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.