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Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map ${\Hom}(A,B)\otimes\mathbb{Z}_l \operatorname{Hom}(A,B)\otimes\mathbb{Z}_l \to \Hom_G(T_l operatorname{Hom}_G(T_l A, T_l B)$ is an isomorphism, where $A,B$ are abelian varieties over $K$, and $T_l A$ is the Tate module of $A$. Such a statement was proved for finite fields by Tate, for global function fields by Zarhin and for number fields by Faltings.

I'm interested in the case where $K$ is a $p$-adic field. The statement is then generally false, but is sometimes true. It holds e.g. when $A,B$ are elliptic curves with bad reduction and $l = p$ (Serre) or when $A,B$ have the same (good) reduction and again $l=p$ (Serre-Tate). It certainly fails if $A,B$ have good reduction and $l \ne p$.

What I don't have is a clear picture of the various possibilities for the reductions and for which cases the statement holds or fails. So that's the question.

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# In which ways can the isogeny theorem fail for local fields?

Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map ${\Hom}(A,B)\otimes\mathbb{Z}_l \to \Hom_G(T_l A, T_l B)$ is an isomorphism, where $A,B$ are abelian varieties over $K$, and $T_l A$ is the Tate module of $A$. Such a statement was proved for finite fields by Tate, for global function fields by Zarhin and for number fields by Faltings.

I'm interested in the case where $K$ is a $p$-adic field. The statement is then generally false, but is sometimes true. It holds e.g. when $A,B$ are elliptic curves with bad reduction and $l = p$ (Serre) or when $A,B$ have the same (good) reduction and again $l=p$ (Serre-Tate). It certainly fails if $A,B$ have good reduction and $l \ne p$.

What I don't have is a clear picture of the various possibilities for the reductions and for which cases the statement holds or fails. So that's the question.