Selected papers, preprints and talks

BibTeX entries


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  1. Joint with John W. Jones, Andrew V. Sutherland and John Voight:
    The L-Functions and Modular Forms Database
    To appear in the AMS Notices Early Career Series: preprint, June 2021.
  2. Joint with Manjul Bhargava, Tom Fisher, and Stevan Gajović:
    The density of polynomials of degree \(n\) over \(\mathbb{Z}_p\) having exactly \(r\) roots in \(\mathbb{Q}_p\)
    ArXiv 2101.09590 (23 January 2021; revised 25 March 2021)
  3. Joint with Filip Najman:
    \(\mathbb{Q}\)-curves over odd degree number fields
    Research in Number Theory 7:62 (2021), DOI.
    ArXiv 2004.10054 (revised 26 January 2021)
  4. Joint with Aurel Page and Andrew V. Sutherland:
    Sorting and labelling integral ideals in a number field
    ArXiv 2005.09491 (19 May 2020)
  5. Joint with Manjul Bhargava and Tom Fisher:
    The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point
    International Journal of Number Theory, Vol. 17, No. 04, pp. 903-923 (2021) DOI
    ArXiv 2004.12085 (revised 31 July 2020)
  6. Joint with Mohammad Sadek:
    Local and global densities for Weierstrass models of elliptic curves
    ArXiv 2003.08454 (revised 23 March 2020)
  7. Joint with Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, Maarten Derickx, David Lowry-Duda, Min Lee, David Roe, Andrew V. Sutherland, John Voight:
    Computing classical modular forms
    To appear in a Simons Symposium volume.
    ArXiv 2002.04717 (revised 11 October 2020)
  8. Joint with Nuno Freitas:
    Global methods for the symplectic type of congruences between elliptic curves
    To appear in Revista Matemática Ibroamericana.
    ArXiv 1910.12290 (revised 4 August 2020)
  9. Joint with Lassina Dembélé, Ariel Pacetti, Ciaran Schembri, John Voight:
    On rational Bianchi newforms and abelian surfaces with quaternionic multiplication
    To appear in a Simons Symposium volume.
    ArXiv 1907.12103 (revised 4 August 2020)
  10. Joint with Ariel Pacetti:
    On Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one
    Proceedings of the LMS Vol.118 no.5 (2019), 1245-1276. DOI.
    ArXiv 1711.02170
  11. Joint with Alejandro Argáez García:
    Black Box Galois Representations
    Journal of Algebra 512 (2018), 526--565 (full text with hyperlinks)
    ArXiv 1709.06430 (40 pages, revised May 2018; includes code for examples)
  12. The L-functions 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, 16(6), pp.1541-1553, 2016.
    Altametric data
  13. 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.
  14. 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)
    IMRN Volume 2016, Issue 12, 2016, Pages 3828–3848.
  15. Joint with Barinder Banwait:
    Tetrahedral Elliptic Curves and the local-to-global principle for Isogenies
    Algebra & Number Theory 8-5 (2014), 1201--1229. 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.
  16. Joint with M. T. Aranes:
    Congruence subgroups, cusps and Manin symbols over number fields
    pdf (19 pages, 2013-03-01)
    "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.
  17. 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)
    Mathematics of Computation 84 No.292 (2015), 895-922 ( offprint )
  18. Numerical evidence for the Birch Swinnerton-Dyer Conjecture:
    slides for a talk given in Cambridge at the BSD 50th anniversary conference, May 2011.
  19. 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
    slides for a talk about this.
  20. 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.
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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.
  30. 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.
  31. 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
  32. 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
  33. 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.
  34. 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)
  35. 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.
  36. 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.
  37. Joint with Michael Stoll:
    Minimal models for 2-coverings of Elliptic Curves: BibTeX
  38. 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.
  39. Review of Henri Cohen's Book "Advanced Topics in Computational Number Theory", for the LMS: ps, pdf.
  40. Joint with Barry Mazur:
    Visualizing elements in the Shafarevich-Tate group (20 pages): preprint in dvi, ps format
    Experimental Mathematics 9:1 (2000), pages 13-28.
    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.
  41. 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.
  42. Reduction of binary cubic and quartic forms (31 pages):
  43. 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)
  44. Review of Ian Stewart's Book "Nature's Numbers", AMS Notices December 1997.
  45. Higher Descents on Elliptic Curves (9 pages): dvi, ps (notes for talk).
  46. Classical Invariants and 2-descent on Elliptic Curves BibTeX
  47. Joint with P. Serf:
    Computing the rank of elliptic curves over real quadratic fields of class number 1 BibTeX
  48. Computing Periods of Cusp Forms and modular elliptic curves (12 pages)
    Experimental Mathematics 6:2 (1997), pp.97--107
  49. 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.
  50. The Arithmetic of Elliptic Curves (13 pages): dvi or ps (slides for talk).
  51. 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
  52. 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.
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. (with P.W.Fowler) Fullerenes containing fused triples of pentagonal rings
    J. Chem. Soc., Faraday Trans., 1997, 93, 2255 - 2262, DOI: 10.1039/a701271c
  2. (with P.W.Fowler and J.I.Steer) Systematics of bonding in non-icosahedral carbon clusters
    Theoretica Chimica Acta Volume 73, Number 1 / January, 1988, pp. 1-26 (DOI: 10.1007/BF00526647)
  3. (with E. Whitley) Periods of cusp forms and elliptic curves over imaginary quadratic fields
    Mathematics of Computation 62 No. 205 (1994), 407-429. [BibTeX]
  4. The analytic order of $Ш$ for modular elliptic curves
    Journal de Théorie des Nombres de Bordeaux, Tome 5 (1993) no. 1, pages 179-184 (DOI).
  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 Susan Landau) Shrinking Lattice Polyhedra
    SIAM J. Discrete Math., 3(3), 338--348 (DOI).
  7. (with R.W.K.Odoni) A generalization of a result of Iwasawa on the capitulation problem,
    Math. Proc. Camb. Phil. Soc. 107 (1990), pp.1--3.
  8. (with R.W.K.Odoni) Some density results for negative Pell equations; an application of graph theory
    Journal London Math. Soc. 39 (1989) 16 - 28. [BibTeX]
  9. On GL(n) of a Dedekind Domain
    Quarterly Journal of Mathematics 39 (1988) 423-426. [BibTeX]
  10. Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields
    Compositio Mathematica, 51 no. 3 (1984), p. 275-324 [BibTeX]
    Addendum and errata:
    Compositio Mathematica, 63 no. 2 (1987), 271-272 [BibTeX]

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