A three dimensional ball quotient

TL;DR

This paper identifies a specific Picard modular variety of general type associated with a ball quotient in the unitary group U(1,3), detailing its modular form ring and algebraic relations.

Contribution

It explicitly determines the ring of modular forms for a particular Picard modular variety, including generators, relations, and their geometric significance.

Findings

Ring of modular forms has 25 generators.

Forms include Borcherds products of weights 1 and 2.

Squares of cuspidal forms define holomorphic differentials.

Abstract

In connection with our previous investigation about Siegel threefolds which admit a Calabi--Yau model, we consider ball quotients which belong to the unitary group $\U(1,3)$. In this paper we determine a very particular example of a Picard modular variety of general type. Really we determine the ring of modular forms. This algebra has 25 generators, 15 modular forms $B_i$ of weight one and ten modular forms $C_j$ of weight 2. Both will appear as Borcherds products. We determine the ideal of relations. The forms $C_i$ are cuspidal. Their squares define holomorphic differential forms on the non-singular models.