Simply transitive quaternionic lattices of rank 2 over F_q(t) and a non-classical fake quadric

TL;DR

This paper constructs infinite series of special lattices in a product of p-adic groups using quaternion algebras, leading to new algebraic surfaces with unique geometric properties over finite fields.

Contribution

It introduces a novel method to build simply transitive lattices in p-adic groups via quaternion algebras, resulting in the first examples of non-classical fake quadrics over finite fields.

Findings

Constructed infinite series of lattices in PGL_2 over local fields.

Produced algebraic surfaces with Chern ratio 2 and trivial Albanese.

Identified minimal Zariski-Euler characteristic surface as a non-classical fake quadric.

Abstract

We construct an infinite series of simply transitive irreducible lattices in PGL_2(F_q((t))) \times PGL_2(F_q((t))) by means of a quaternion algebra over F_q(t). The lattices depend on an odd prime power q = p^r and a parameter \tau\ in F_q^* different from 1, and are the fundamental group of a square complex with just one vertex and universal covering T_{q+1} \times T_{q+1}, a product of trees with constant valency q + 1. Our lattices give rise via non-archimedian uniformization to smooth projective surfaces of general type over F_q((t)) with ample canonical class, Chern ratio (c_1)^2/c_2 = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value \chi = 1: the surface is a non-classical fake quadric.