Dimension formulas for automorphic forms of coabelian hyperbolic type. (English) Zbl 1093.14033

Holzapfel, Rolf-Peter

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].

Reviewer: A. Muhammed Uludag (Istanbul)

MSC:

14G35	Modular and Shimura varieties
11G15	Complex multiplication and moduli of abelian varieties
11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14J15	Moduli, classification: analytic theory; relations with modular forms
11G18	Arithmetic aspects of modular and Shimura varieties
11F55	Other groups and their modular and automorphic forms (several variables)

Keywords:

Picard modular lattice; ball quotient; complex hyperbolic