Selected papers, preprints and talks

The Lfunctions and modular forms database project
Based on a plenary lecture given at FoCM'14, December 2014,
Montevideo, Uruguay
ArXiv 1511.04289 (14 pages, November 2015) (minor revision January 2016)
Foundations of Computational Mathematics, February 2016.

Joint with Manjul Bhargava and Tom Fisher:
The proportion of plane cubic curves over
$\mathbb{Q}$ that everywhere locally have a point
ArXiv 1311.5578 (13 pages, November 2013) (revised July 2015)
International
Journal of Number Theory Vol.~12, No.~04.

Joint with Manjul Bhargava, Tom Fisher, Nick Jones and Jon Keating:
What is the probability that a random
integral quadratic form in $n$ variables has an integral zero?
ArXiv 1502.05992 (17 pages, February 2015, supercedes
ArXiv 1311.5543 (November 2013) which has been withdrawn)
IMRN (published online, 9 September
2015); full
text; doi:10.1093/imrn/rnv251

Joint with Barinder Banwait:
Tetrahedral Elliptic Curves and the
localtoglobal principle for Isogenies
Algebra & Number Theory 85 (2014), 12011229. DOI 10.2140/ant.2014.8.1201
ArXiv 1305.6818 (23 pages, June 2013; revised August 2013 and May 2014)
Details of Sage computation of the three genus 3 modular curves of level 13.
BibTeX

Joint with M. T. Aranes:
Congruence subgroups, cusps and Manin symbols over number fields
pdf (19 pages, 20130301)
"Computations with Modular Forms" Proceedings of the Summer School and Conference
Heidelberg August/September 2011, eds. G. Boeckle, G. Wiese.
Springer 2014, Contributions
in Mathematical and Computational Sciences Volume 6.

Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael
Stoll:
Explicit ndescent on elliptic curves III. Algorithms
ArXiv 1107.3516 (43 pages, revised August 2012)
Mathematics
of Computation 84 No.292 (2015), 895922
( offprint )
BibTeX

Numerical evidence for the Birch SwinnertonDyer
Conjecture:
slides for a talk given in Cambridge at
the BSD 50th anniversary conference, May 2011.

Joint with Thotsaphon Thongjunthug:
The complex AGM, periods of elliptic
curves over $\mathbb{C}$ and complex elliptic logarithms
arXiv:1011.0914v1 [math.NT]
The Journal of Number Theory
Volume 133, Issue 8,
August 2013, Pages 28132841
slides for a talk about this.
BibTeX

Joint with Tom Fisher and Michael Stoll:
Minimisation and reduction of 2, 3 and
4coverings of elliptic curves
ArXiV
math.NT/0908.1741, 56 pages
Journal of Algebra & Number Theory 4 No.6 (2010), 763820.
BibTeX

Joint with Andrew V. Sutherland:
On a theorem of Mestre and Schoof:
pdf (4 pages, 20081231);
arXiv:0901.0120v1 [math.NT]
Journal de Théorie des nombres de Bordeaux, Tome 22 No.2
(2010), pages 353358.
BibTeX

Joint with Martin Edjvet:
Cyclically Presented Groups and
Resultants
pdf (15 pages, 20100313).
IJAC
(International Journal of Algebra and Computation)
2010 Volume
No.20, Issue No. 3, pp. 417435 , DOI: 10.1142/S0218196710005777.
BibTeX

Reduction of binary forms over imaginary
quadratic fields:
slides for a talk given in Bordeaux at
the conference "Explicit Methods in Number Theory" in honour of Henri
Cohen, October 2007.

Unimodular Integer Circulants:
preprint, 14 pages (20070228, minor
corrections 20070516, revision 20070628, minor revision 20070726)
Mathematics of Computation 77 (2008), 16391652.
BibTeX

Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael
Stoll:
Explicit ndescent on elliptic curves II. Geometry:
ArXiv
math.NT/0611606, 24 pages
Journal für die reine und angewandte Mathematik 632 (2009), 6384
BibTeX

Joint with Tom Fisher:
On the equivalence of binary quartics:
pdf, 13 pages (20081113)
Journal of Symbolic Computation, 44 (2009) 673682. DOI: doi:10.1016/j.jsc.2008.09.004
BibTeX

Computing in component groups of elliptic curves:
preprint (pdf)
ANTS
VIII Proceedings: A. van der Poorten, A. Stein (eds.), ANTS VIII,
Lecture Notes in Computer Science 5011 (2008), pages 118124.
Available online from SpringerLink
here
Slides from the talk given at ANTS: pdf
BibTeX

Joint with Mark Lingham:
Finding all elliptic curves with good reduction outside a given set
of primes:
pdf, 15 pages (Revised 20061214)
Experimental Mathematics 16 No.3 (2007), 303312.
Slides for a talk
BibTeX

Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael
Stoll:
Explicit ndescent on elliptic curves I. Algebra:
ArXiv
math.NT/0606580, 37 pages
Journal für die reine und angewandte Mathematik, Vol.
615 (2008), 121155.
BibTeX
 Appendix to a paper by Amod Agashe, Ken Ribet and William Stein:
The Manin Constant
Pure and Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617636.
Preprint available here.
BibTeX

Joint with Samir Siksek:
Computing a Lower Bound for the Canonical Height on Elliptic Curves
over Q:
pdf [revised 20060407].
ANTS
VII Proceedings: F.Hess, S.Pauli and M.Pohst (eds.), ANTS VII,
Lecture Notes in Computer Science 4076 (2006), pages 275286.
Available online from SpringerLink here
BibTeX

The elliptic curve database for conductors to 130000:
ANTS
VII Proceedings: F.Hess, S.Pauli and M.Pohst (eds.), ANTS VII,
Lecture Notes in Computer Science 4076 (2006), pages 1129.
Preprint (with some corrections)
Slides for talk, either with or without page pauses (with some corrections)
Available online from SpringerLink here
BibTeX

Joint with Mark van Hoeij:
Solving Conics over Function Fields:
pdf (revised 21 July 2006).
Journal de Théorie des Nombres de Bordeaux 18(3) 2006, 595606.
BibTeX

Joint with Martin Prickett and Samir Siksek:
Height Difference Bounds For Elliptic Curves over Number Fields:
pdf.
Journal of Number Theory 116(1) (2006), 4268.
(Available from Science Direct)
BibTeX

Joint with Samir Siksek:
On the Diophantine Equation $x^2+7=y^m$:
ps, pdf.
Acta Arithmetica 109.2 (2003), 143149.
Slides for a talk on this.
BibTeX

Joint with Michael Stoll:
On the reduction theory of binary forms (20 pages):
ps, pdf.
Journal für die reine und angewandte Mathematik 565 (2003), 7999.
Slides for a talk on this.
BibTeX

Joint with Michael Stoll:
Minimal models for 2coverings of Elliptic Curves:
BibTeX

On the computation of MordellWeil and 2Selmer Groups of
Elliptic Curves (11 pages): ps, pdf.
Rocky Mountain Journal of Mathematics (2002) vol. 32, no. 3, pp.953967.
BibTeX

Review of Henri Cohen's Book "Advanced Topics in Computational Number
Theory", for the LMS: ps, pdf.

Joint with Barry Mazur:
Visualizing elements in the ShafarevichTate group (20 pages):
preprint in dvi, ps format
Experimental Mathematics 9:1 (2000), pages 1328.
BibTeX
See also
A. Agashe, W. Stein, J. Cremona, B. Mazur:
Visible
evidence for the Birch and SwinnertonDyer conjecture for
modular abelian varieties of analytic rank zero,
Mathematics of Computation, Vol 74, Number 249, 2005, pages 455484.

Joint with Dave Rusin:
Efficient solution of rational conics:
final version, August 2001, (minor revisions Jan 2002) dvi, ps, pdf (23 pages)
Mathematics of Computation, 72 (2003), no. 243, pages 14171441.
BibTeX

Reduction of binary cubic and quartic forms (31 pages):
BibTeX

Modular Forms and Elliptic Curves over Imaginary Quadratic Number
Fields (12 pages): dvi, ps
Talk given at ESF conference, Spain 1997 (extended abstract).
Updated version (2008)

Review of Ian Stewart's Book "Nature's Numbers", AMS Notices
December 1997.

Higher Descents on Elliptic Curves (9 pages): dvi, ps (notes for talk).

Classical Invariants and 2descent on Elliptic Curves
 slides for talk (15 pages dvi).

preprint:
dvi,
ps (17 pages).

Published version:
Journal of Symbolic Computation (Proceedings of the
Second Magma Conference, Milwaukee, May 1996), Jan/Feb 2001, pages 7187.
BibTeX

Joint with P. Serf:
Computing the rank of elliptic curves over real quadratic fields of
class number 1
BibTeX

Computing Periods of Cusp Forms and modular elliptic curves (12
pages)
Experimental
Mathematics 6:2 (1997), pp.97107
BibTeX

Modular Symbols and the computation of Modular Elliptic Curves
(31 pages): dvi, ps,
gzip'd ps,
Lecture notes for short course given in Bordeaux, May 1995
This is essentially an abbreviated form of Chapter 2 of the book which is now online.

The Arithmetic of Elliptic Curves (13 pages): dvi or ps (slides for talk).
 Computing the Degree of the Modular Parametrization of a
Modular Elliptic Curve (16 pages):
Preprint version of
Math. Comp. 62 No. 205 (Jan. 1994) pp. 407429
BibTeX
 Modular symbols for $\Gamma_1(N)$ and
elliptic curves with everywhere good reduction
pdf
Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 199218.
BibTeX
Some earlier papers are available online; some of these may require
some kind of authorization. Feel free to email me for offprints.
The following is not a complete list.
 Hyperbolic tessellations, modular symbols, and elliptic curves
over complex quadratic fields
Compositio Mathematica, 51 no. 3
(1984), p. 275324 [BibTeX]
 Addendum and errata to previous paper
Compositio Mathematica, 63 no. 2 (1987), 271272 [BibTeX]
 On GL(n) of a Dedekind Domain
Quarterly Journal of Mathematics 39 (1988) 423426. [BibTeX]
 Some density results for negative Pell
equations; an application of graph theory
Journal London Math. Soc. 39 (1989) 16  28. [BibTeX]
 Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields
Journal
of the London Mathematical Society 45 (1992) 402416 [BibTeX]
 (with E. Whitley) Periods of cusp forms and elliptic curves
over imaginary quadratic fields
Mathematics of Computation 62 No. 205 (1994), 407429.
[BibTeX]
 (with P.W.Fowler)
Systematics of bonding in nonicosahedral carbon clusters
Theoretica Chimica Acta
Volume 73, Number 1 / January, 1988, pp. 126 (DOI: 10.1007/BF00526647)
 (with P.W.Fowler)
Fullerenes containing fused triples of pentagonal rings
J. Chem. Soc., Faraday Trans., 1997, 93, 2255  2262, DOI: 10.1039/a701271c
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