Let $p$ be a prime number. Denote by $G$ the absolute Galois group of (a finite extension of) $\mathbb{Q}_p$. Let $\ell$ be a prime number.

For $\ell= p$, I guess it is well known that the irreducible continuous and finite dimensional representations of $G$ with coefficients in $\bar{\mathbb{F}}_p$ are induced from a character. More precisely, let $d$ be the dimension of the representation which we denote by $\rho$ and let $\mathbb{Q}_{p^d}$ be the unramified extension of $\mathbb{Q}_p$ of degree $d$, with absolute Galois group $G_d$. Then there exist a character $\omega : G_d \to \bar{\mathbb{F}}_p^{\times}$ such that $\rho$ is isomorphic to $Ind_{G_d}^{G} (\omega)$ (and I think $\omega$ is a power of a fundamental character of "niveau" $d$ as defined by Serre).

Now can we describe the irreducible representations of $G$ with coeffients in $\bar{\mathbb{F}}_{\ell}$ when $\ell \neq p$ ?


1 Answer 1


Not in any simple way, no. There are lots of non-abelian finite groups $G$ that can appear as Galois groups of finite extensions of $\mathbb Q_p$. (E.g. the group $A_4$ can appear as the Galois group of an ext. of $\mathbb Q_2$.)

If $\ell \not\mid |G|$, then the rep. theory of $G$ in char. $\ell$ is the same as in char. zero. So, basically, any irrep. of the Galois gp. of any finite ext. of $\mathbb Q_p$ can occur as a mod $\ell$ irrep. for many choices of $\ell$, and there are lots of such possible groups (and hence lots of such irreps.).

To get a more sophisticated answer to this question, you could look at the mod $\ell$ version of the local Langlands correspondence.


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