Independence of $\ell$ for Frobenius conjugacy classes attached to abelian varieties

Rong Zhou, Imperial College London

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Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$ and let $v$ be a place of good reduction lying over a prime $p$. For a prime $\ell\neq p$, a result of Deligne implies that upon replacing $E$ by a finite extension, the Galois representation on the $\ell$-adic Tate module of $A$ factors as $\rho_\ell:\mathrm{Gal}(\overline{E}/E)\rightarrow G_A$, where $G_A$ is the Mumford--Tate group of $A_{\mathbb{C}}$. For $p>2$, we prove that the conjugacy class of  $\rho_\ell(\mathrm{Frob}_v)$ is defined over $\mathbb{Q}$ and independent of $\ell$. This is joint work with Mark Kisin.