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 (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.)

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

## Keywords

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

© Springer Science+Business Media New York 1983