My collected works (last updated March 2012)

Download MathSciNet reviews of some of my papers: PDF, DVI, or HTML.

- Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks (22 pages), with Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer and Jamie Weigandt, 2016, submitted.
- A
*p*-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties [arxiv] (33 pages), with Jennifer S. Balakrishnan and J. Steffen Muller, 2013, accepted. Published version [pdf]. -
*p*-adic Heights of Heegner Points and Anticyclotomic Lambda-Adic Regulators (34 pages), with Jennifer S. Balakrishnan and Mirela Ciperiani, 2013, accepted. Published version [pdf]. - Non-commutative Iwasawa theory for modular forms (40 pages), with John Coates, Tim Dokchitser, Zhibin Liang, and Ramdorai Sujatha, 2012, accepted.
- A Database Of Elliptic Curves Over Q(sqrt(5)) -- First Report (16 pages), with Jonathan Bober, Alyson Deines, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana, Ashwath Rabindranath, Paul Sharaba, 2012, in ANTS.
- Numerical computation of Chow-Heegner points associated to pairs of elliptic curves (12 pages), 2011. Published version [pdf].
- Sage: Creating a Viable Free Open Source Alternative to Magma, Maple, Mathematica, and MATLAB (9 pages), 2011, to appear in the FoCM 2011 proceedings.
- Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory (46 pages), with Christian Wuthrich, 2011, appeared
*Mathematics of Computation*. Published version [pdf]. - Kolyvagin's Conjecture for Some Specific Higher Rank Elliptic Curves (40 pages), 2011, submitted.
- Heegner Points and the Arithmetic of Elliptic Curves over Ring Class Extensions (20 pages), with Robert Bradshaw, 2012, in Journal of Number Theory.
- The Sage Project: Unifying Free Mathematical Software to Create a Viable Alternative to Magma, Maple, Mathematica and Matlab, with Burcin Erocal, 2010, for my plenary talk at the 2010 International Congress of Mathematical Software in Japan.
- The Modular Degree, Congruence Primes and Multiplicity One, with Amod Agashe and Ken Ribet (16 pages), 2009, to appear in a volume in honor of Serge Lang.
- Toward a Generalization of the Gross-Zagier Conjecture (33 pages), 2009, IMRN.
- 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.
- 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.
- Elementary Number Theory: Primes, Congruences, and Secrets (book), 2008 published by Springer-Verlag
- Three Lectures about Explicit Methods in Number Theory Using Sage (38 pages), 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. - Open Source Mathematical Software (opinion piece) with David Joyner, November 2007. The link is in the upper right corner of the site.
- The Birch and Swinnerton-Dyer Conjecture, a Computational Approach, (70 pages), 2007, free online book.
- 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.
- Modular Forms: A Computational Approach (free online book) or buy it from the AMS or buy it from Amazon.com; with an appendix by Paul Gunnells (282 pages), AMS Graduate Studies in Mathematics, Vol. 79.
- Average Ranks of Elliptic Curves, with Baur Bektemirov, Barry Mazur and Mark Watkins (19 pages), 2007, appeared in the Bulletins of the AMS.
- The Manin Constant, with Amod Agashe and Ken Ribet (22 pages), 2006, accepted.
- Computation of p-Adic Heights and Log Convergence,
with Barry Mazur and John Tate (36 pages), 2005, appeared.

- Visualizing Elements of Shafarevich-Tate Groups at Higher Level,
with Dimitar Jetchev (28 pages), 2007, Documenta Mathematica
- Visibility of Mordell-Weil Groups (20 pages), 2007, Documenta
Mathematica
- A Brief Introduction To Classical and Adelic Algebraic Number Theory (190 pages), free online book.
- Conjectures About Discriminants of Hecke Algebras of Prime Level, with
Frank Calegari (ANTS VI Proceedings, 2004).
- Modular Degrees of Neumann-Setzer Curves, with Mark Watkins (IMRN 2004, no 27, 1395-1405).
- 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).
- A Database of Elliptic Curves---First Report,
with Mark Watkins (ANTS V proceedings, Sydney, Australia, 2002).
- 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).
- Shafarevich-Tate Groups of Nonsquare Order
(Progress in Math.,
**224**(2004), 277-289, Birkhauser). - Approximation of Infinite-Slope Modular Eigenforms
By Finite-Slope Eigenforms, with Robert Coleman (Dwork Proceedings).
- J
_{1}(p) Has Connected Fibers, with Brian Conrad and Bas Edixhoven (Documenta Mathematica,**8**(2003), pages 331-408). - 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).
- 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]

- Component Groups of Purely Toric Quotients of Semistable
Jacobians, with Brian Conrad
(Math.
Res. Letters,
**8**(2001) no. 5-6, 745-766) - Appendix on Generating the Hecke Algebra,
with Amod Agashe (Experimental Math.,
**11**(2002), no. 2, 303-311). - The field generated by the points of small prime order
on an elliptic curve, with Loic Merel
(IMRN,
2001, no. 20, 1075-1082.)
- An introduction to computing modular forms using
modular symbols (in an MSRI proceedings).
- Lectures on Serre's conjectures, with Ken Ribet (Arithmetic
Algebraic Geometry, IAS/Park City Math. Inst. Series, Vol. 9, 143-232).
- Cuspidal modular symbols are transportable, with
Helena Verrill (LMS Journal of Computation and Mathematics,
**4**(2001), 170-181). - There are genus one curves over
**Q**of every odd index (J. Reine Angew. Math.**547**(2002), 139-147). - A mod five approach to modularity of icosahedral Galois
representations, with Kevin Buzzard
(Pac. J. Math.
**203**(2002), no. 2, 265-282) - Explicit approaches to modular abelian varieties
(UC Berkeley Ph.D. thesis)
- Component groups of quotients of J
_{0}(N), with David Kohel (ANTS IV proceedings) - 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).