Constructing cubic curves with involutions (Q2094275)

In the paper under review, the authors give a proof of Schroeter's construction. Schroeter's construction is a method of generating points on a cubic curve via line involutions using only a ruler. The proof relies on Chasles' Theorem and geometric properties of line involutions and addition on elliptic curves. Possible applications of the construction are presented. For instance, it can be used to obtain a parametrisation of elliptic curves with torsion group \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}\) suitable to study some properties of pythagorean pairs (see [\textit{L. Halbeisen} and \textit{N. Hungerbühler}, J. Number Theory 233, 467--480 (2022; Zbl 1484.11104)]). In addition, Schroeter's construction provides a few examples of configurations whose points belong to an elliptic curve.