Automorphisms of Compact Riemann Surfaces

I read a statement that for a compact Riemann surface $C$ with genus $g\geq 2$, one has for the Jacobian $J(C)$ of the curve $C$: $$Aut (J(C))\sim Aut C$$ when $C$ is hyperelliptic and $$Aut(J(C))\sim Aut(C)\times Z_2$$ when $C$ is not hyperelliptic. I suppose this is easy from Torelli's theorem, but what is the proof? Here $Aut(C)$ etc. means the automorphism group. Thanks in advance for any help.

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it would take me some time to research this, but in principle i do not understand why any proof of Torelli would not yield the more precise result. although Serre says otherwise, e.g., of the Andreotti proof, it seems to me that an isomorphism of polarized Jacobians should yield an isomorphism of the dual curves and hence an isomorphism of curves. it remains of course to trace the composition of these functors to see that in one direction it yields an injection and in the other it equates isoms of Jac that are negatives of each other. Perhaps Serre or someone else will explain why not. – roy smith Dec 28 '12 at 8:09
According to Serre, the precise result is essentially contained in Weil's proof, and I seem to recall a similarly precise statement in Matsusaka's paper on Torelli. So i would consult those sources. I have also been told that Torelli's original proof established a functor from Jacobians to curves, and Ciliberto has a more recent article (LNM 997) on Torelli type proofs, with inputs from Comessati that may shed some light. – roy smith Dec 28 '12 at 8:35