Tim Dokchitser

Research

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

(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  of my Ph.D. ancestors  (light) and some of  my Ph.D. relatives (dark).  Most of the data comes from the Mathematics Genealogy Project   at Minnesota State University.

My Ph.D. genealogy tree

Birch-Swinnerton-Dyer conjecture (Instructional course in Leiden, April 2009)

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.