Modular Forms and Arithmetic

Abstract:

In his landmark 1976 paper "Modular curves and the Eisenstein ideal," Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level. He proved a great deal about these congruences, deducing a spectacular theorem about elliptic curves. He also posed a question: how many cusp forms of a given level are congruent to the Eisenstein series? The aim of this talk is to introduce modular forms and congruences between them, and to illustrate some of the connections between modular forms and questions of arithmetic. In particular, joint work with Preston Wake will be featured, answering Mazur's question.