Suppose $C$ is a hyperelliptic curve. Then the set of two-torsion points on the jacobians is generated by the set of difference of Weierstrass points.

Suppose $C'$ is another hyperelliptic curve. Is it correct that the set of two-torsion points on the fibre product $C\times _{{\mathbb P}^1} C'$ is generated by pullback of the two-torsion line bundles on $C$ and $C'$.

Using Riemann-Hurwitz formula, there seems to be some relation coming. Is it possible to obtain the relation. Thanks.