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- Title
Simply transitive quaternionic lattices of rank 2 over $\mathbb{F}$q(t) and a non-classical fake quadric.
- Authors
STIX, JAKOB; VDOVINA, ALINA
- Abstract
By means of a quaternion algebra over $\mathbb{F}$q(t), we construct an infinite series of torsion free, simply transitive, irreducible lattices in PGL2($\mathbb{F}$q((t))) × PGL2($\mathbb{F}$q((t))). The lattices depend on an odd prime power q = pr and a parameter τ ∈ $\mathbb{F}$q×, τ ≠ 1, and are the fundamental group of a square complex with just one vertex and universal covering Tq+1 × Tq+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 $\mathbb{F}$q((t)) with ample canonical class, Chern numbers c12 = 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.
- Subjects
QUATERNIONS; INFINITE series (Mathematics); CHERN classes; ZARISKI surfaces; LATTICE theory; FUNDAMENTAL groups (Mathematics)
- Publication
Mathematical Proceedings of the Cambridge Philosophical Society, 2017, Vol 163, Issue 3, p453
- ISSN
0305-0041
- Publication type
Article
- DOI
10.1017/S0305004117000056