| DPMMS | Robinson | Research | Number theory seminar |
ComputeL | BSD in Leiden |
LMS Spitalfields Day "Potential Modularity" 30 October 2009 |
Tim DokchitserRoyal Society University Research Fellow
Dept of Pure Maths and Math Statistics Robinson College, Cambridge, CB3 9AN DPMMS office C0.11 |
|
Most of my work concerns elliptic curves and the "standard conjectures" for L-functions associated to arithmetic varieties. For my algorithms to compute special values of these L-functions see ComputeL (Pari-based), Magma (starting from V2.12, July 2005, see L-functions chapter) and W. Stein's SAGE
Recent publications/preprints:
(arxiv)
A note on the Mordell-Weil rank modulo n, with V. Dokchitser, preprint, October 2009
(arxiv) Root numbers and parity of ranks of elliptic curves, with V. Dokchitser, preprint, June 2009
(arxiv) A note on Larsen's conjecture and ranks of elliptic curves,
with V. Dokchitser, to appear in Bull. London Math. Soc.
(abs,
pdf)
(arxiv) Elliptic curves with all quadratic twists of positive rank, with V. Dokchitser, Acta Arith. 137 (2009), 193-197
(journal)
(arxiv) Regulator constants and the parity conjecture, with V. Dokchitser, Invent. Math. 178, no. 1 (2009), 23-71
(journal)
[Regulator constants - group-theoretic version]
(arxiv) Quotients of functors of Artin rings,
Proc. Cam. Phil. Soc. 146 (2009), 531-534
(abs,
pdf)
(arxiv) Self-duality of Selmer groups, with V. Dokchitser,
Proc. Cam. Phil. Soc. 146 (2009), 257-267 (pdf)
(arxiv) On the Birch-Swinnerton-Dyer quotients modulo squares, with V. Dokchitser, to appear in Annals of Math.
[final
journal version with a proof of the Parity Conjecture for Selmer ranks over Q]
(arxiv) Root numbers of elliptic curves in residue characteristic 2, with V.
Dokchitser
Bull. London Math. Soc. 40 (2008), 516-524
(abs, pdf)
(arxiv) Parity of ranks for elliptic curves with a cyclic isogeny, with V. Dokchitser, J. Number Theory 128 (2008), 662-679
(pdf)
(arxiv) Ranks of elliptic curves in cubic extensions,
Acta Arith. 126 (2007), 357-360
(arxiv) Computations in non-commutative Iwasawa theory, with V. Dokchitser
and appendix by J. Coates and R. Sujatha, Proc. London Math. Soc. (3) 94 (2006), 211-272
(arxiv) Numerical verification of Beilinson's conjecture for K2 of hyperelliptic curves,
with R. de Jeu and D. Zagier
Compositio Math. 142, Issue 02 (2006), 339-373
(pdf)
(arxiv) LLL & ABC, J. Number Theory 107, No.1 (2004), 161-167
(pdf)
for additional algebraic ABC examples see
ABC conjecture home page
(arxiv)
Computing special values of motivic L-functions, Exper. Math. 13, No.2 (2004), 137-149
(pdf)
| My Ph.D. thesis (2000,
University of Utrecht, The Netherlands) is
``Deformations of p-divisible groups and p-descent on elliptic curves'' This is a link to the
thesis, and on the right is
a link to the genealogy tree |
My Ph.D. genealogy tree |
Exercise sheet 1
Exercise sheet 2
Exercise sheet 3
Exercise sheet 4
I mentioned in the course that the following `embarrassing' question is
unsolved: given a number field, prove that there is an elliptic curve over it of
rank r<100. As of 23.04.09 this is no longer true! An extremely interesting
paper by Barry Mazur and Karl Rubin settles this question (even with r=0) and
many other things:
Ranks of twists of elliptic curves and Hilbert's Tenth Problem.