ℤ/nℤ  ℤ\ℂ^{∗} × D^{∗} × ℤ/mℤ
(t, q_{n}, c)  ↦ (t^{n/m}, q_{n}^{n/m}, c mod m)

extends to a finite morphism Tate_{n} → Tate_{m}. But this is easily checked using the charts U_{i} used to define Tate_{n}; the morphism in question is

(x_{i}, y_{i+1}, q_{n}) ∈ U_{i} ⊂Tate_{n}
↓
(x_{i}^{n/m}, y_{i+1}^{n/m}, q_{n}^{n/m})∈U_{imodm}⊂Tate_{m}

whiceh extends to a proper morphism for Tate_{n} to Tate_{m} with finite fibres; we have only to check what happens at q_{n} = 0, and there it is obtained by passage to the quotient from the morphism

\begin{matrix}ℙ^{1}(ℂ) × ℤ/nℤ & ⟶ & ℙ^{1}(ℂ) × ℤ/mℤ \\ (t, i) & ↦ & (t^{n/m}, i mod m). \end{matrix}

We can therefore form the inverse limit

\overline{X} =^{def} lim_{⟵}_{n}(\overline{X}_{n}, σ_{n,m})

which comes with a morphism f̄: \overline{X} → \overline{M} which is the inverse limit of proper morphisms. What can we say about its fibres at cusps? For x ∈ M^{∞}, the fibre f̄^{−1}(x) is an inverse limit of Néron polygons, and the level structure α̂^{univ} : (ℤ̂)_{M}^{2} ↪ X extends to an embedding (ℤ̂)2 ; \overline{M} ↪ \overline{X}. This does not however extend to a morphism 𝔸_{f/\overline{M}}^{2} → \overline{X} (even though such a morphism exists over

M) simply because the fibres of \overline{X}/\overline{M} do not have enough components—the group of irreducible components of f̄^{−1}(x) for x ∈ M^{∞} is isomorphic to ℤ̂. This means also the the G_{f}-action on X does not extend to \overline{X}. What we are seeing here is that the concept of regular model of elliptic curve does not adapt easily to a base ring with a non-discrete valuation.

We can form a bigger object over \overline{M} on which G_{f} acts. It is a group scheme

ˆX̂/\overline{M}. Unlike \overline{X} it is not pro-proper over \overline{M}, nor is it quasicompact. We will only sketch the construction here, as it is not needed in the sequel.

Let us consider \overline{X}^{reg} =^{def} lim_{⟵}_{n} \overline{X}_{n}^{reg}. Its fibre over x ∈ M is a pro-elliptic curve.

Over x ∈ M^{∞} the fibre is isomorphic to 𝔾̃_{m} × ℤ̂, and [×n] is an isomorphism

on 𝔾̃_{m} but not on ℤ̂. Define ˆ X̂ to be the direct limit:

ˆX̂ =^{def} lim_{⟶}_{n}((\overline{X}^{reg})_{n}, [×n/m]).

The connecting morphisms [×n/m] are open immersions, identifying \overline{X}^{reg} (at level n) with the open subgroup scheme of \overline{X}^{reg} of points which are divisible by n/m at each fibre over M^{∞} (at level m). Therefore the direct limit trivially

exists. Over M we have done nothing: ˆ X̂ × M = X. However if x ∈ M^{∞} then the fibre of ˆ X̂/\overline{M} at X is isomorphic to 𝔾̃_{m} × 𝔸_{f}. Since

lim_{⟶}_{n}((ℤ̂^{2})_{n}, [×n/m]) = 𝔸_{f}^{2}

the level structure on \overline{X} really extends to a homomorphism of group schemes

𝔸_{f/\overline{M}}^{2} ⟶ ˆ X̂

and we can show that G_{f} acts on ˆ X̂, compatibly with its action on X. We will

content ourselves with describing the complex points of ˆ X̂ in the neighbourhood of the cusps.

To simplify matters we introduce some further notation. We denote a general element of

ℂ̃^{∗} = lim_{⟵}_{n}((ℂ^{∗})_{n}, t ↦ t^{n/m})

as t_{∼} = (t_{n}); so t_{n} ∈ ℂ^{∗} are elements with t_{n}^{n/m} = t_{m} whenever m|n. Likewise, a general element of

D̃ = lim_{⟵}_{n}((D)_{n}, q ↦ q^{n/m})

will be denoted q_{∼} = (q_{n}) (in accordance with earlier notation). If t_{∼} ∈ ℂ̃^{∗} and

r ∈ ℚ then the power t_{∼}^{r} ∈ ℂ̃^{∗} makes sense (simply set t_{∼}^{a/b} = t_{∼}′, with t′_{n} = t_{nb}^{a}, for a, b ∈ ℤ with b > 0).

If z ∈ ℂ or 𝔸_{f} then we write

exp_{∼}(2πiz) = (e^{2πiz/n})_{n} ∈ ℂ̃^{∗}, ζ_{∼} = exp_{∼}(1) = (e^{2πi/n})_{n} ∈ ℤ̂(1) ⊂ ℂ̃^{∗}.

The pro-Tate curve over D̃ is the quotient

(pro-Tate) = ℂ̃^{∗} × D̃ × 𝔸_{f}/ ∼

where the equivalence relation is

(t_{∼}, q_{∼}, c) ∼ (t_{∼}′, q_{∼}′, c′)  ⇔  q_{∼} = q_{∼}′ ≠ 0 and for some r ∈ ℚ, t_{∼}′ = t_{∼}q_{∼}^{−r} and c′ = c + r.

It is a topological group over D̃, with a canonical level structure

𝔸_{f}^{2} × D̃ ⟶ (pro-Tate)
(\binom{c_{1}}{c_{2}}, q_{∼}) ↦ (ζ_{∼}^{−c_{1}}, q_{∼}, c_{2}).

The fibre over q_{∼} ∈ D̃^{∗} is a pro-elliptic curve over ℂ:

\begin{aligned}ℂ̃^{∗} × 𝔸_{f}/\{(q_{∼}^{−r}, r) | r ∈ ℚ\} & ⟶^{∼} & ℂ × 𝔸_{f}^{2}/\{(r_{1} + r_{2}τ, \binom{r_{1}}{r_{2}}) | r_{i} ∈ ℚ\} \\ (exp_{∼} 2πi(u − c_{1}), c_{2}) & ← & (u, \binom{c_{1}}{c_{2}}) \end{aligned}

if q_{∼} = exp_{∼} 2πi(τ + s) for some τ ∈ H, s ∈ 𝔸_{f}. The fibre at q_{∼} = 0 is the product

ℂ̃^{∗} × 𝔸_{f}.

The level structure gives a classifying maps which extend to the horizontal maps in the following Cartesian square:

(pro-Tate) ⟶^{ψ_{(pro-Tate)}} ˆ X̂(ℂ)
↓  ↓
D̃  ⟶^{ϕ_{(pro-Tate)}} \overline{M}(ℂ)

Over D̃^{∗} the map ψ_{(pro-Tate)} is given in coordinates as:

ψ_{(pro-Tate)}
ℚ\ℂ̃^{∗} × D̃^{∗} × 𝔸_{f} ⟵^{∼} N_{ℚ}^{(1)}\ℂ × H × N_{f}^{(1)} ⟶ G_{ℚ}^{(1)}\ℂ × H^{±} × G_{f}^{(1)}
(t_{∼}, q_{∼}, c)  ←  (u, τ, c, \binom{1 s}{1})

where t_{∼} = exp_{∼} 2πi(u − c_{1} + sc_{2}), q_{∼} = exp_{∼} 2πi(τ − s), and c = c_{2}.

If (\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ G_{f} with a, d ∈ ℚ and ad > 0 we compute:

ψ_{(pro-Tate)}(t_{∼}, q_{∼}, c) (\begin{matrix}a & b \\ 0 & d \end{matrix}) = (u, τ, c, (\begin{matrix}a & b + ds \\ 0 & d \end{matrix}))

∼ (\frac{u}{a}, \frac{dτ}{a} , \binom{a^{−1}c_{1}}{d^{−1}c_{2}}, \binom{1 (b + ds)/a}{1})

= ψ_{(pro-Tate)}(t_{∼}′, q_{∼}′, c′)

Last revision 4 December 1996 5:33 where t_{∼}′ = t_{∼}^{1/a}ζ_{∼}^{bc/ad}, q_{∼}′ = q_{∼}^{d/a}ζ_{∼}^{−b/a}, c′ = d^{−1}c. So we can just as before consider the map

(pro-Tate) ×G_{f} ⟶ ˆ X̂
(x, g)  ↦ ψ_{(pro-Tate)}(x)g.

Passing to the fibre over q_{∼} = 0 we recover the G_{f}-equivariant isomorphism

M^{∞}(ℂ) ⟶^{∼} N_{f}A_{ℚ}^{+}\G_{f}.

The fibre over the cusps is then

ˆX̂ ×\overline{M} M^{∞}(ℂ) ⟶^{∼} N_{f}A_{ℚ}^{+}\ℂ̃^{∗} × 𝔸_{f} × G_{f}

where the quotient is taken with respect to the action

(\begin{matrix}a & b \\ 0 & d \end{matrix}) : (t_{∼}, c, g) ↦ (t_{∼}^{a}ζ_{∼}^{−bc}, cd, (\begin{matrix}a & b \\ 0 & d \end{matrix}) g)

and the connected component containing the identity of the fibres at infinity is

X̂ ×\overline{M} M^{∞}(ℂ) ⟶^{∼} N_{f}A_{ℚ}^{+}\ℂ̃^{∗} × G_{f}

where N_{f}A_{ℚ}^{+} acts by

(\begin{matrix}a & b \\ 0 & d \end{matrix}) : (t_{∼}, g) ↦ (t_{∼}^{a}, (\begin{matrix}a & b \\ 0 & d \end{matrix}) g)

An invariant differerential from on the pro-Tate curve is

ω_{can} = \frac{dt_{1}}{t_{1}} = n\frac{dt_{n}}{t_{n}} for any n ≥ 1.

The space of invariant differential forms on X̂×\overline{M} M^{∞}(ℂ) is therefore isomorphic as a G_{f}-module to

\{ϕ(g)ω_{can} | ϕ: G_{f} → ℂ locally constant, with ϕ((\begin{matrix}a & b \\ 0 & d \end{matrix}) g) = a^{−1}ϕ(g) for every (\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ N_{f}A_{ℚ}^{+}.\}