Tom Fisher's Home Page
My research interests are in arithmetical algebraic geometry and
computational number theory. In particular I work on elliptic curve
descent calculations, and the construction of explicit elements
in the Tate-Shafarevich group.
I am compiling a list of genus one curves that are counter-examples
to the Hasse principle and have Jacobian of small conductor.
The list so far covers elements of Sha of
order 3
and
order 5.
The PhD theses of my students Mohammad Sadek,
Graham Sills,
Monique van Beek, Zexiang Chen, Lazar Radičević and Jiali Yan are available by following these links.
Publications and Preprints
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Minimisation of 2-coverings of genus 2 Jacobians, joint with M. Liu, preprint.
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Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian, joint with J. Yan, preprint.
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The density of integral quadratic forms having a k-dimensional totally isotropic subspace, joint with L. Drakengren, to appear in Rocky Mountain J. Math.
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On binary quartics and the Cassels-Tate pairing,
(proceedings of ANTS XV),
Res. Number Theory 8 (2022), Art. 74, 13 pages.
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Computing structure constants for rings of finite rank from minimal free resolutions, joint with L. Radičević, to appear in Math. Res. Lett.
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On pairs of 17-congruent elliptic curves, preprint.
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The density of polynomials of degree n over Zp having exactly r roots in Qp, joint with M. Bhargava, J.E. Cremona and S. Gajovic, Proc. Lond. Math. Soc. (3) 124 (2022), no. 5, 713-736.
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The proportion of genus one curves over Q defined by a
binary quartic that everywhere locally have a point,
joint with M. Bhargava and J.E. Cremona, Int. J. Number Theory 17 (2021), no. 4, 903-923.
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On families of 13-congruent elliptic curves, preprint.
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Everywhere local solubility for hypersurfaces in products of projective spaces, joint with W. Ho and J. Park,
Res. Number Theory 7 (2021), no. 1, Art. 6, 27 pages.
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Explicit moduli spaces for congruences of elliptic curves, Math. Zeit. 295 (2020), 1337-1354.
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Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations, joint with M. van Beek, Acta Arith. 185 (2018), no. 4, 367-396.
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On some algebras associated to genus one curves, J. Algebra 518 (2019), 519-541.
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Some minimisation algorithms in arithmetic invariant theory, joint with
L. Radičević, J. Théor. Nombres Bordeaux 30 (2018), no. 3, 801-828.
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Visibility of 4-covers of elliptic curves, joint with N. Bruin,
Res. Number Theory 4 (2018), no. 1, Art. 11, 29 pages.
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Visualising elements of order 7 in the Tate-Shafarevich group of
an elliptic curve, LMS J. Comput. Math. 19 (2016),
proceedings of ANTS XII, 100-114.
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A formula for the Jacobian of a genus one curve of arbitrary degree,
Algebra & Number Theory 12 (2018), no. 9, 2123-2150.
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The proportion of plane cubic curves over Q that everywhere locally have a point, joint with M. Bhargava and J.E. Cremona,
Int. J. Number Theory 12 (2016), no. 4, 1077-1092.
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What is the probability that a random integral quadratic form
in n variables has an integral zero?
joint with M. Bhargava, J.E. Cremona, N.G. Jones and J.P. Keating,
Int. Math. Res. Not. (2016), no. 12, 3828-3848.
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Higher descents on an elliptic curve with a rational 2-torsion point,
Math. Comp. 86 (2017), no. 307, 2493–2518.
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On families of 9-congruent elliptic curves,
Acta Arith. 171 (2015), no. 4, 371-387.
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Ranks of quadratic twists of elliptic curves,
joint with M. Watkins, S. Donnelly, N.D. Elkies,
A. Granville and N.F. Rogers, Pub. math. de Besançon, 2014/2, 63-98.
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On genus one curves of degree 5 with square-free discriminant,
joint with M. Sadek,
J. Ramanujan Math. Soc. 31 (2016), no. 4, 359-383.
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Minimal models for 6-coverings of elliptic curves,
LMS J. Comput. Math. 17 (2014), proceedings
of ANTS XI, 112-127.
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On families of 7 and 11-congruent elliptic curves,
LMS J. Comput. Math. 17 (2014), no. 1, 536-564.
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Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve, joint with R. Newton, Int. J. Number Theory 10 (2014), no. 7, 1881-1907.
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Invariant theory for the elliptic normal quintic, II. The covering map,
preprint.
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Invisibility of Tate-Shafarevich groups in abelian surfaces,
Int. Math. Res. Not. (2014), no. 15, 4085-4099.
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Explicit 5-descent on elliptic curves, in ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, E.W. Howe, K.S. Kedlaya (eds.),
Open Book Series, 1, Mathematical Sciences Publishers, Berkeley, 2013.
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Minimisation and reduction of 5-coverings of elliptic curves,
Algebra & Number Theory 7 (2013), no. 5, 1179–1205.
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Invariant theory for the elliptic normal quintic, I. Twists of X(5),
Math. Ann. 356 (2013), no. 2, 589-616.
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Explicit n-descent on elliptic curves, III. Algorithms,
joint with J.E. Cremona, C. O'Neil, D. Simon and M. Stoll,
Math. Comp. 84 (2015), no. 292, 895–922.
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The Hessian of a genus one curve,
Proc. Lond. Math. Soc. (3) 104 (2012) 613-648.
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Local solubility and height bounds for coverings of elliptic curves, joint with G.F. Sills,
Math. Comp. 81 (2012), no. 279, 1635-1662.
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Some bounds on the coefficients of covering curves, Enseign. Math. (2) 58
(2012) 99-124.
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Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, joint with J.E. Cremona and M.Stoll, Algebra & Number Theory 4 (2010),
no. 6, 763-820.
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Some improvements to 4-descent on an elliptic curve,
in Algorithmic number theory,
A. van der Poorten, A. Stein (eds.),
Lecture Notes in Comput. Sci., 5011, Springer, 2008.
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The yoga of the Cassels-Tate pairing, joint with E.F. Schaefer and M.Stoll,
LMS J. Comput. Math. 13 (2010) 451-460.
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Finding rational points on elliptic curves
using 6-descent and 12-descent,
Journal of Algebra 320 (2008), no. 2, 853-884.
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Explicit n-descent on elliptic curves, II. Geometry,
joint with J.E. Cremona, C. O'Neil, D. Simon and M. Stoll,
J. reine angew. Math. 632 (2009) 63-84.
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On the equivalence of binary quartics,
joint with J.E. Cremona,
Journal of Symbolic Computation 44 (2009) 673-682.
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The invariants of a genus one curve,
Proc. Lond. Math. Soc. (3) 97 (2008) 753-782.
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Pfaffian presentations of elliptic normal curves,
Trans. Amer. Math. Soc. 362 (2010), no. 5, 2525-2540.
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Explicit n-descent on elliptic curves, I. Algebra,
joint with J.E. Cremona, C. O'Neil, D. Simon and M. Stoll,
J. reine angew. Math. 615 (2008) 121-155.
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A new approach to minimising binary quartics and ternary cubics,
Math. Res. Lett. 14 (2007) Issue 4, 597--613.
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Testing equivalence of ternary cubics,
in Algorithmic number theory, F. Hess, S. Pauli, M. Pohst (eds.),
Lecture Notes in Comput. Sci., 4076, Springer, 2006, 333-345.
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Genus one curves defined by Pfaffians, preprint.
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The higher secant varieties of an elliptic normal curve, preprint.
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A counterexample to a conjecture of Selmer, in
Number theory and algebraic geometry, M. Reid, A. Skorobogatov (eds.),
LMS Lecture Note Series 303, CUP 2003.
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The Cassels-Tate pairing and the Platonic solids,
J. Number Theory 98 (2003) 105-155.
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Descent calculations
for the elliptic curves of conductor 11, Proc. Lond. Math. Soc.
(3) 86 (2003) 583-606.
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Diagonal cubic equations in four variables with prime coefficients, joint with C.L. Basile, in Rational points on algebraic varieties, E. Peyre, Y. Tschinkel (eds.), Progress in Mathematics, Birkhauser, 2001, 1-12.
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Some examples of 5
and 7 descent for elliptic curves over Q, J. Eur. Math.
Soc. 3 (2001) Issue 2, 169-201.
PhD Thesis
On 5 and 7 descents for elliptic curves
Last updated 22nd January 2024