The densest lattices in PGL3(Q2)
Daniel Allcock, Fumiharu Kato
TL;DR
This paper determines the minimal covolume lattices in PGL3(Q2), explicitly describes the two such lattices, and explores their connections to Mumford's fake projective plane and a new 2-adic uniformization.
Contribution
It identifies the smallest covolume lattices in PGL3(Q2), characterizes them explicitly, and links them to known constructions of fake projective planes.
Findings
Exactly two lattices have minimal covolume in PGL3(Q2)
One lattice is related to Mumford's fake projective plane
A new 2-adic uniformization of a fake projective plane is presented
Abstract
We find the smallest possible covolume for lattices in PGL3(Q2), show that there are exactly two lattices with this covolume, and describe them explicitly. They are commensurable, and one of them appeared in Mumford's construction of his fake projective plane. We also discuss a new 2-adic uniformization of another fake projective plane.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology