Basic results on theta functions in several variables

Abstract

We seek a generalization of the function ϑ (z,τ) of Chapter I where z ɛ ℂ is replaced by a g-tuple $\vec z = (z_1 , \cdots ,z_g ) \varepsilon \mathbb{C}^g$, 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 be the set of such Ω. Thus is an open subset in ℂ^g(g+1)/2. It is called the Siegel upper-half-space. The fundamental definition is:

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