Tata Lectures on Theta I pp 118-235 | Cite as

# Basic results on theta functions in several variables

Chapter

## 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 Open image in new window be the set of such Ω. Thus Open image in new window is an open subset in ℂ^{g(g+1)/2}. It is called the Siegel upper-half-space. The fundamental definition is:$$
\vartheta (\vec z,\Omega ) = \sum\limits_{\vec n \in \mathbb{Z}^g } {\exp } (\pi i^t \vec n\Omega \vec n + 2\pi i^t \vec n \cdot \vec z).
$$

## Keywords

Functional Equation Holomorphic Function Meromorphic Function Modular Form Theta Function
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## Copyright information

© Birkhäuser Boston 2007