Abstract
Let $A$ be an abelian variety over a number field $\mathrm{E}\subset \mathbb{C}$ and let $\mathbf{G}$ denote the Mumford–Tate group of $A$. After replacing $\mathrm{E}$ by a finite extension, the action of the absolute Galois group $\mathrm{Gal}(\overline{\mathrm{E}}/\mathrm{E})$ on the $\ell$-adic cohomology $\mathrm{H}_{\mathrm{ét}}(A_{\overline{\mathrm{E}}},\mathbb{Q}_\ell)$ factors through $\mathbf{G}(\mathbb{Q}_\ell)$. We show that for $v$ an odd prime of $\mathrm{E}$ where $A$ has good reduction, the conjugacy class of Frobenius $\mathrm{Frob}_v$ in $\mathbf{G}(\mathbb{Q}_\ell)$ is independent of $\ell$. Along the way, we prove that under certain hypotheses, every point in the $\mu$-ordinary locus of the special fiber of Shimura varieties has a special point lifting it.
