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# Basic results on theta functions in several variables

• David Mumford
Chapter
Part of the Progress in Mathematics book series (PM, volume 28)

## 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⊂ℂg. 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 logg be the set of such Ω. Thus logg is an open subset in ℂg(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.)

## Keywords

Functional Equation Holomorphic Function Meromorphic Function Modular Form Theta Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1983

## Authors and Affiliations

• David Mumford
• 1
1. 1.Department of MathematicsHarvard UniversityCambridgeUSA