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Dimension formulas for automorphic forms of coabelian hyperbolic type. (English) Zbl 1093.14033
Mladenov, Ivaïlo M.(ed.) et al., Proceedings of the 6th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 3–10, 2004. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-9-5/pbk). 240-251 (2005).
A lattice ${\Gamma }$ acting on a 2-dimensional complex hyperbolic ball is called coabelian if the compactification of its orbit space is birational to an abelian surface ${A}_{{\Gamma }}$. Let ${\Gamma }$ be a coabelian lattice on a 2-ball (then ${\Gamma }$ is Picard modular in all known cases). The purpose of the paper under review is to compute the dimensions of spaces of ${\Gamma }$-automorphic functions of fixed weight. The paper gives dimension formulas for neat lattices ${\Gamma }$ with ${A}_{{\Gamma }}$ being isomorphic to $E×E$ for a suitable elliptic curve $E$. (It is also proved in the paper that in general ${A}_{{\Gamma }}$ isogeneous to $E×E$ for a suitable $E$.) In particular, the dimension of the space of cusp forms is computed explicitly.
##### MSC:
 14G35 Modular and Shimura varieties 11G15 Complex multiplication and moduli of abelian varieties 11F67 Special values of automorphic $L$-series, etc 14J15 Analytic moduli, classification (surfaces) 11G18 Arithmetic aspects of modular and Shimura varieties 11F55 Groups and their modular and automorphic forms (several variables)
##### Keywords:
Picard modular; ball quotient; complex hyperbolic