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



Mod l local Galois representations (l different from p)
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.


