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 \(A_{-1}\).


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