Fix a field $K$ with absolute Galois group $G$. By an isogeny theorem over $K$, I mean the statement that the map $\operatorname{Hom}(A,B)\otimes\mathbb{Z}\_l \to \operatorname{Hom}\_G(T\_l A, T\_l B)$ 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.



In which ways can the isogeny theorem fail for local fields?
I believe for $l\ne p$ the theorem is 'always' false, in the sense that for every positive-dimensional abelian variety $A$ one can find many $B$s for which your map is not onto, independently of the reduction type:


Fix $A$ and take any non-constant family of abelian varieties in which $A$ is a fibre, e.g. some neighbourhood of $A$ in the moduli space (choosing a polarization or whatever). Then all nearby fibers $A'$ in the family have the same $l$-adic representation, $V\_l A\cong V\_l A'$. This is a special case of a very general theorem of Mark Kisin that $l$-adic representations are locally constant in families in the $p$-adic topology (Math. Z. 230, 569-593 (1999), [Link](https://doi.org/10.1007/PL00004802), esp. lines 4-5 from the bottom). So there are uncountably many abelian varieties with the same $l$-adic representation as $A$, but only countably many isogenous ones, contradicting surjectivity.


This is just an extension of what you mentioned in the question: in the case of good reduction, $V\_l A$ is a trivial inertia module so it is determined as a Galois representation by the characteristic polynomial of Frobenius. So, e.g., in a family of elliptic curves if $E$ and $E'$ are close enough so that they have the same reduction mod $p$, then $V\_l E\cong V\_l E'$; but, of course, there are uncountably many such 'nearby' $j$-invariants in any non-isotrivial family, so for any $E$ you can find lots of $E'$s for which the theorem fails.


(I don't know what happens for $l=p$, so this is only a partial answer, but it won't fit in a comment box.)


