This research is in the area of number theory, one of the oldest branches of mathematics, which is concerned with questions about the integers. The research described in this proposal aims to understand the properties of the integer solutions to systems of polynomials, via some analytic, algebraic, and geometric tools. Recent advances in cryptography and coding theory have depended on solutions to these problems. For instance, most cell phone calls are protected by a code based on elliptic curves over a prime, one of the primary objects of study in the PI's field.
The PI will continue his work on integral p-adic Hodge theory. The first aim is to improve the understanding of integral structures (lattices) in semi-stable p-adic Galois representations, and then use this improvement to study the reduction of these representations. As the expected consequences, certain parts of automorphic lifting theorems and Serre's conjecture will be extended to more general settings. The second aim is to study Galois representations associated to automorphic representations and certain algebraic varieties. In particular, the PI and his collaborators will focus on the Galois representations arising from modular forms of noncongruence subgroups and certain torsion representations constructed (by Scholze) from Shimura varieties.
