# Is there a finite number of supersingular genus 2 curves?

Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a projective reduced irreducible smooth genus 2 curve over $k$. It is called supersingular if the Jacobian $J_C$ is supersingular.

Is there a finite number of supersingular genus 2 curves over $k$ up to $k$-isomorphism?

• There are infinitely many pairwise non-isomorphic supersingular abelian surfaces over $\overline{\mathbb{F}_p}$. (You even have a family parametrized by $\mathbb P^1$, but peu importe.) Is the Torelli map $\mathcal{M}_2\to \mathcal{A}_2$ surjective? Is the pull-back of the supersingular locus (contained in) the supersingular locus of $\mathcal{M}_2$? – Ariyan Javanpeykar Aug 31 '17 at 10:22
• @Ariyan Javanpeykar I think it contains everything but the products of two elliptic curves, of which only finitely many are supersingular. – Will Sawin Sep 3 '17 at 14:23
• @WillSawin Yes. You're right. This answers the question right? – Ariyan Javanpeykar Sep 3 '17 at 19:17