curve database
ComputeL Curves and L-functions
Trieste Aug-Sep 2017
He, Li, Ag
Algebra 2 TCC Galois

Tim Dokchitser

Heilbronn Chair in Arithmetic/Algebraic Geometry

Department of Mathematics
University Walk

Office: 2.7 Howard House
tel 0117 954 56 23



Most of my work concerns elliptic curves, the Birch-Swinnerton-Dyer conjecture, Galois representations and L-functions. All of my algorithms are in Magma: L-functions, Artin Representations, and Galois Representations (new); the original Pari-based package ComputeL to compute special values of L-functions is also available through William Stein's SAGE, and a database of Artin representations is included in LMFDB/Artin representations. I am an editor of the Glasgow Mathematical Journal and of the International Journal of Number Theory (IJNT).

Recent publications/preprints:

(arxiv) Quotients of hyperelliptic curves and etale cohomology, with V. Dokchitser, November 2015
(arxiv) Rational representations and permutation representations of finite groups, with A. Bartel, Math. Ann. 364, Issue 1 (2016), 539-558 (abs)
(arxiv) A positive proportion of hyperelliptic curves have odd/even 2-Selmer rank, with V. Dokchitser, October 2013, appendix to
     M. Bhargava, B. H. Gross, X. Wang "Pencils of quadrics and the arithmetic of hyperelliptic curves", to appear in J. Amer. Math. Soc.
(arxiv) Euler factors determine local Weil representations, with V. Dokchitser, Crelle, Issue 717 (2016), 35-46 (abs)
(arxiv) Growth of Ш in towers for isogenous curves, with V. Dokchitser, Compositio Math. 151 (2015), 1981-2005 (abs)
(arxiv) Local invariants of isogenous elliptic curves, with V. Dokchitser, Trans. Amer. Math. Soc. 367 (2015), 4339-4358 (abs, pdf)
(arxiv) A remark on Tate's algorithm and Kodaira types, with V. Dokchitser, Acta Arith. 160 (2013), 95-100 (abs, pdf)
(arxiv) Non-commutative Iwasawa theory for modular forms, with J. Coates, Z. Liang, W. Stein and R. Sujatha,
    Proc. London Math. Soc. 107(3), 2013, 481-516 (abs, pdf)
(arxiv) Brauer relations in finite groups II - quasi-elementary groups of order paq, with A. Bartel, J. Group Theory 17, Issue 3 (2014), 381-393. (abs)
(arxiv) Surjectivity of mod 2n representations of elliptic curves, with V. Dokchitser, Math. Z., Vol. 272, Issue 3-4 (2012), 961-964 (abs, pdf)
(arxiv) Brauer relations in finite groups, with A. Bartel, J. Eur. Math. Soc. 17 (2015), 2473-2512
(arxiv) Identifying Frobenius elements in Galois groups, with V. Dokchitser, Algebra and Number Theory 7 (2013), no. 6, 1325-1352 (abs)
(arxiv) Notes on the parity conjecture, September 2010, CRM Barcelona Advanced Courses in Mathematics
     "Elliptic Curves, Hilbert Modular Forms and Galois Deformations", Birkhauser, 2013 (publisher, amazon)
(arxiv) Solomon's induction in quasi-elementary groups, J. Group Theory 14 (2011), 49-51 (abs)
(arxiv) A note on the Mordell-Weil rank modulo n, with V. Dokchitser, J. Number Theory 131 (2011), 1833-1839 (abs)
(arxiv) Root numbers and parity of ranks of elliptic curves, with V. Dokchitser, Crelle, Issue 658 (2011), 39-64 (abs)
(arxiv) A note on Larsen's conjecture and ranks of elliptic curves, with V. Dokchitser, Bull. London Math. Soc. 41 no. 6 (2009), 1002-1008 (abs, pdf)
(arxiv) Elliptic curves with all quadratic twists of positive rank, with V. Dokchitser, Acta Arith. 137 (2009), 193-197 (abs)
(arxiv) Regulator constants and the parity conjecture, with V. Dokchitser, Invent. Math. 178, no. 1 (2009), 23-71 (abs, pdf)
     [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, Annals of Math. 172 no. 1 (2010), 567-596 (abs, pdf)
     [final version (Apr 2007) with the 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 (abs)
(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 (abs, pdf)
(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.

  PhD genealogy tree

Future/past events

Curves and L-functions
Summer school and workshop, ICTP Trieste, August 28 - September 8, 2017

British Mathematical Colloquium
Bristol, March 21-24, 2016

LMS-CMI Summer school on Diophantine Equations
Baskerville Hall, September 14-20, 2015

Elliptic curves, modular forms and Iwasawa theory
Conference in honour of the 70th birthday of John Coates, Cambridge, March 25-27, 2015

Algebraic Geometry
TCC graduate lecture course, with lecture notes, Spring 2015.

2nd EU/US Summer School and Workshop on Automorphic Forms and Related Topics
Bristol, June 30-July 14, 2014 - L-functions course lecture notes (3 lectures)

Average ranks of elliptic curves (after Bhargava and Shankar)
Short lecture course in Poznan, May 2014, lecture notes by Zędrzej Garnek

LMS Regional meeting
"Arithmetic of L-functions", Bristol, 1-3 October, 2012

"Selmer Groups, Descent and Rank Distributions", Warwick, 24-28 September, 2012
workshop home page; part of the EPSRC Warwick Number Theory Symposium

School and Workshop on Computational Algebra and Number Theory, Trieste, June 18-29, 2012
school website

BSD Summer school, Sardinia, June 26-July 3, 2011
notes from all 3 courses by Jim Brown.

BSD Conference, Cambridge, 4-6 May, 2011
program, participants, talk notes

SAGE Days 22: Computing with elliptic curves, MSRI Berkeley, June 2010
program and videos

Birch-Swinnerton-Dyer and Parity (Short course in CRM Barcelona, December 2009)
Course notes and references

Potential Modularity Day, 30 October 2009
program and videos

Alex Bartel Julio Brau Vladimir Dokchitser Christopher Doris Adam Morgan Rachel Newton