Jacobi Theta Embedding of a Hyperbolic 4-Space with Cusps

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 Gauß numbers the constructed basic modular forms define a Baily-Borel embedding in $\mathbb{P}^{22}$. The relations yield explicit homogeneous equations for the Picard modular image surface.