Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve inside $A$. Is there a way to know the possible degrees of $f$ if we fix some invariants of $C$ and of $A$? For example I know that for number fields if we fix the modular heights of $A$ and of $Jac_C$ one can write a bound for the degrees in terms of the heights if such configuration exists. But I don't know which degrees appears. I would be interested to know when one can find $A$ and $C$ such that $f$ is an isogeny of degree coprime to the residual characteristic of $K$ in the local case.

1