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fix blueprint?
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kbuzzard committed Jan 19, 2024
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Expand Up @@ -8,7 +8,7 @@ \section{Hardly ramified representations}

We make the following definition (this is not in the literature but it is a useful concept for us). We discuss the meaning of some of the concepts involved afterwards.

\begin{definition} Let $p\geq5$ be a prime. A representation $\rho: \GQ\to \GL_2(\Z/p\Z)$ is said to be \emph{hardly ramified} if it satisfies the following four axioms:
\begin{definition}\label{hardly_ramified} Let $p\geq5$ be a prime. A representation $\rho: \GQ\to \GL_2(\Z/p\Z)$ is said to be \emph{hardly ramified} if it satisfies the following four axioms:
\begin{enumerate}
\item $\det(\rho)$ is the mod $p$ cyclotomic character;
\item $\rho$ is unramified outside $2p$;
Expand All @@ -19,5 +19,5 @@ \section{Hardly ramified representations}

The theorem we want to discuss in this section is:

\begin{theorem} If $\rho$ is the Galois representation on the $p$-torsion of the Frey curve coming from a Frey package, then $\rho$ is hardly ramified.
\begin{theorem}\label{frey_curve_hardly_ramified}\uses{hardly_ramified} If $\rho$ is the Galois representation on the $p$-torsion of the Frey curve coming from a Frey package, then $\rho$ is hardly ramified.
\end{theorem}

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