# Linear Theory of Theta Functions

Chapter

## Abstract

Let V be a complex vector space of dimension n, real dimension 2n. Let D be a lattice in V, that is, a discrete subgroup of real dimension 2n, so that the factor group V/D is a complex torus. We define a theta function on V, with respect to D (or on V/D), to be a quotient of entire functions (called a meromorphic function for this chapter), not identically zero, and satisfying the relation where

$$F(x + u) = F(x){e^{2\pi i[L(x,u) + J(u)]}},{\text{ }}all{\text{ }}x{\text{ }} \in {\text{V,u}} \in {\text{D}}$$

(1)

*L*is C-linear in*x*, and no specifications are made on its dependence on*u*, or on the dependence of the function*J*on*u*. However, we note that we can change*J*by a**Z**-valued function on*D*without changing the above equation. Also, we shall see below that any such*L*and*J*must satisfy additional conditions which can be deduced from this equation. We note that the theta functions form a multiplicative group.### Keywords

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## Copyright information

© Springer-Verlag New York Inc. 1972