The quasi-periods of the Weierstrass zeta-function

Abstract

We study the ratio $p={\eta }_1/{\eta }_2$ of the quasi-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.