The quasi-periods of the Weierstrass zeta-function

TL;DR

This paper explores the relationship between the pseudo-period ratios of the Weierstrass zeta-function and the lattice generators, providing a geometric description and explaining classical theorems about their value distribution.

Contribution

It offers an explicit geometric description of the map from lattice ratios to pseudo-period ratios, clarifying classical results and serving as an accessible introduction to these topics.

Findings

The ratio p attains every value in the Riemann sphere infinitely often.

Provides an explicit geometric description of the map τ→p(τ).

Clarifies classical theorems about the distribution of pseudo-period ratios.

Abstract

We study the ratio $p=\eta_1/\eta_2$ of the pseudo-periods of the Weierstrass $\zeta$-function in dependence of the ratio $\tau=\omega_1/\omega_2$ of the generators of the underlying rank-2 lattice. We will give an explicit geometric description of the map $\tau\mapsto p(\tau)$. 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.