Modular Forms on Ball Quotients of Non-Positive Kodaira Dimension

Abstract

The Baily-Borel compactification $\widehat{{\mathbb B} / \Gamma}$ of an arithmetic ball quotient admits projective embeddings by $\Gamma$-modular forms of sufficiently large weight. We are interested in the target and the rank of the projective map $\Phi$, determined by $\Gamma$-modular forms of weight one. This paper concentrates on the finite $H$-Galois quotients ${\mathbb B} / \Gamma _H$ of a specific ${\mathbb B} / \Gamma _{-1}^{(6,8)}$, birational to an abelian surface $A_{-1}$. Any compactification of ${\mathbb B} / \Gamma _H$ has non-positive Kodaira dimension. The rational maps $\Phi ^H$ of $\widehat{{\mathbb B} / \Gamma _H}$ are studied by means of the $H$-invariant abelian functions on $A_{-1}$.