GroupNames Regular models
of curves
Inverse Galois Problem
Park City GSS 26.7-30.7.21
curve database
ComputeL Bristol
He, Li, Ag

Tim Dokchitser

Heilbronn Chair in Arithmetic/Algebraic Geometry

University of Bristol
Fry Building
Woodland Road
Bristol BS8 1UG

tel 0117 428 4878 (school manager)



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. My most recent projects are and Regular models of curves.

Recent publications/preprints:

(arxiv) Character formula for Weil representations in terms of Frobenius traces, with V. Dokchitser, January 2022, to appear in Arithmetic of L-series, EMS Series of Congress Reports.
(arxiv) Character formula for conjugacy classes in a coset, with V. Dokchitser, May 2021, to appear in Arithmetic of L-series, EMS Series of Congress Reports.
(arxiv) Parity reduction to 2-Sylow, with V. Dokchitser, appendix to
     V. Dokchitser, C. Maistret "On the parity conjecture for abelian surfaces", Proc. London Math. Soc. 127(2), 2023, 295-365 (abs, pdf)
(arxiv) Tame torsion and the tame inverse Galois problem, with M. Bisatt, Math. Annalen 381(3) (2021), 1439-1453. (abs, pdf)
(arxiv) Models of curves over discrete valuation rings, Duke Math. J. 170, no. 11 (2021), 2519-2574 (abs, pdf)
(arxiv) Tate module and bad reduction, with V. Dokchitser and A. Morgan, Proc. Amer. Math. Soc. 149 (2021), 1361-1372 (abs, pdf)
(arxiv) Arithmetic of hyperelliptic curves over local fields ("M2D2"), with V. Dokchitser, C. Maistret and A. Morgan,
    Math. Ann. 385 (2023), 1213-1322, doi: 10.1007/s00208-021-02319-y (abs, pdf)
(arxiv) 3-torsion and conductor of genus 2 curves, with C. Doris, Math. Comp. 88 (2019), 1913-1927 (abs, pdf)
(arxiv) Semistable types of hyperelliptic curves, with V. Dokchitser, C. Maistret and A. Morgan, Contemporary Math., vol. 724 (2019)
(arxiv) Quotients of hyperelliptic curves and etale cohomology, with V. Dokchitser, Quaterly J. Math. 69, issue 2 (2018), 747-768 (abs, pdf)
(arxiv) A positive proportion of hyperelliptic curves have odd/even 2-Selmer rank, with V. Dokchitser, appendix to
     M. Bhargava, B. H. Gross, X. Wang "Pencils of quadrics and the arithmetic of hyperelliptic curves", J. Amer. Math. Soc. 30 (2017), 451-493 (abs, pdf)
(arxiv) Rational representations and permutation representations of finite groups, with A. Bartel, Math. Ann. 364, Issue 1 (2016), 539-558 (abs)
(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);
The 49a1→49a2 example in Remark 1.10 is wrong. All curves in the 49a isogeny class have the same BSD period, while Remark 1.10 claims that there is a factor 2 difference for 49a1 and 49a2. In fact, as John Coates pointed out, such an example would condtradict the MH(G) conjecture of Coates-Fukaya-Kato-Sujatha-Venjakob. --- Found by John Coates
(arxiv) Local invariants of isogenous elliptic curves, with V. Dokchitser, Trans. Amer. Math. Soc. 367 (2015), 4339-4358 (abs, pdf);
Thm A.5: K(√d)/K is supposed to be a ramified quadratic extension.
--- Found by Filip Najman and Antonela Trbović
(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, pdf)
(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);
Section 1.6 (Notation), definition of Cv(A/K,ω): the constant 2 in front of the integral for K=C should be 2dim A.
Proposition 3.23 (published version) = 3.26 (arxiv version). The representation τ must be assumed to be self-dual.
   [Regulator constants - group-theoretic version]
(arxiv) Quotients of functors of Artin rings, Proc. Cam. Phil. Soc. 146 (2009), 531-534 (abs)
(arxiv) Self-duality of Selmer groups, with V. Dokchitser, Proc. Cam. Phil. Soc. 146 (2009), 257-267 (pdf);
Proof of Theorem 2.3, second paragraph. It is not true that every abelian variety (over a number field) is isogenous to a principally polarised one. Instead, to reduce to the principally polarised case, use Zarhin's trick, which asserts that A4 is always principally polarised.
(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);
Proposition 2 assumes c4≠0 (otherwise γ(x) is not square-free). --- Found by Wrenna Robson
(arxiv) Parity of ranks for elliptic curves with a cyclic isogeny, with V. Dokchitser, J. Number Theory 128 (2008), 662-679 (abs, pdf)
(arxiv) Ranks of elliptic curves in cubic extensions, Acta Arith. 126 (2007), 357-360 (abs, pdf)
(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 (abs, pdf)
(arxiv) LLL & ABC, J. Number Theory 107, No.1 (2004), 161-167 (abs)
    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

Inverse Galois Problem
Park City online lectures, July 26-30, 2021.

Algebraic Geometry
TCC graduate lecture course, with lecture notes, Autumn 2020.

Computational Number Theory
CMI-HIMR Summer school, Bristol, 17-28 June 2019

Curves and groups in families
Summer school, Rennes, 13-17 May 2019

Arithmetic of Curves
Workshop in Baskerville Hall, Wales, 13-17 August 2018

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

BSD Data @ Bristol
One day workshop, 28 March 2017

Galois Representations
TCC graduate lecture course, Autumn 2016

British Mathematical Colloquium
Bristol, 21-24 March 2016

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

Elliptic curves, modular forms and Iwasawa theory
Conference in honour of the 70th birthday of John Coates, Cambridge, 25-27 March 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, 30 June-14 July 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, 18-29 June 2012
school website

BSD Summer school, Sardinia, 26 June-3 July 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 Matthew Bisatt Julio Brau Nirvana Coppola Vladimir Dokchitser Christopher Rowley
Pip Goodman Celine Maistret Adam Morgan Simone Muselli Bartosz Naskręcki Rachel Newton