Compact Riemann Surfaces

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.