A Computational Framework for the Moduli of Supersingular Abelian Varieties

Abstract

We establish structure theorems for polarised flag type quotients arising in the study of the moduli space of supersingular abelian varieties. In particular, we prove a refined normal form for the top level of these flags and derive an explicit, computable descent criterion for quasi-polarisations. These results provide a complete classification of the first and last step of these flags. As an application, we compute polarised flag type quotients in dimensions $g\le 5$, describing the fibers of the truncation morphisms as classification objects of supersingular abelian varieties.

37 pages, comments welcome! v2: added the non-supergeneral case in dimension $g=5$, changed the title, and made minor corrections