Diophantine Stability for Elliptic Curves on Average (with Anwesh Ray)

Hilbert's tenth problem in Anticyclotomic towers of number fields (with Anwesh Ray), Transactions of the American Mathematical Society 377 (2024), 3577-3597.

Explicit reciprocity laws and Iwasawa theory for modular forms (with Matthew Emerton and Robert Pollack), to appear in Duke Mathematical Journal.

Class group statistics for torsion fields generated by elliptic curves  (with Anwesh Ray), Journal of the Australian Mathematical Society (2024).

Arithmetic statistics for Galois deformation rings (with Anwesh Ray), Ramanujan Journal 64 (2024), 685-708.

Unobstructed deformation problems for GSp(4) (with Michael Broshi, Mohammed Zuhair Mullath and Claus Sorensen)

On the freeness of anticyclotomic Selmer groups of modular forms (with Chan Ho Kim and Robert Pollack), International Journal of Number Theory 13 (2017), 1443-1455. 

Mazur-Tate elements of non-ordinary modular forms (with Robert Pollack), Duke Mathematical Journal 156 (2011), 349-385.

On anticyclotomic μ-invariants of modular forms (with Robert Pollack), Compositio Mathematica 147 (2011), 1353-1381.

Power residues of Fourier coefficients of elliptic curves with complex multiplication (with Elena Zaurova), International Journal of Number Theory 5 (2009), 109-124.

Local torsion on elliptic curves and the deformation theory of Galois representations (with Chantal David), Mathematical Research Letters 15 (2008), 599-611.

Kida's formula and congruences (with Robert Pollack), Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006), 615-630.

Variation of Iwasawa invariants in Hida families (with Matthew Emerton and Robert Pollack), Inventiones Mathematicae 163 (2006), 523-580.

Iwasawa invariants of Galois deformations, Manuscripta Mathematica 118 (2005), 161-180.

Power residues of Fourier coefficients of modular forms, Canadian Journal of Mathematics 57 (2005), 1102-1120.

Explicit unobstructed primes for modular deformation problems of squarefree level, Journal of Number Theory (special issue in honor of Arnold Ephraim Ross) 110 (2005), 199-218.

Unbostructed modular deformation problems, American Journal of Mathematics 126 (2004), 1237-1252.

Geometric Euler systems for locally isotropic motives, Compositio Mathematica 140 (2004), 317-332.

Kummer theory of abelian varieties and reduction of Mordell-Weil groups, Acta Arithmetica 110 (2003), 77-88.

Selmer groups and Chow groups of self-products of algebraic varieties, International Mathematics Research Notices 2002, no. 39, 2063-2073.

Algebraic cycles, modular forms and Euler systems, Journal fur die Reine und Angewandte Mathematik 543 (2002), 103-145.

An overview of a theorem of Flach, appendix to Deformations of Galois representations by Fernando Gouvêa, in Arithmetic algebraic geometry (Park City, UT 1999), IAS/Park City Mathematics Series 9, 2001.

On Selmer groups of geometric Galois representations, Ph.D. thesis, 2000.

Educational/Employment History

S.B. in theoretical mathematics (Massachusetts Institute of Technology, 1996)

A.M. in mathematics (Harvard University, 1997)

Ph.D. in mathematics (Harvard University, 2000, under the supervision of Barry Mazur)

T.H. Hildebrant Assistant Professor and NSF Postdoctoral Fellow (University of Michigan, 2000-02, under the supervision of Chris Skinner)

Visiting Assistant Professor and NSF Postdoctoral Fellow (Univerisity of California, Berkeley, 2002-03, under the supervision of Ken Ribet)

Visiting Assistant Professor (Amherst College, 2003-04)

Assistant Professor (University of Massachusetts Amherst, 2004-09)

Associate Professor (University of Massachusetts Amherst, 2009-23)

Professor (University of Massachusetts Amherst, 2023-present)

Department Head, Mathematics and Statistics (University of Massachusetts Amherst, 2024-present)

Ph.D. Students

Penny Ridgdill, On The Frequency of Finitely Anomalous Elliptic Curves, 2010.

Adam Gamzon, Local Torsion on Abelian Surfaces , 2012.

Jeff Hatley, Obstruction criteria for modular deformation problems, 2015.

Vy Nguyen, Elliptic Curves And Power Residues, 2019.

Tori Day, On the universal ordinary deformation ring for ordinary modular deformation problems, 2021.

Rylan Gajek-Leonard, On the Iwasawa Invariants of Nonordinary Modular Forms, 2022.

Bartu Bingol, Obstructed Deformation Problems and (mod l) Congruences Between Elliptic Curves, 2025.

Jiangyi Qiu, On 3-Diophantine Stability for Elliptic Curves, 2025.

Expository papers

An introduction to cobordism theory.  My minor thesis, written under the supervision of Raoul Bott and Franklin Peterson. Nothing terribly original here, although I think that, as these things go, it's a fairly painless introduction to the subject.

The Banach-Tarski paradox.  An extra-explicit exposition of this well-known result. The highlight, from a purely visual standpoint, is pages 10 and 11.

The inflation-restriction sequence: an introduction to spectral sequences. A quick introduction to spectral sequences. My most popular paper, for some reason. I suspect that it's the pictures.

The modular curves X0(11) and X1(11).  An introduction to modular curves focused on the motivating problem of analyzing rational 11-torsion on elliptic curves. It originated in some notes of Matthew Emerton and some computations of Keith Conrad and Robert Pollack, and ostensibly had some connection to the 1999 Park City Mathematics Institute.

L-functions and cyclotomic units.  A write-up of a lecture on Euler systems I gave at the 2001 Arizona Winter School.

The Euler system of Heegner points.  An exposition of Kolyvagin's work on the conjecture of Birch and Swinnerton-Dyer, inspired by a student project I supervised at the 2001 Arizona Winter School.

Lectures on the Dirichlet class number formula for imaginary quadratic fields.  A write-up of some lectures I gave at the Ross program in the summer of 2004.

Course notes

Algebraic number theory.  Course notes from a tutorial I gave at Harvard in 1999. Not brilliant, and I really dislike some parts, but you could do a lot worse. A couple of somewhat more recent supplementary articles:

• An introduction to local fields.  A quick introduction to local fields and their Galois theory.

• The idelic approach to number theory.  An introduction to adeles and ideles, working towards a proof of the Dirichlet unit theorem and the finiteness of the class group. Similar to the relevant portion of Lang's Algebraic number theory, but with more details and more of an emphasis on the underlying topology.

Euler systems in arithmetic geometry.  My course notes from Barry Mazur's 1998 course on Euler systems. Any errors in the notes are, of course, probably my fault.

Barry Mazur's 1998 notes on Euler systems.  Mazur's own notes on that course. Much less extensive than mine, but somehow much more charming.

Barry Mazur's 1999 notes on Euler systems.  Mazur's notes from his 1999 course on Euler systems.