Question

Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I\subset G$ the inertia group and $P$ the maximal pro-$p$ subgroup of $I$. Let $I_t:=I/P$.

Let $A_0$ be an abelian scheme over $R$ with generic fibre $A$. Then $A[p]$ is an $I$-module. Let $V$ be a Jordan-Hölder quotient of the $I$-module $A[p]$. I am interested in the representation $I\to Aut(V)$.

Question (*): Is it true that $P$ acts trivially on $V$?

(I have seen that there are results of Raynaud and Serre on the "action of $I_t$ on $V$". I want to study these things, but I am already stuck with Question (*) at the moment, i.e. with the question whether $I_t$ acts at all.)

Maybe someone can help?

Answer 1

$V$ is an irreducible $\mathbb{F}_p$-representation of $I$. As $P$ is a pro-$p$ group, $V^P\ne 0$, and $P$ is normal in $I$ so $V^P$ is stable under $I$. Therefore $V^P=V$. You might like to look at Tate's article on finite flat group schemes in "Modular Forms and Fermat's Last Theorem" for more information.