A fake plane via 2-adic uniformization with torsion

TL;DR

This paper constructs a new fake projective plane using 2-adic uniformization with torsion, expanding the understanding of non-Archimedean uniformization and its applications in algebraic geometry.

Contribution

It introduces a novel fake projective plane derived from a lattice with torsion in PGL3(Q2), differing from known examples and advancing the theory of non-Archimedean uniformization.

Findings

Constructed a smooth surface as a fake projective plane with torsion

Computed the homotopy type of the Berkovich space of the plane

Demonstrated the plane's distinctness from previously known fake planes

Abstract

We adapt the theory of non-Archimedean uniformization to construct a smooth surface from a lattice in PGL3(Q2) that has nontrivial torsion. It turns out to be a fake projective plane, commensurable with Mumford's fake plane yet distinct from it and the other fake planes that arise from 2-adic uniformization by torsion-free groups. As part of the proof, and of independent interest, we compute the homotopy type of the Berkovich space of our plane.