Let $p$ be a rational prime number, and let $K$ be a finite extension of $\mathbb Q\_p$. Let $A$ be an abelian variety over $K$. For any rational prime number $\ell$, let $K(A[\ell])$ be the field of definition of the $\ell$-torsion points of $A$.


Suppose $A$ and $A'$ are $K$-isogenous abelian varieties over $K$. 


Is there an odd prime number $\ell\neq p$ such that $K(A[\ell])=K(A'[\ell])$?


Are there even infinitely many such $\ell$?



Division fields of isogenous abelian varieties over a p-adic field
Let $f:A\to A′$ be an isogeny over $K$. Then there exists an isogeny $g:A'\to A$ such that the composite of $f$ with $g$ is multiplication by the degree of $f$ (see any book on abelian varieties). It follows that, for every prime $l$ not dividing its degree, $f$ defines an isomorphism from $A[l]$ onto $A'[l]$. This isomorphism automatically commutes with the action of the Galois group.


