"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."
The methods of p-adic Hodge theory have emerged as a powerful tool for studying Galois representations arising from algebraic varieties over p-adic fields. This proposal investigates integral p-adic Hodge theory, which is the study of integral and torsion structures associated to semi-stable Galois representations and how they relate to automorphic forms and various integral and torsion p-adic cohomology groups of algebraic varieties. The first goal in this proposal is to provide new classifications of lattices in semi-stable representations and torsion semi-stable representations, and to study invariants (such as Weil-Deligne representations) attached to them. The second goal is to use these results to study Galois representations associated to automorphic forms. Examples of such applications are to prove the compatibility between the local Langlands correspondence and Fontaine's construction for Hilbert modular forms, and the bounded denominator conjecture for certain modular forms on noncongruence subgroups (joint with W. Li and L. Long).
This research is in the area of number theory, one of the oldest branches of mathematics, which is concerned with questions about the integers. From the time of Euclid, research in number theory has been active and productive but difficult. The research described in this proposal aims to understand the behavior and properties of integer solutions to systems of polynomials when focusing on properties relative to a fixed prime, via some analytic, algebraic, and geometric tools. Recent advances in cryptography and coding theory have depended on solutions to problems of this type.
