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Thursday 30 Nov 2017Paramodularity of abelian surfaces

Tobias Berger - University of Sheffield

H103 14:30-16:30

The key ingredient in Wiles' proof of Fermat's last theorem was to establish the modularity of elliptic curves. Despite many impressive advances in the Langlands programme the analogous question of modularity for abelian varieties of dimension 2 is far from settled. I will report on work in progress with Kris Klosin on the modularity of Galois representations $G_{\mathbf{Q}}  \to {\rm GSp}_4(\mathbf{Q}_p)$ that are residually reducible. I will explain, in particular, how this can be used in certain cases to verify Brumer and Kramer's paramodular conjecture for abelian surfaces over Q with a rational torsion point of order p.

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