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

2013/04/19 by Jakob Stix, Stix, Jakob, Alina Vdovina +1
Mathematics · #14J29 #20E08 #20F05 #20G30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1304.5549

openalex publication_date 2013/04/19 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We construct an infinite series of simply transitive irreducible lattices in PGL₂(F_q((t))) × PGL₂(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 τ 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 × 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 (c₁)²/c₂ = 2, trivial Albanese variety and non-reduced Picard scheme. For q = 3, the Zariski-Euler characteristic attains its minimal value χ= 1: the surface is a non-classical fake quadric.

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