Quasi-isogeny groups of supersingular abelian surfaces via pro-\'etale fundamental groups

TL;DR

This paper explores the structure of quasi-isogeny groups of supersingular abelian surfaces using pro-étale fundamental groups, establishing isomorphisms and classifying torsors in the context of algebraic geometry.

Contribution

It introduces a new approach to understanding quasi-isogeny groups via pro-étale torsors and classifies certain torsors in terms of fundamental groups, advancing the theory of supersingular abelian surfaces.

Findings

Established an isomorphism between a free group and the group of self-quasi-isogenies.

Classified pro-étale torsors using fundamental groups.

Computed pro-étale fundamental groups of curves.

Abstract

We consider a $J_b(\mathbb{Q}_p)$-torsor 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-\'etale torsors in terms of the pro-\'etale fundamental group, describe the category of geometric covers of non-normal schemes, and use this to compute pro-\'etale fundamental groups of curves.