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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 $E\times E$ for a suitable elliptic curve $E$. (It is also proved in the paper that in general ${A}_\Gamma$ isogenous to $E\times E$ for a suitable $E$.) In particular, the dimension of the space of cusp forms is computed explicitly. For the entire collection see [Zbl 1066.53003].
14G35Modular and Shimura varieties
11G15Complex multiplication and moduli of abelian varieties
11F67Special values of automorphic $L$-series, etc
14J15Analytic moduli, classification (surfaces)
11G18Arithmetic aspects of modular and Shimura varieties
11F55Groups and their modular and automorphic forms (several variables)