Independence of $l$ for Frobenius conjugacy classes attached to abelian varieties

Abstract: 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 $l\neq p$, a theorem of Deligne implies that upon making a finite extension of $E$, the Galois representation on the $l$-adic Tate module factors as $\rho_l:\Gamma_E\rightarrow G_A(\mathbb{Q}_l)$, where $G_A$ is the Mumford-Tate group of $A$. We prove that the conjugacy class of $\rho_l(Frob_v)$ is defined over $\mathbb{Q}$ and independent of $l$. This is joint work with Mark Kisin.

number theory

Audience: researchers in the topic

London number theory seminar

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