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?).
Thanks
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