## Modular forms on ball quotients of non-positive Kodaira dimension.(English)Zbl 1268.32006

Summary: 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 1. The note concentrates on the finite $$H$$-Galois quotients $$\mathbb B/\Gamma_H$$ of a specific $$\mathbb B/\Gamma^{(6,8)}_{-1}$$, 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 $$\widehat{\mathbb B/\Gamma}$$ are studied by the means of $$H$$-invariant abelian functions on $$A_{-1}$$.

### MSC:

 32N10 Automorphic forms in several complex variables 11F99 Discontinuous groups and automorphic forms