Cusps of Picard modular surfaces

TL;DR

This paper calculates the number of cusps on minimal Picard modular surfaces and related surfaces, showing finiteness of classes with a given number of cusps and exploring higher-rank analogues.

Contribution

It provides a precise count of cusps for minimal Picard modular surfaces and establishes finiteness results for classes with fixed cusp counts.

Findings

Number of cusps for minimal Picard modular surfaces determined

Finiteness of commensurability classes with N cusps established

Discussion of higher-rank analogues included

Abstract

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also discuss a higher-rank analogue.