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Zbl 1093.14033
Holzapfel, Rolf-Peter
Dimension formulas for automorphic forms of coabelian hyperbolic type. (English)
[A] Mladenov, Iva\"ilo 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. 240-251 (2005). ISBN 954-84952-9-5/pbk

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 $E\times E$ for a suitable elliptic curve $E$. (It is also proved in the paper that in general ${A}_\Gamma$ isogeneous to $E\times E$ for a suitable $E$.) In particular, the dimension of the space of cusp forms is computed explicitly.
[A. Muhammed Uludag (Istanbul)]

MSC 2000:: *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

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