Jacobi theta embedding of a hyperbolic 4-space with cusps

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 53

MSC 2000

11F27 Theta series; Weil representation

11F55 Other groups and their modular and automorphic forms

11G15 Complex multiplication and moduli of abelian varieties

11H56 Automorphism groups of lattices

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)

14E20 Coverings

14G35 Modular and Shimura varieties

14H30 Coverings, fundamental group

14H45 Special curves and curves of low genus

14H52 Elliptic curves

14J25 Special surfaces

16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)

20H05 Unimodular groups, congruence subgroups

Abstract
Starting from a fixed elliptic curve with complex multiplication we compose lifted quotients of elliptic Jacobi theta functions to abelian functions in higher dimension. In some cases, where complete Picard-Einstein metrics have been discovered on the underlying abelian surface (outside of cusp points), we are able to transform them to Picard modular forms. Basic algebraic relations of basic forms come from different multiplicative decompositions of these abelian functions in simple ones of the same lifted type. In the case of Gau\ss\ numbers the constructed basic modular forms define a Baily-Borel embedding in $\mathbf{P}^{22}$. The relations yield explicit homogeneous equations for the Picard modular image surface.

Keywords:elliptic curve, abelian surface, Shimura variety, arithmetic group, Picard modular surface, Gau\ss\ numbers, complex multiplication, theta functions, modular forms, K\