Mod l local Galois representations (l different from p)

2010-08-18T22:35:44Z

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

Answer by Tony Scholl for Mod l local Galois representations (l different from p)

2010-08-18T22:58:43Z

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.

Mod l local Galois representations (l different from p) - MathOverflow

Mod l local Galois representations (l different from p)

Answer by Tony Scholl for Mod l local Galois representations (l different from p)