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Research publications
Papers Publications on elliptic curves, modular forms, arithmetic geometry, computational number theory, and open mathematical software.
51 listed works
39 research papers
4 research areas
This broad scholarly catalog includes papers, books, a thesis, and exposition. Archive links preserve source, computations, drafts, and supporting data.
2023 Torsion points on elliptic curves over number fields of small degree With Maarten Derickx, Sheldon Kamienny, and Michael Stoll. Algebra & Number Theory 17 (2023), 267-308. Paper Elliptic curves and BSD DOI arXiv 2016 Beyond the black box With Jeroen Demeyer and Ursula Whitcher. Notices of the AMS 63 (2016), 928-929. Article Number theory PDF arXiv DOI 2016 Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks With Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, and James Weigandt. LMS Journal of Computation and Mathematics 19 (2016), 351-370. Paper Elliptic curves and BSD DOI arXiv Data 2016 A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties With Jennifer S. Balakrishnan and J. Steffen Muller. Mathematics of Computation 85 (2016), no. 298, 983-1016. Paper Elliptic curves and BSD PDF arXiv DOI 2015 p-adic heights of Heegner points and Lambda-adic regulators With Jennifer S. Balakrishnan and Mirela Ciperiani. Mathematics of Computation 84 (2015), no. 292, 923-954. Paper Elliptic curves and BSD PDF Draft 2013 Non-commutative Iwasawa theory for modular forms With John Coates, Tim Dokchitser, Zhibin Liang, and Ramdorai Sujatha. Proceedings of the London Mathematical Society 107 (2013), 481-516. Paper Elliptic curves and BSD PDF arXiv DOI Archive 2013 Algorithms for arithmetic elliptic curves using Iwasawa theory With Christian Wuthrich. Mathematics of Computation 82 (2013), no. 283, 1757-1792. Paper Computation and software PDF DOI Archive 2012 A database of elliptic curves over Q(sqrt(5)): first report With Jonathan Bober, Alyson Deines, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana, Ashwath Rabindranath, and Paul Sharaba. ANTS X proceedings (2012). Paper Elliptic curves and BSD PDF arXiv Data Archive 2012 Numerical computation of Chow-Heegner points associated to pairs of elliptic curves 2012 preprint; a modified version was published as an appendix in Mathematics of Computation. Exposition Computation and software Published PDF Draft 2012 Heegner Points and the Arithmetic of Elliptic Curves over Ring Class Extensions With Robert Bradshaw. Journal of Number Theory 132 (2012), no. 8, 1707-1719. Paper Elliptic curves and BSD PDF DOI Archive 2012 The Modular Degree, Congruence Primes and Multiplicity One With Amod Agashe and Kenneth Ribet. Number theory, analysis and geometry (2012), 19-49. Paper Modular forms PDF TeX Archive 2011 Sage: Creating a Viable Free Open Source Alternative to Magma, Maple, Mathematica, and MATLAB Foundations of Computational Mathematics 2011 proceedings. Paper Computation and software PDF TeX Archive 2011 Kolyvagin's Conjecture for Some Specific Higher Rank Elliptic Curves 2011 preprint (40 pages). Paper Elliptic curves and BSD PDF TeX Archive 2010 The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and Matlab With Burcin Erocal. Plenary paper for the 2010 International Congress on Mathematical Software. Paper Computation and software PDF 2010 Fast Computation of Hermite Normal Forms of Random Integer Matrices (9 pages), with Clement Pernet, Volume 130, Issue 7, July 2010, Pages 1675-1683, Journal of Number Theory. Paper Computation and software PDF TeX Archive 2009 Toward a Generalization of the Gross-Zagier Conjecture (33 pages), 2009, IMRN. Paper Number theory PDF TeX Archive 2009 Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves with G. Grigorov, A. Jorza, S. Patrikis, and C. Patrascu (29 pages), Math. Comp. 78 (2009), 2397-2425. Paper Elliptic curves and BSD Archive 2008 Elementary Number Theory: Primes, Congruences, and Secrets (book), 2008 published by Springer-Verlag Book Number theory Archive 2008 Three Lectures about Explicit Methods in Number Theory Using Sage (38 pages), 2008. Exposition Computation and software Archive 2008 On the generation of the coefficient field of a newform by a single Hecke eigenvalue with Koopa Koo and Gabor Wiese (11 pages), appeared in Journal de Theorie des Nombres de Bordeaux 20 (2008), 373-384 . Paper Modular forms Archive 2007 Open Source Mathematical Software (opinion piece) with David Joyner, November 2007. The link is in the upper right corner of the site. Article Computation and software External 2007 The Birch and Swinnerton-Dyer Conjecture, a Computational Approach, (70 pages), 2007, free online book. Book Computation and software Archive 2007 Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich-Tate group with Dimitar Jetchev and Kristin Lauter (18 pages), 2007, accepted. Paper Elliptic curves and BSD Archive 2007 Average Ranks of Elliptic Curves with Baur Bektemirov, Barry Mazur and Mark Watkins (19 pages), 2007, appeared in the Bulletins of the AMS. Paper Elliptic curves and BSD Archive 2007 Visualizing Elements of Shafarevich-Tate Groups at Higher Level with Dimitar Jetchev (28 pages), 2007, Documenta Mathematica Paper Elliptic curves and BSD Archive 2007 Visibility of Mordell-Weil Groups (20 pages), 2007, Documenta Mathematica Paper Number theory Archive 2006 The Manin Constant with Amod Agashe and Ken Ribet (22 pages), 2006, accepted. Paper Modular forms Archive 2005 Computation of p-Adic Heights and Log Convergence with Barry Mazur and John Tate (36 pages), 2005, appeared. Paper Computation and software Archive 2004 Conjectures About Discriminants of Hecke Algebras of Prime Level with Frank Calegari ( ANTS VI Proceedings, 2004). Paper Modular forms Archive 2004 Modular Degrees of Neumann-Setzer Curves with Mark Watkins ( IMRN 2004, no 27, 1395-1405). Paper Modular forms Archive 2004 Shafarevich-Tate Groups of Nonsquare Order ( Progress in Math., 224 (2004), 277-289, Birkhauser ). Paper Elliptic curves and BSD Archive 2003 J 1 (p) Has Connected Fibers with Brian Conrad and Bas Edixhoven ( Documenta Mathematica, 8 (2003), pages 331-408 ). Paper Number theory Archive 2002 A Database of Elliptic Curves---First Report with Mark Watkins ( ANTS V proceedings, Sydney, Australia, 2002). Paper Elliptic curves and BSD Archive 2002 Visibility of Shafarevich-Tate Groups of Abelian Varieties with Amod Agashe ( J. of Number Theory , 97 (2002), no. 1, 171-185.) Visibility of Shafarevich-Tate Groups [streaming MSRI video] Paper Elliptic curves and BSD Archive 2002 Appendix on Generating the Hecke Algebra with Amod Agashe ( Experimental Math. , 11 (2002), no. 2, 303-311). Paper Modular forms Archive 2002 There are genus one curves over Q of every odd index ( J. Reine Angew. Math. 547 (2002), 139-147). Paper Number theory Archive 2002 A mod five approach to modularity of icosahedral Galois representations with Kevin Buzzard ( Pac. J. Math. 203 (2002), no. 2, 265-282 ) Paper Modular forms Archive 2001 Component Groups of Purely Toric Quotients of Semistable Jacobians with Brian Conrad ( Math. Res. Letters , 8 (2001) no. 5-6, 745-766) Paper Number theory Archive 2001 The field generated by the points of small prime order on an elliptic curve with Loic Merel ( IMRN , 2001, no. 20, 1075-1082 .) Paper Elliptic curves and BSD Archive 2001 Cuspidal modular symbols are transportable with Helena Verrill ( LMS Journal of Computation and Mathematics , 4 (2001), 170-181 ). Paper Modular forms Archive 2001 Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves with five coauthors ( Mathematics of Computation , 70 (2001), no . 236, 1675-1697). Paper Elliptic curves and BSD Archive Earlier Modular Forms: A Computational Approach With an appendix by Paul Gunnells. Graduate Studies in Mathematics 79, American Mathematical Society. Book Computation and software HTML PDF Earlier A Brief Introduction To Classical and Adelic Algebraic Number Theory (190 pages), free online book. Book Number theory Archive Earlier Studying the Birch and Swinnerton-Dyer Conjecture for Modular Abelian Varieties Using MAGMA (appeared as a chapter in a book for Springer-Verlag edited by John Cannon ). Exposition Computation and software Archive Earlier Constructing Elements in Shafarevich-Tate Groups of Modular Motives with Neil Dummigan and Mark Watkins ( "Number theory and algebraic geometry--to Peter Swinnerton-Dyer on his 75th birthday", Ed. M. Reid and A. Skorobogatov, pages 91-118 ). Paper Elliptic curves and BSD Archive Earlier Approximation of Infinite-Slope Modular Eigenforms By Finite-Slope Eigenforms with Robert Coleman (Dwork Proceedings). Paper Modular forms Archive Earlier Visible Evidence for the Birch and Swinnerton-Dyer Conjecture for Rank 0 Modular Abelian Varieties with Amod Agashe , and an appendix by Mazur and Cremona (in Math. Comp., Vol 74, Number 249, pages 455-484 ). Paper Elliptic curves and BSD Archive Earlier An introduction to computing modular forms using modular symbols (in an MSRI proceedings). Exposition Modular forms Archive Earlier Lectures on Serre's conjectures with Ken Ribet (Arithmetic Algebraic Geometry, IAS/Park City Math. Inst. Series, Vol. 9, 143-232). Exposition Number theory Archive Earlier Explicit approaches to modular abelian varieties ( UC Berkeley Ph.D. thesis) Thesis Modular forms Archive Earlier Component groups of quotients of J 0 (N) with David Kohel ( ANTS IV proceedings) Paper Number theory Archive No publications match this search.
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Collected works and reviews Collected works through 2012 and selected MathSciNet reviews are preserved alongside the individual project directories.