Principal polarizations of supersingular abelian surfaces

Abstract

We consider supersingular abelian surfaces $A$ over a field $k$ of characteristic $p$ which are not superspecial. For any such fixed $A$, we give an explicit formula of numbers of principal polarizations $\lambda$ of $A$ up to isomorphisms over the algebraic closure of $k$. We also determine all the automorphism groups of $(A, \lambda)$ over algebraically closed field explicitly for every prime $p$. When $p \geq 5$, any automorphism group of $(A,\lambda)$ is either $\mathbb{Z}/2\mathbb{Z} = \{\pm 1\}$ or $\mathbb{Z}/10\mathbb{Z}$. When $p=2$ or 3, it is a little more complicated but explicitly given. The number of principal polarizations having such automorphism groups is counted exactly. In particular, for any odd prime $p$, we prove that the automorphism group of any generic $(A, \lambda)$ is $\{\pm 1\}$. This is a part of a conjecture by Oort that the automorphism group of any generic principally polarized supersingular abelian variety should be $\{\pm 1\}$. On the other hand, we prove that the conjecture is false for $p=2$ in case of dimension two by showing that the automorphism group of any $(A, \lambda)$ (with $\dim A = 2$) is never equal to $\{\pm 1\}$.