Selected papers, preprints and talks

BibTeX entries

ORCID id: orcid.org/0000-0002-7212-0162

1. Joint with Andrew V. Sutherland:
   Computing the endomorphism ring of an elliptic curve over a number field
   ArXiV preprint, January 2023.
2. On the equivalence of binary cubic forms
   ArXiV preprint, December 2022; revised November 2023.
3. Joint with John W. Jones, Andrew V. Sutherland and John Voight:
   The L-Functions and Modular Forms Database
   AMS Notices Early Career Series, October 2021.
   preprint, June 2021.
4. 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 \(\mathbb{Q}_p\)
   Proceedings of the LMS, 17 March 2022
   ArXiv 2101.09590 (23 January 2021; revised 16 December 2021)
5. 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)
6. Joint with Aurel Page and Andrew V. Sutherland:
   Sorting and labelling integral ideals in a number field
   ArXiv 2005.09491 (19 May 2020)
7. 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 (2021) pp. 903-923. DOI
   ArXiv 2004.12085 (revised 31 July 2020)
8. Joint with Mohammad Sadek:
   Local and global densities for Weierstrass models of elliptic curves
   Mathematics Research Letters 30 No. 2 (2023), pp. 413-461. DOI
   Preprint ArXiv 2003.08454 (includes two SageMath Jupyter notebooks)
9. 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
   In Arithmetic Geometry, Number Theory, and Computation (J. Balakrishnan, N.Elkies, B. Hassett, B. Poonen, A.V. Sutherland, J. Voight, eds.), Simons Symposium Series, Springer 2022. DOI
   ArXiv 2002.04717
10. Joint with Nuno Freitas:
    Global methods for the symplectic type of congruences between elliptic curves
    Revista Matemática Ibroamericana 38 No.1 (2022), 1--32. DOI
    Preprint: ArXiv 1910.12290
11. Joint with Lassina Dembélé, Ariel Pacetti, Ciaran Schembri, John Voight:
    On rational Bianchi newforms and abelian surfaces with quaternionic multiplication
    In Arithmetic Geometry, Number Theory, and Computation (J. Balakrishnan, N.Elkies, B. Hassett, B. Poonen, A.V. Sutherland, J. Voight, eds.), Simons Symposium Series, Springer 2022. DOI
    ArXiv 1907.12103
12. 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
13. Joint with Alejandro Argáez García:
    Black Box Galois Representations
    Journal of Algebra 512 (2018), 526--565 (full text with hyperlinks)
    DOI: https://doi.org/10.1016/j.jalgebra.2018.05.017
    ArXiv 1709.06430 (40 pages, revised May 2018; includes code for examples)
    
14. 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
    DOI: http://dx.doi.org/10.1007/s10208-016-9306-z
    BibTeX
15. 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.
    DOI: https://doi.org/10.1142/S1793042116500664
    BibTeX
16. 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.
    DOI: https://doi.org/10.1093/imrn/rnv251
    BibTeX
17. 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)
    DOI: http://dx.doi.org/10.2140/ant.2014.8.1201
    Details of Sage computation of the three genus 3 modular curves of level 13.
    BibTeX
18. 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.
    DOI: https://doi.org/10.1007/978-3-319-03847-6_4
    BibTeX
19. 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 )
    DOI: https://doi.org/10.1090/S0025-5718-2014-02858-5
    BibTeX
20. Numerical evidence for the Birch Swinnerton-Dyer Conjecture:
    slides for a talk given in Cambridge at the BSD 50th anniversary conference, May 2011.
21. 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.
    DOI: http://dx.doi.org/10.1016/j.jnt.2013.02.002
    BibTeX
22. 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
23. 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.
    BibTeX
24. 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
25. 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.
26. 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
27. 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
    BibTeX
28. 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
29. 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
30. 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
31. 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
32. 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
33. 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
34. 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
35. 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
36. 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
37. 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
38. 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
39. Joint with Michael Stoll:
    Minimal models for 2-coverings of Elliptic Curves:
    • LMS JCM 5 (2002), 220--243
    • Preprint version in ps or pdf formats.
    BibTeX
40. 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
41. Review of Henri Cohen's Book "Advanced Topics in Computational Number Theory", for the LMS: ps, pdf.
42. 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.
    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.
43. 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
44. Reduction of binary cubic and quartic forms (31 pages):
    • LMS JCM 2 (1999), 62--92
    • Preprint with corrections (updated 2010-04-21): pdf.
    
    BibTeX
45. 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)
46. Review of Ian Stewart's Book "Nature's Numbers", AMS Notices December 1997.
47. Higher Descents on Elliptic Curves (9 pages): pdf (notes for talk).
48. Classical Invariants and 2-descent 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 71-87.
    BibTeX
49. Joint with P. Serf:
    Computing the rank of elliptic curves over real quadratic fields of class number 1
    • original longer version (22 pages dvi)
    • published version (revised, shorter) in Mathematics of Computation 68 (1999), pages 1187--1200
    BibTeX
50. Computing Periods of Cusp Forms and modular elliptic curves (12 pages)
    Experimental Mathematics 6:2 (1997), pp.97--107
    BibTeX
51. 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.
52. The Arithmetic of Elliptic Curves (13 pages): dvi or ps (slides for talk).
53. 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
54. 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

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]