Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups

Abstract

We consider a $J_b({\mathbb{Q}}_p)$ on the supersingular locus of the Siegel threefold constructed by Caraiani–Scholze, and show that it induces an isomorphism between a free group on a finite number of generators, and the group of self-quasi-isogenies of a supersingular abelian surface, respecting a principal polarization and a prime-to-$p$ level structure. Along the way, we classify certain pro-étale torsors in terms of the pro-étale fundamental group, describe the category of geometric covers of non-normal schemes, and use this to compute pro-étale fundamental groups of curves.