STNB - Seminari de Teoria de Nombres de Barcelona - STNB2020(34th edition) - Tate module tensor decompositions and Sato-Tate conjecture for varieties potentially of $\mathrm{GL}

STNB2020(34th edition)

Tate module tensor decompositions and Sato-Tate conjecture for varieties potentially of $\mathrm{GL}_2$-type

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Abstract

We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic and potentially of $\mathrm{GL}_2$-type. We use this decomposition as a fundamental tool to describe the Sato-Tate group of $A_0$ and to prove the Sato-Tate conjecture in certain cases. This is joint work with Francesc Fité.

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