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

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations