Compact Riemann Surfaces

Raghavan Narasimhan and Yves Nievergelt
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Raghavan Narasimhan: University of Chicago, Department of Mathematics
Yves Nievergelt: Eastern Washington University, Department of Mathematics

Chapter Chapter 9 in Complex Analysis in One Variable, 2001, pp 351-359 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-1-4612-0175-5_22

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