RolfPeter Holzapfel
Preprint series: Institut für Mathematik, HumboldtUniversität
zu Berlin (ISSN 08630976), 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 (PicardLefschetz, 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,
termrewriting)

20H05 Unimodular groups, congruence subgroups
ZDM:
32M15
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 PicardEinstein 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 BailyBorel
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\