If $\rho:\Gal_K\to G(\F_{\ell})$ is an mod-$\ell$ Galois representation with splitting field $L$, then a prime $\frak{p}$ of $K$ is ramified if it is ramified in $L/K$.
Equivalently, a prime is ramified if the inertia subgroup of $\Gal_K$ for a prime above $\frak{p}$ is not contained in the kernel of $\rho$.
A prime which is not ramified is unramified.
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- Last edited by John Cremona on 2023-03-24 12:04:35
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