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.
Publications and Preprints
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Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves, joint with J.E. Cremona and M.Stoll, preprint.
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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,
preprint.
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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 Hessian of a genus one curve,
preprint.
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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 12th August 2009