Let $\rho:\Gal_K\to G(\F_{\ell})$ be a mod-$\ell$ Galois representation. The Frobenius order of a prime $\frak{p}$ of $K$ at which $\rho$ is unramified is the order of $\rho(\text{Frob}_{\frak{p}})$ as an element of the finite group $G(\F_{\ell})$.
Note that unramified primes $\frak{p}$ of $K$ are, by definition, unramified in the splitting field $L/K$ of $\rho$, so the Frobenius automorphism $\text{Frob}_{\frak{p}}$ is well-defined up to conjugacy in $\Gal(L/K)$, and hence the order of $\rho(\text{Frob}_{\frak{p}})$ is well-defined.
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- Last edited by John Cremona on 2023-03-24 11:42:34
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