Linear Theory of Theta Functions

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


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


Entire Function Linear Theory Null Space Theta Function Function Field 
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Copyright information

© Springer-Verlag New York Inc. 1972

Authors and Affiliations

  • Serge Lang
    • 1
  1. 1.Department of MathematicsYale UniversityNew HavenUSA

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