Algebraic number theory
My research is in Galois theory. This is the branch of mathematics concerned with symmetries of solutions of equations. There is an object that encodes all symmetries of solutions to all equations, the absolute Galois group of the rational numbers. I study this object and its relations with number theory. The study of these symmetries has gained an increasingly important role in number theory in recent years. In particular, Galois theory played an important role in the solution of Fermat's Last Theorem.
Quantitative level lowering for Galois representations (with N. Fakhruddin and C. Khare) J. Lond. Math. Soc. (2) 103 (2021), no. 1, 250-287.
Infinite class field towers of number fields of prime power discriminant (with F. Hajir and C. Maire), Advances in Mathematics (2020).
- Some supercongruences occurring in truncated hypergeometric series (with L. Long). Adv. Math. 290 (2016), 773–808.
- Deformations of certain reducible Galois representations (with S. Hamblen). II. Amer. J. Math. 130 (2008), no. 4, 913–944.
- Constructing semisimple p-adic Galois representations with prescribed properties (with C. Khare and M. Larsen), American Journal of Mathematics 127 (2005), 709–734.
- Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur, Ann. of Math. (2) 156 no. 1 (2002), 115–154.