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 N.

He proved a great deal about these congruences, and also posed some questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. In this talk, we will introduce this subject and these product structures. In particular, we will discuss how these products reflect delicate algebraic number-theoretic interactions between the primes N and p and can be related to analytic objects.