Basic results on theta functions in several variables

Abstract

We seek a generalization of the function ϑ (z, τ) of Chapter I where z є ℂ is replaced by a g-tuple $\vec z = \left( {{z_1}, \cdots ,{z_g}} \right) \in {\mathbb{C}^g}$, and which, like the old ϑ, is quasi-periodic with respect to a lattice L but where L⊂ℂ^g. The higher-dimensional analog of τ is not so obvious. It consists in a symmetric g×g complex matrix Ω whose imaginary part is positive definite: why this is the correct generalization will appear later. Let log_g be the set of such Ω. Thus log_g is an open subset in ℂ^g(g+l)/2 It is called the Siegel upper-half-space. The fundamental definition is:

$$\vartheta \left( {\vec z,\Omega } \right) = \sum\limits_{\vec n \in {\mathbb{Z}^g}} {\exp \left( {\pi {\kern 1pt} {i^t}\vec n\Omega \vec n + 2\pi {i^t}\vec n \cdot \vec z} \right)} .$$

(Here $\vec n,\vec z$ are thought of as column vectors, so $\mathop n\limits^{t \to }$ is a row vector, $\mathop n\limits^{t \to } \cdot \vec z$ is the dot product, etc.; we shall drop the arrow where there is no reason for confusion between a scalar and a vector.)