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Compact Riemann Surfaces

  • Raghavan Narasimhan
  • Yves Nievergelt

Abstract

Exercise 322. For each τ ∈ ℂ with ℑm(τ) > 0, define the lattice ⋀ τ := ℤ × τℤ ⊂ ℂ, and define the complex torus X τ := ℂ/⋀ τ . For two such complex numbers τ 1, τ 2 ∈ ℂ with ℑm(τ 1) > 0 and ℑm(τ 2) > 0, assume that there exist a holomorphic isomorphism \( f:{X_{{\tau _1}}} \to {X_{{\tau _2}}} \) and an entire function g : ℂ → ℂ such that the following diagram commutes:
$$ {p_1}\matrix{ c & {\buildrel g \over \longrightarrow } & c \cr \downarrow & {} & \downarrow \cr {{X_{{\tau _1}}}} & {\mathrel{\mathop{\kern0pt\longrightarrow} \limits_f} } & {{X_{{\tau _2}}}} \cr } {p_2} $$
where each \( {p_k}:c \to {X_{{\tau _k}}} = c/{\Lambda _{{\tau _k}}} \) is the canonical projection.

Keywords

Riemann Surface Meromorphic Function Complex Manifold Compact Riemann Surface Linear Fractional Transformation 
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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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Raghavan Narasimhan
    • 1
  • Yves Nievergelt
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsEastern Washington UniversityCheneyUSA

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