Lethbridge Number Theory and Combinatorics Seminar: Clifton Cunningham

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The modularity theorem tells us that for every elliptic curve E over the field Q of rational numbers, there is a modular form f such that the L-function L(s,f) for f coincides with the L-function L(s,rho) for the Galois representation on the Tate module of E. In fact, f is a cusp form and its level is determined by the conductor of rho. Since the modular form f determines an automorphic representation pi of GL(2,A) (where A is the ring of adeles of Q) with the same L-function as f, we have L(s,rho) = L(s,pi).

The modularity conjecture for abelian varieties is the obvious generalization of this theorem from the case of one-dimensional abelian varieties: For every abelian variety A over Q there is an automorphic representation pi of a group G(A) such that L(s,rho) = L(s,pi), where rho is the Galois representation on the Tate module of A.

In this talk I will describe recent joint work with Lassina Dembélé giving new instances of the modularity conjecture for abelian varieties over Q. Where do we hunt for the representation pi of A? What is the group G over Q? What is the level of pi? Can we find a generalized modular form f from which pi can be built? I will explain how we use work of Benedict Gross, Freydoon Shahidi and others to answer these questions. I will also explain how thesis work by Majid Shahabi illuminates the level of pi.

This is joint work with Lassina Dembélé.