Special Theta Relations

Abstract

§My topic arises out of my efforts to respond to a question raised last summer by W. L. Hoyt at the AMS Summer Institute on Theta Functions: How can theta nullwerte be used to describe Hilbert modular surfaces, or, more generally, Humbert surfaces, as subvarieties of the variety of moduli of abelian surfaces (e. g., with principal polarization)? I want to show that an easy very slight re-packaging of the presentation by Mumford [4] of the subject of Riemann’s relations, which are general quartic relations satisfied by the homogeneous coordinates (for suitable projective embeddings) of the moduli of all abelian varieties, leads to special quadratic relations for these homogeneous coordinates when the moduli represent abelian varieties that are special in a certain sense.