The quasi-periods of the Weierstrass zeta-function

2022/12/14 by Mario Bonk, Bonk, Mario
Mathematics · #11F03 #30C20 #33C75 #33E05 #Advanced Mathematical Identities #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #History and Theory of Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2212.07012

openalex publication_date 2022/12/14 · openalex created_date 2022/12/27 · openalex updated_date 2026/07/28

Abstract

We study the ratio p=η₁/η₂ of the pseudo-periods of the Weierstrass ζ-function in dependence of the ratio τ=ω₁/ω₂ of the generators of the underlying rank-2 lattice. We will give an explicit geometric description of the map τ↦ p(τ). As a consequence, we obtain an explanation of a theorem by Heins who showed that p attains every value in the Riemann sphere infinitely often. Our main result is implicit in the classical literature, but it seems not to be very well known. Essentially, this is an expository paper. We hope that it is easily accessible and may serve as an introduction to these classical themes.

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