Basic results on theta functions in several variables
David B. Mumford · Progress in mathematics · 1983
We seek a generalization of the function ϑ (z, τ) of Chapter I where z є ℂ is replaced by a g-tuple z → =( z 1 ,⋯, z g )∈ ℂ g $$\vec z = \left( {{z_1}, \cdots ,{z_g}} \right) \in {\mathbb{C}^g}$$ , 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: ϑ( z → ,Ω )= ∑ n → ∈ ℤ g exp( π i t n → Ω n → +2π i t n → ⋅ z → ) . $$\vartheta \left( {\vec z,\Omega } \right) = \sum\limits_{\vec n \in {\mathbb{Z}^g}} {\exp \left( {\pi {\kern 1pt} {i^t}\vec n\Omega \vec n + 2\pi {i^t}\vec n \cdot \vec z} \right)} .$$ (Here n → , z → $$\vec n,\vec z$$ are thought of as column vectors, so n t→ $$\mathop n\limits^{t \to } $$ is a row vector, n t→ ⋅ z → $$\mathop n\limits^{t \to } \cdot \vec z$$ is the dot product, etc.; we shall drop the arrow where there is no reason for confusion between a scalar and a vector.)