  Faltings's isogeny theorem states that two abelian varieties are isogenous
over a number field precisely when the characteristic polynomials of the
reductions at almost all prime ideals of the number field agree. This implies
that two abelian varieties over $\mathbb{Q}$ with the same $L$-series are
necessarily isogenous, but this is false over a general number field. Let $A$
and $A'$ be two abelian varieties, defined over number fields $K$ and $K'$
respectively. Our main result is that $A$ and $A'$ are isogenous after a
suitable isomorphism between $K$ and $K'$ if and only if the Dirichlet
character groups of $K$ and $K'$ are isomorphic and the $L$-series of $A$ and
$A'$ twisted by the Dirichlet characters match.
