# Linear Theory of Theta Functions

• Serge Lang
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 89)

## 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
$$F(x + u) = F(x){e^{2\pi i[L(x,u) + J(u)]}},{\text{ }}all{\text{ }}x{\text{ }} \in {\text{V,u}} \in {\text{D}}$$
(1)
where 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

Entire Function Linear Theory Null Space Theta Function Function Field
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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