Theta Functions and Divisors

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


Let M be a complex manifold. In the sequel, M will either be Cn or Cn/D where D is a lattice (discrete subgroup of real dimension 2n). Let U i be an open covering of M, and let ϕi be a meromorphic function on Ui. If for each pair of indices (i, j) the function ϕi /ϕj is holomorphic and invertible on UiUj, then we shall say that the family (Ui, ϕi) represents a divisor on M. If this is the case, and (U, ϕ) is a pair consisting of an open set U and a meromorphic function ϕ on U, then we say that (U, ϕ) is compatible with the family (Ui, ϕi) if ϕi/ϕ is holomorphic invertible on UUi. If this is the case, then the pair (U, ϕ) can be adjoined to our family, and again represents a divisor. Two families (Ui, ϕi) and (Vk, ψk) are said to be equivalent if each pair (Vk, ψk) is compatible with the first family. An equivalence class of families as above is called a divisor on M. Each pair (U, ϕ) compatible with the families representing the divisor is also said to represent the divisor on the open set U.


Meromorphic Function Open Covering Theta Function Abelian Variety Positive Divisor 
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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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