Obstructed deformation rings and modularity of Galois representations

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

This project investigates the relationship between algebra (Galois representations), analysis (automorphic forms), and geometry (motives). The study of this relationship in the past has had a major impact in solving classical problems in number theory, for example in the solution by Andrew Wiles in 1995 of Fermat's Last Theorem, a problem that had remained unresolved for more than 350 years. The work in this project will advance the paradigmatic Langlands program, which drives a lot of the current research in modern number theory. This has broad implications which might also be useful to applications of number theory where reciprocity laws can provide a powerful computational tool. The project provides training opportunities for graduate students. One of the broad themes of algebraic number theory is to prove reciprocity laws which give a way to understanding the splitting behavior of primes in number fields, or L-functions of varieties, in terms of more computable and apparently unrelated objects like automorphic forms. Such reciprocity laws go back to Gauss and his Law of Quadratic Reciprocity, and have a continuing life in current mathematics in the guise of the Langlands program. The methods of this proposal will prove novel cases of such reciprocity laws and establish new relations between the much studied triad of motives, Galois representations and automorphic forms. Reciprocity laws have implications for Diophantine geometry that studies solutions of polynomial equations in integers. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.