BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Modular forms for abelian varieties Shahabi, Majid


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.

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