Projective Embeddings of Ball Quotients, Birational to a Bi-elliptic Surface

Abstract

For a neat lattice $$\Gamma < SU(1,2)$$, whose quotient $${\mathbb B} / \Gamma$$ is birational to a bi-elliptic surface, we compute the dimensions of the cuspidal $$\Gamma$$-modular forms $$[ \Gamma,n]_o$$ and all modular forms $$[ \Gamma, n]$$ of weight $$n \geq 2. $$ The work provides a sufficient condition for a subspace $$V \subset [ \Gamma, n]$$ to determine a regular projective embedding of the Baily-Borel compactification $$\widehat{ {\mathbb B} / \Gamma}$$ and applies this criterion to a specific example.