Quaternionic Arithmetic Lattices of Rank 2 and a Fake Quadric in Characteristic 2

TL;DR

This paper constructs a torsion-free arithmetic lattice in a product of p-adic groups from a quaternion algebra over a function field, leading to a new example of a fake quadric surface in characteristic 2.

Contribution

It introduces a novel torsion-free arithmetic lattice in a product of p-adic groups derived from a quaternion algebra, and constructs a fake quadric surface over a field of characteristic 2.

Findings

Constructed a torsion-free arithmetic lattice with minimal Euler characteristic.

Produced a fake quadric surface via non-archimedean uniformization.

Lattice arises from a quaternion algebra over a function field.

Abstract

We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over $\mathbb{F}_2(\!(t)\!)$ by means of non-archimedean uniformization.