  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
$\mathrm{H}^1_{\mathrm{\acute{e}t}}(A_{\overline{\mathrm E}},\mathbb Q_\ell)$
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 every point in the $\mu$-ordinary locus of the
special fiber of Shimura varieties has a special point lifting it.
