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Modular forms for abelian varieties

Banff International Research Station for Mathematical Innovation and Discovery

As the modularity theorem shows, classical modular forms are connected to Tate modules of elliptic curves over $\mathbb{Q}$ through their $L$-functions. This connection is built through the automorphic representations of GL(2) and its subgroups. This talk concerns a generalization of this story to abelian varieties. The Langlands program predicts that for abelian varieties $A$ over $\mathbb{Q}$, there should be an automorphic representation of GSpin over $\mathbb{Q}$ such that the $L$-function of the automorphic representation coincides with the $L$-function coming from the Tate module of the abelian variety $A$. Recently, Gross has refined this prediction for certain abelian varieties $A$, showing exactly how to describe the weight and level of a type-$B$ modular form $f_A$ whose $L$-function matches the $L$-function of the Tate module of $A$. In this talk, I will review some of this story and will describe my own work on the group scheme of the level of the GSpin modular forms that arise in Gross' conjecture.

Mathematics; Number theory; Algebraic geometry

video/mp4

Author affiliation: University of Calgary

2017-09-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63037

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/