Selected papers, preprints and talks

BibTeX entries

  1. Joint with M. T. Aranes:
    Congruence subgroups, cusps and Manin symbols over number fields
    pdf (19 pages, 2013-03-01)
    To appear in the proceedings of the Summer School and Conference "Computations with Modular Forms" (Heidelberg 2011), edited by G. Boeckle, G. Wiese and J. Voight, Springer 2013 (expected).
  2. Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael Stoll:
    Explicit n-descent on elliptic curves III. Algorithms
    ArXiv 1107.3516 (43 pages, revised August 2012)
  3. Numerical evidence for the Birch Swinnerton-Dyer Conjecture:
    slides for a talk given in Cambridge at the BSD 50th anniversary conference, May 2011.
  4. 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 2813-2841
  5. Joint with Tom Fisher and Michael Stoll:
    Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves
    ArXiV math.NT/0908.1741, 56 pages
    Journal of Algebra & Number Theory 4 No.6 (2010), 763-820.
    BibTeX
  6. Joint with Andrew V. Sutherland:
    On a theorem of Mestre and Schoof:
    pdf (4 pages, 2008-12-31); arXiv:0901.0120v1 [math.NT]
    Journal de Théorie des nombres de Bordeaux, Tome 22 No.2 (2010), pages 353-358.
  7. Joint with Martin Edjvet:
    Cyclically Presented Groups and Resultants: pdf (15 pages, 2010-03-13).
    IJAC (International Journal of Algebra and Computation) 2010 Volume No.20, Issue No. 3, pp. 417-435 , DOI: 10.1142/S0218196710005777.
    BibTeX
  8. 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.
  9. Unimodular Integer Circulants: preprint, 14 pages (2007-02-28, minor corrections 2007-05-16, revision 2007-06-28, minor revision 2007-07-26)
    Mathematics of Computation 77 (2008), 1639-1652.
    BibTeX
  10. Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael Stoll:
    Explicit n-descent on elliptic curves II. Geometry: ArXiv math.NT/0611606, 24 pages
    Journal für die reine und angewandte Mathematik 632 (2009), 63--84.
    Available online here
    BibTeX
  11. Joint with Tom Fisher:
    On the equivalence of binary quartics: pdf, 13 pages (2008-11-13)
    Journal of Symbolic Computation, 44 (2009) 673--682. DOI: doi:10.1016/j.jsc.2008.09.004
    BibTeX
  12. 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 118-124.
    Available online from SpringerLink here
    Slides from the talk given at ANTS: pdf
    BibTeX
  13. Joint with Mark Lingham:
    Finding all elliptic curves with good reduction outside a given set of primes: pdf, 15 pages (Revised 2006-12-14)
    Experimental Mathematics 16 No.3 (2007), 303-312.
    Slides for a talk
    BibTeX
  14. Joint with Tom Fisher, Catherine O'Neil, Denis Simon and Michael Stoll:
    Explicit n-descent on elliptic curves I. Algebra: ArXiv math.NT/0606580, 37 pages
    Journal für die reine und angewandte Mathematik, Vol. 615 (2008), 121-155.
    BibTeX
  15. 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. 617-636.
    Preprint available here.
    BibTeX
  16. Joint with Samir Siksek:
    Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q: pdf [revised 2006-04-07].
    ANTS VII Proceedings: F.Hess, S.Pauli and M.Pohst (eds.), ANTS VII, Lecture Notes in Computer Science 4076 (2006), pages 275-286.
    Available online from SpringerLink here
    BibTeX
  17. 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 11-29.
    Preprint (with some corrections)
    Slides for talk, either with or without page pauses (with some corrections)
    Available online from SpringerLink here
    BibTeX
  18. 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, 595--606.
    BibTeX
  19. Joint with Martin Prickett and Samir Siksek:
    Height Difference Bounds For Elliptic Curves over Number Fields: pdf.
    Journal of Number Theory 116(1) (2006), 42-68. (Available from Science Direct)
    BibTeX
  20. Joint with Samir Siksek:
    On the Diophantine Equation $x^2+7=y^m$: ps, pdf.
    Acta Arithmetica 109.2 (2003), 143--149.
    Slides for a talk on this.
    BibTeX
  21. Joint with Michael Stoll:
    On the reduction theory of binary forms (20 pages): ps, pdf.
    Journal für die reine und angewandte Mathematik 565 (2003), 79--99.
    Slides for a talk on this.
    BibTeX
  22. Joint with Michael Stoll:
    Minimal models for 2-coverings of Elliptic Curves: BibTeX
  23. On the computation of Mordell-Weil and 2-Selmer Groups of Elliptic Curves (11 pages): ps, pdf.
    Rocky Mountain Journal of Mathematics (2002) vol. 32, no. 3, pp.953--967.
    BibTeX
  24. Review of Henri Cohen's Book "Advanced Topics in Computational Number Theory", for the LMS: ps, pdf.
  25. Joint with Barry Mazur:
    Visualizing elements in the Shafarevich-Tate group (20 pages): preprint in dvi, ps format (Abstract)
    Experimental Mathematics 9:1 (2000), pages 13-28.
    BibTeX
    See also
    A. Agashe, W. Stein, J. Cremona, B. Mazur:
    Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero,
    Mathematics of Computation, Vol 74, Number 249, 2005, pages 455-484.
  26. 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 1417-1441.
    BibTeX
  27. Reduction of binary cubic and quartic forms (31 pages):
    BibTeX
  28. 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)
  29. Review of Ian Stewart's Book "Nature's Numbers", AMS Notices December 1997.
  30. Higher Descents on Elliptic Curves (9 pages): dvi, ps (notes for talk).
  31. Classical Invariants and 2-descent on Elliptic Curves BibTeX
  32. Joint with P. Serf:
    Computing the rank of elliptic curves over real quadratic fields of class number 1 BibTeX
  33. Computing Periods of Cusp Forms and modular elliptic curves (12 pages)
    Experimental Mathematics 6:2 (1997), pp.97--107
    BibTeX
  34. 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.
  35. The Arithmetic of Elliptic Curves (13 pages): dvi or ps (slides for talk).
  36. 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. 407--429
    BibTeX
  37. Modular symbols for $\Gamma_1(N)$ and elliptic curves with everywhere good reduction pdf
    Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 199--218.
    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.
  1. Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields
    Compositio Mathematica, 51 no. 3 (1984), p. 275-324 [BibTeX]
  2. Addendum and errata to previous paper
    Compositio Mathematica, 63 no. 2 (1987), 271-272 [BibTeX]
  3. On GL(n) of a Dedekind Domain
    Quarterly Journal of Mathematics 39 (1988) 423-426. [BibTeX]
  4. Some density results for negative Pell equations; an application of graph theory
    Journal London Math. Soc. 39 (1989) 16 - 28. [BibTeX]
  5. Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields
    Journal of the London Mathematical Society 45 (1992) 402-416 [BibTeX]
  6. (with E. Whitley) Periods of cusp forms and elliptic curves over imaginary quadratic fields
    Mathematics of Computation 62 No. 205 (1994), 407-429. [BibTeX]
  7. (with P.W.Fowler) Systematics of bonding in non-icosahedral carbon clusters
    Theoretica Chimica Acta Volume 73, Number 1 / January, 1988, pp. 1-26 (DOI: 10.1007/BF00526647)
  8. (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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