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 𝜆 of 𝐴 up to isomorphisms over the algebraic closure of 𝑘. We also determine all the automorphism groups of (𝐴, 𝜆) over algebraically closed field explicitly for every prime 𝑝. When 𝑝 ≥ 5, any automorphism group of (𝐴,𝜆) is either ℤ/2ℤ = {± 1} or ℤ/10ℤ. 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, 𝜆) is {± 1}. This is a part of a conjecture by Oort that the automorphism group of any generic principally polarized supersingular abelian variety should be {± 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, 𝜆) (with dim 𝐴 = 2) is never equal to {± 1}.