Tom Fisher's Home Page

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.

Publications and Preprints

• Visible 2-torsion in the Tate-Shafarevich group of an elliptic curve, preprint.
• Minimisation of 2-coverings of genus 2 Jacobians, joint with M. Liu, Manuscripta Math. 176 (2025), no. 1, Paper No. 9, 15 pages.
• Computing the Cassels-Tate pairing on the 2-Selmer group of a genus 2 Jacobian, joint with J. Yan, to appear in Algebra & Number Theory.
• The density of integral quadratic forms having a k-dimensional totally isotropic subspace, joint with L. Drakengren, Rocky Mountain J. Math. 56 (2026), no. 2, 403–421.
• On binary quartics and the Cassels-Tate pairing, (proceedings of ANTS XV), Res. Number Theory 8 (2022), Art. 74, 13 pages.
• Computing structure constants for rings of finite rank from minimal free resolutions, joint with L. Radičević, Math. Res. Lett. 30 (2023), no. 4, 1011–1044.
• On pairs of 17-congruent elliptic curves, preprint.
• The density of polynomials of degree n over Z_p having exactly r roots in Q_p, joint with M. Bhargava, J.E. Cremona and S. Gajovic, Proc. Lond. Math. Soc. (3) 124 (2022), no. 5, 713-736.
• 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.
• On families of 13-congruent elliptic curves, preprint.
• 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.
• Explicit moduli spaces for congruences of elliptic curves, Math. Zeit. 295 (2020), 1337-1354.
• 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.
• On some algebras associated to genus one curves, J. Algebra 518 (2019), 519-541.
• Some minimisation algorithms in arithmetic invariant theory, joint with L. Radičević, J. Théor. Nombres Bordeaux 30 (2018), no. 3, 801-828.
• Visibility of 4-covers of elliptic curves, joint with N. Bruin, Res. Number Theory 4 (2018), no. 1, Art. 11, 29 pages.
• 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.
• A formula for the Jacobian of a genus one curve of arbitrary degree, Algebra & Number Theory 12 (2018), no. 9, 2123-2150.
• 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.
• 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.
• Higher descents on an elliptic curve with a rational 2-torsion point, Math. Comp. 86 (2017), no. 307, 2493–2518.
• On families of 9-congruent elliptic curves, Acta Arith. 171 (2015), no. 4, 371-387.
• 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.
• 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.
• Minimal models for 6-coverings of elliptic curves, LMS J. Comput. Math. 17 (2014), proceedings of ANTS XI, 112-127.
• On families of 7 and 11-congruent elliptic curves, LMS J. Comput. Math. 17 (2014), no. 1, 536-564.
• 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.
• Invariant theory for the elliptic normal quintic, II. The covering map, preprint.
• Invisibility of Tate-Shafarevich groups in abelian surfaces, Int. Math. Res. Not. (2014), no. 15, 4085-4099.
• 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.
• Minimisation and reduction of 5-coverings of elliptic curves, Algebra & Number Theory 7 (2013), no. 5, 1179–1205.
• Invariant theory for the elliptic normal quintic, I. Twists of X(5), Math. Ann. 356 (2013), no. 2, 589-616.
• 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.
• The Hessian of a genus one curve, Proc. Lond. Math. Soc. (3) 104 (2012) 613-648.
• Local solubility and height bounds for coverings of elliptic curves, joint with G.F. Sills, Math. Comp. 81 (2012), no. 279, 1635-1662.
• Some bounds on the coefficients of covering curves, Enseign. Math. (2) 58 (2012) 99-124.
• 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.
• 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.
• The yoga of the Cassels-Tate pairing, joint with E.F. Schaefer and M.Stoll, LMS J. Comput. Math. 13 (2010) 451-460.
• Finding rational points on elliptic curves using 6-descent and 12-descent, Journal of Algebra 320 (2008), no. 2, 853-884.
• 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.
• On the equivalence of binary quartics, joint with J.E. Cremona, Journal of Symbolic Computation 44 (2009) 673-682.
• The invariants of a genus one curve, Proc. Lond. Math. Soc. (3) 97 (2008) 753-782.
• Pfaffian presentations of elliptic normal curves, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2525-2540.
• 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.
• A new approach to minimising binary quartics and ternary cubics, Math. Res. Lett. 14 (2007) Issue 4, 597--613.
• 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.
• Genus one curves defined by Pfaffians, preprint.
• The higher secant varieties of an elliptic normal curve, preprint.
• 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.
• The Cassels-Tate pairing and the Platonic solids, J. Number Theory 98 (2003) 105-155.
• Descent calculations for the elliptic curves of conductor 11, Proc. Lond. Math. Soc. (3) 86 (2003) 583-606.
• 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.
• 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