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My question is referred to the statement and proof of Prop. 2.4 of Diamond's article "An extension of Wiles' Results", in Modular Forms and Fermat Last Theorem, page 479.

More precisely: fix $l$ and $p$ two distinct primes, with $l$ odd. Let $\sigma$ be an irreducible, continuous, degree 2 representation of the absolute Galois group $G_{p}$ of $Q_{p}$, with coefficients in $k$, an algebraic closure of the finite field with $l$ elements. Proposition 2.4 states that if the restriction of $\sigma$ to the inertia subgroup of $G_{p}$ is irreducible and $p$ is odd, then $\sigma$ is isomorphic to the representation induced from a character of the Galois group of a quadratic ramified extension $M$ of $Q_{p}$. The proof given works if the restriction of $\sigma$ to the wild inertia of $G_{p}$ is reducible (I think there's a typo in the first line of the proof). What if $\sigma$ is irreducible on wild inertia (and $p$ is always odd)? It seems to me that this case is not covered in the proof of the Proposition, but maybe I'm not seeing something obvious.. If such a representation exists, it cannot be induced from a quadratic extension $M$ as above, so how does it fit in the description given by the Proposition? Can one say something about such a representation (for example something about its projective image?).


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up vote 8 down vote accepted

The image of wild inertia is a finite $p$-group, and if $d$ is the degree of an irreducible representation of a $p$-group over an algebraically closed field of characteristic $\ne p$, then $d$ is a power of $p$. So for $p$ odd the image of wild inertia is always reducible.

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