Consider two hyperlliptic curves $C_1,C_2$ over $\mathbb{Q}$, and a morphism $\phi:C_1 \rightarrow C_2$. Lifting this morphism on the Jacobians of $C_1,C_2$ and taking its kernel defines a Prym variety $\mathcal{V}$. Given a point in $\mathcal{V}$, I wish to determine if it is of finite order.
For Jacobians, there is a theorem saying that a good reduction mod $p$ on divisors restricted to torsion divisors of order relatively prime with $p$ is an isomorphism. This allows to test if a divisor is of torsion by looking for the torsion order of its reduction.
Does such result exists for Prym varieties, and what would be the meaning of ``good reduction'' in this case? Is it also true for even more general Abelian variety?
