My research interests mainly center around the arithmetic theory of modular forms, including the question of reciprocity in the Langlands programme and the cohomology of arithmetic groups.


Cuspidal cohomology classes for GL_n(Z).
(with G. Boxer and T. Gee)

The Ramanujan and Sato-Tate conjectures for Bianchi modular forms.
(with G. Boxer, T. Gee, J. Newton, and J. Thorne)

The unbounded denominators conjecture.
(with V. Dimitrov and Y. Tang)

Published Papers

Reciprocity in the Langlands program since Fermat's Last Theorem.
(survey article) Proceedings of the ICM Vol. II. Plenary lectures, 610–651,EMS Press, Berlin, 2023.

Potential Automorphy over CM fields.
(with P. Allen, A. Caraiani, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne), Video lecture I, Video lecture II. Annals of Mathematics, (2) 197 (2023), no. 3, 897-1113.

Bloch Groups, Algebraic K-Theory, Units, and Nahm's Conjecture.
(with S. Garoufalidis and D. Zagier). Annales Scientifiques de l'École Normale Supérieure, (4) 56 (2023), no. 2, 383-426.

Rationality of twists of the Siegel modular variety of genus 2 and level 3.
(with S. Chidambaram). (magma code and magma output.) Proceedings of the AMS, 150 (2022), no.5, 1975-1984.

Globally realizable components of local deformation rings.
(with M. Emerton and T. Gee), Journal de l'Institut de Math de Jussieu. 21 (2022), no. 2, 533-602.

Abelian Surfaces over totally real fields are potentially modular.
(with G. Boxer, T. Gee, and V. Pilloni). Video lecture I, Video lecture II. Publ. Math. Inst. Hautes Études Sci. 134 (2021), 153–501.

Vanishing Fourier coefficients of Hecke eigenforms.
(with Naser T. Sardari), Math. Annalen, 381 (2021), no. 3-4, 1197–1215.

Abelian Surfaces with fixed 3-torsion.
(with with S. Chidambaram and D. Roberts), Proceedings of the Fourteenth Algorithmic Number Theory Symposium (ANTS-XIV), p 91-108, 2020.

Some modular abelian surfaces.
(with S. Chidambaram and A. Ghitza), Mathematics of Computation, 89 (2020), no.321, p 387-394.

Modularity lifting for non-regular symplectic representations.
(with D. Geraghty), Duke Math, 169 Number 5 (2020), 801-896.

Bloch-Kato conjectures for automorphic motives.
(with D. Geraghty and M. Harris), this is an appendix to the paper above. Video Lecture.

Compatible Systems of Galois Representations associated to the exceptional group E6.
(with G. Boxer, M. Emerton, B. Levin, K. Madapusi Pera, and S. Patrikis), Forum of Mathematics, Sigma. 7 (2019), e4, 29p.

Semistable modularity lifting over imaginary quadratic fields.
(I do not plan to submit this paper at this time.) Video lecture.

Pseudo-Representations of weight one are unramified. (with J. Specter), Algebra & Number Theory 13 (2019), no. 7, p 1583–1596.

A Torsion Jacquet-Langlands correspondence, Asterisque, 409 (2019). (Longer Version from 2012). (with A. Venkatesh)

Non-minimal modularity lifting in weight one.
J. Reine Angew. Math. 740 (2018), p.41-62.

Modularity lifting beyond the Taylor-Wiles method.
(with D. Geraghty). Inventiones Mathematicae, 211 (2018). No. 1, 297-433.

Abelian Spiders and Real Cyclotomic Integers.
(with Z. Guo). Transactions of the American Mathematical Society, 370 (2018), No. 9, 6515-6533.

Counting Algebraic Integers by Absolute Value.
(with Z. Huang). Journal of the London Mathematical Society, 96 (2017), No. 1, 181-200.

Explicit Serre weights for two-dimensional Galois representations.
(with M. Emerton, T. Gee, and Lambros Mavrides), Compositio Mathematica. 153 (2017), no. 9, 1893–1907.

Homological stability for completed homology.
(with M. Emerton) Mathematische Annalen, 364 (2016), 1025-1041.

The Stable Homology of Congruence Subgroups.
Geometry & Topology. 19 (2015), 3149-3191.

Finiteness of unramified deformation rings.
(with P. Allen) Algebra & Number Theory, 8 (2014), No. 9, 2263-2272.

The Artin Conjecture for some S_5-extensions.
Mathematische Annalen, 356 (2013), 191-207.

Irreducibility of automorphic Galois representations of GL(n), n at most 5.
(erratum now added) (with T. Gee). Annales de l'Institut Fourier, Vol. 63 no. 5 (2013), p. 1881-1912.

Completed Cohomology, a survey.
(with M. Emerton) Non-abelian Fundamental Groups and Iwasawa Theory, London Math. Soc. Lecture Note Ser., (2012) 393, 239–257, CUP.

Even Galois Representations and the Fontaine-Mazur Conjecture. II.
Journal of the American Mathematical Society, 25 (2012), 533-554. Video lecture.

Even Galois Representations and the Fontaine-Mazur Conjecture.
Inventiones Mathematicae, 185 (2011), 1-16.

Mod-p Cohomology Growth in p-adic Analytic Towers of 3-Manifolds.
(with M. Emerton) Groups, Geometry and Dynamics, 5 (2011), no. 2, 355-366, Volume in honour of Fritz Grunewald.

Cyclotomic Integers, Fusion Categories, and Subfactors.
(with S. Morrison and N. Snyder), Communications in Mathematical Physics, Volume 303, Issue 3 (2011), Page 845-896.

Elliptic Curves of odd modular degree.
(with M. Emerton) Israel Journal of Mathematics, 169 (2009), 417-444.

Bounds for Multiplicities of Unitary Representations of Cohomological Type in Spaces of Cusp Forms.
(with M. Emerton) Annals of Mathematics, 170 (2009), 1437-1446.

Nearly Ordinary Galois Deformations over Arbitrary Number Fields.
(with B. Mazur) Journal de l'Institut de Math de Jussieu. (2009) 8(1), 99-177.

The Coleman-Mazur Eigencurve is Proper at Integral Weights.
Algebra & Number Theory. 2 (2008), No. 2, 209-215.

Automorphic forms and Rational homology 3-spheres.
(with N. Dunfield) Geometry & Topology. 10 (2006), 295-329.

Eisenstein Deformation Rings.
Compositio Mathematica, 142 (2006), no. 1, 63-83.

The p = 2, N = 1 eigencurve is proper.
(with K. Buzzard) Documenta Mathematica, Special Volume in honour of John Coates' 60th Birthday. (2006), 211-232.

Mod p representations on Elliptic Curves.
Pacific Journal of Mathematics. 225 (2006), no. 1, 1-11.

Irrationality of certain p-adic periods for small p.
International Mathematics Research Notices, (2005), no. 20, 1235-1249.

On the ramification of Hecke Algebras at Eisenstein primes.
(with M. Emerton) Inventiones Mathematicae, 160, (2005), no. 1, 97-144.

Slopes of Overconvergent 2-adic Modular Forms.
(with K. Buzzard) Compositio Mathematica, 141 (2005), no 3, 591-604.

A counterexample to the Gouvea-Mazur conjecture.
(with K. Buzzard) Comptes Rendus Mathematique, 338 (2004), no. 10, 751-753.

The Hecke Algebra has Large Index.
(with M. Emerton) Math Research Letters, 11 (2004), no. 1, 125-137

Semistable Abelian Varieties over Q.
Manuscripta Mathematica, 113 (2004), no. 4, 507-529.

Almost Rational Torsion Points on Elliptic Curves.
International Mathematics Research Notices, (2001), no. 10, 487-503.

Informal Notes and Other Materials

This sections includes various expositional materials as well as a book review.

Motives and L-functions.
Notes from my lectures at the Current Developments in Mathematics Conference, November 2018. Video Lecture I, Video Lecture II.

Congruences Between Modular Forms.
Notes from my lectures at the Arizona Winter School, March 2013. Video lectures.

Even Galois Representations.
Notes from a talk given during the Galois trimester at Institut Henri Poincaré, March, 2010.

Book Review: "A First Course in Modular Forms", by F. Diamond and J. Shurman.
Bulletin of the American Mathematical Society, 43 (2006), 415-421.

A remark on a theorem of Folsom, Kent, and Ono.

Conferences Organized

  • The Arithmetic of the Langlands Program, Special Trimester, Bonn, 2020.
  • Workshop on Motives, Galois Representations and Cohomology Around the Langlands Program
  • Deformation Theory, Completed Cohomology, Leopoldt Conjecture and K-theory
  • Workshop on the Cohomology of Arithmetic Groups
  • Workshop on Representation Theory and Arithmetic
  • Current Developments in the Langlands Program
  • Modular Forms and Arithmetic
  • Thematic Program on Galois Representations at the Fields Institute
  • Galois Representations, Shimura Varieties, and Automorphic Forms
  • Torsion in the homology of arithmetic groups