Basic results on theta functions in several variables

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


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.)


Functional Equation Holomorphic Function Meromorphic Function Modular Form Theta Function 
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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

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