### The rank of Mazur's Eisenstein ideal

Heilbronn Number Theory Seminar

12th December 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

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 many cusp forms of a given level are congruent to the Eisenstein series? How big is the extension generated by their coefficients? We give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. The meaning of Mazur’s question will be illustrated through explicit examples.

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