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?



Abelian varieties over local fields
$V$ 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. 


