Let $C$ be a smooth projective curve over $\mathbb{C}$ of genus $g \ge 2$ with Jacobian $J(C)$. By the Torelli theorem, we have that $\text{Aut }J(C)$ is either isomorphic to $\text{Aut }(C)$ or $\text{Aut }(C) \times \mathbb{Z}/2$.
- Is the automorphism group $\text{Aut }J(C)$ considered here just automorphisms of an algebraic variety respecting the principal polarization or are the automorphisms here really compatible with the group structure?
I'm a bit confused since the proofs of finiteness of automorphisms of principally polarized abelian varieties that I've seen seem to assume that they respect the group structure. Otherwise, the automorphisms mentioned above would also take things like translations into account and I'm not sure why the group above should be finite as given by the Hurwitz bound.
- Does an automorphism $\alpha$ of $C$ necessarily induce one on $J(C)$ which is compatible with its group structure? If you use the isomorphism $J(C) \cong \text{Pic}^0(C)$, it seems like the induced map $\tilde{\alpha}$ on $\text{Pic}^0(C)$ would send an element of the form $\text{div }f$ to $\text{div }g$, where $g = f \circ \alpha^{-1}$. Would this mean that the lattice for $J(C)$ would be sent to itself under this induced map in the complex analytic representation?