My collected works (updated March 2012)

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

- 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 Müller, 2012, submitted -
*p*-adic Heights of Heegner Points and Anticyclotomic Lambda-Adic Regulators (34 pages), with Jennifer S. Balakrishnan and Mirela Ciperiani, 2012, submitted - Non-commutative Iwasawa theory for modular forms (40 pages), with John Coates, Tim Dokchitser, Zhibin Liang, and Ramdorai Sujatha, 2012, to appear.
- 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, to appear in ANTS.
- Numerical computation of Chow-Heegner points associated to pairs of elliptic curves (12 pages), 2011, to appear.
- 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, to appear in
*Mathematics of Computation* - 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, to appear 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).