Let $C_{1}$ and $C_{2}$ be two curves with Jacobians $J_{1}$ and $J_{2}$. Let $\theta_{1}$ and $\theta_{2}$ be theta divisors in $J_{1}$ and $J_{2}$, resp. Suppose $C_{1}$ is a degree $n$ covering of $C_{2}$, we have the map $$\phi:C_{1} \to C_{2}$$ which induces an isogeny $$\phi':J_{1} \to J_{2}.$$ Is there any relation between $\theta_{1}$ and $\theta_{2}$ via $\phi'$?
Or more generally, if we forget about curves, let $A$ and $B$ be two isogenous abelian varieties, is there any relation between theta divisors via the isogeny?
If the conditions are not strong enough to deduce a result, by adding some extra conditions, is it possible to say "something" between those theta divisors?
Thank you for your help, any comments/hints/references are appreciated!
Frey and Kani's paper gives some related result.