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

Lookup NU author(s): Dr Alina Vdovina

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This is the authors' accepted manuscript of an article that has been published in its final definitive form by Cambridge University Press, 2017.

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Abstract

Copyright © Cambridge Philosophical Society 2017 By means of a quaternion algebra over (Formula presented.) q (t), we construct an infinite series of torsion free, simply transitive, irreducible lattices in PGL2((Formula presented.) q((t))) × PGL2((Formula presented.) q((t))). The lattices depend on an odd prime power q = pr and a parameter τ ∈ (Formula presented.) q ×, τ ≠ 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 (Formula presented.) q((t)) with ample canonical class, Chern numbers c1 2 = 2 c2, 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.

Publication metadata

Author(s): Stix J, Vdovina A

Publication type: Article

Publication status: Published

Journal: Mathematical Proceedings of the Cambridge Philosophical Society

Year: 2017

Volume: 163

Issue: 3

Pages: 453-498

Print publication date: 01/11/2017

Online publication date: 20/03/2017

Acceptance date: 02/04/2016

Date deposited: 25/05/2017

ISSN (print): 0305-0041

ISSN (electronic): 1469-8064

Publisher: Cambridge University Press

URL: https://doi.org/10.1017/S0305004117000056

DOI: 10.1017/S0305004117000056

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