Automorphic forms, Galois representations and ramification

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The project ``Automorphic forms, Galois representations and ramification'' deals with some of the key issues that occupy modern algebraic number theory. They relate to the study of algebraic extensions of the rationals, and some of their more elusive properties that are not accessible by direct means. One uses automorphic forms to construct algebraic extensions of the rationals, which gives greater control on some of the basic properties that one would like to know about such extensions. The PI proposes to continue this theme of his work by using congruences between modular forms to produce extensions of number fields with controlled ramification. Such a control is related to classical conjectures like the Leopoldt conjecture. He will also continue the study of attaching automorphic forms to Galois representations, and proving more cases of the inverse Galois problem. Work on the connection between automorphic forms and number theory is a very active area of number theory. This has broad implications which might also be useful to applications of number theory in areas like cryptography that are of practical use. The PI also expects to continue to disemminate knowledge by involving students in research projects, by organising conferences, and giving talks aimed at broad audiences about his subject.