Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
Let $C$ be a smooth projective curve. Is it true that $$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$ and in case, what would be a reference for this? Thanks. 


That is certainly not true. Consider the case that $C$ is an elliptic curve. Then $\text{Aut}(C\times C)$ contains $\text{GL}(2,\mathbb{Z})$ as a subgroup. 


It is true if $g(C)\geq 2$. The point is that if you map nontrivially $C$ to $C\times C$, the projections of the image on each factor have degree $0$ or $1$ (say, by Hurwitz formula). Thus the image is either $C\times \{p\} $ or $\{q\}\times C $ for some $p,q\in C$, or it is the graph of an automorphism. Since $\mathrm{Aut}(C)$ is finite for $g(C)\geq 2$, your statement follows easily. 

This is a particular case of a more general rigidity result, whose proof (similar to the one given in abx's answer) can be found in Lemma 3.8 of Fabrizio Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), no. 1, 144.


