Supersingular abelian varieties and their automorphism groups

Let \(A_g\) be the moduli space over \(F_p\) of \(g\)-dimensional principally polarised abelian varieties (where \(p\) is a prime and \(g ≥ 1\)), and let \(S_g\) be the supersingular locus. We study geometric stratifications on \(A_g\) and \(S_g\) by invariants, to deduce information on arithmetic properties of points in \(S_g\), with a particular emphasis on their automorphism groups.
As our main result, we show that if \(g\) is even and \(p ≥ 5\), then every geometric generic member in the maximal supersingular Ekedahl-Oort stratum in \(A_g\) has automorphism group \({±1}\). This confirms Oort’s conjecture in these cases.

This talk is based on joint works with Tomoyoshi Ibukiyama, Akio Tamagawa, and Chia-Fu Yu.