Let $p: C \rightarrow X$ a surjective morphism of smooth algebraic curves over $\mathbb{C}$, ramified, of degree $d$.
What is known about the kernel of the induced pullback on Jacobians $p^*: Jac(X) \rightarrow Jac(C)$?
My only idea is the following: using the Norm map, one sees that $Nm(p^*(L))=dL$, hence the kernel is contained in the $n$-th torsion of $Jac(X)$.