Galois group at Galois point for genus-one curve (Q2908988)

The authors continue the study of Galois points on curve, initiated in [\textit{H. Yoshihara}, J. Algebra 226, No.1, 283--294 (2000; Zbl 0983.11067); J. Algebra 239, No. 1, 340--355 (2001; Zbl 1064.14023)]. In this article, they study the case of genus-one curve over \(\mathbb{C}\), i.e. plane irreducible curve whose smooth model has genus one. They prove that a finite group \(G\) can be the Galois group at a Galois point for some genus-one curve if and only if \(G\) is isomorphic to one of the following \begin{itemize}\item[1)] Abelian cases: \((\mathbb{Z}/2\mathbb{Z})^2\), \((\mathbb{Z}/2\mathbb{Z})^3\), \(\mathbb{Z}/3\mathbb{Z}\),\((\mathbb{Z}/3\mathbb{Z})^2\), \(\mathbb{Z}/4\mathbb{Z}\), \(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}\) or \(\mathbb{Z}/6\mathbb{Z}\). \item[2)] Some infinite families of non-abelian subgroups of the automorphism group of an elliptic curve (as variety) defined in the article.NEWLINENEWLINENote that the case where the curve is a smooth plane cubic corresponds to the case \(\mathbb{Z}/3\mathbb{Z}\). In all abelian cases, the authors give defining equations for the corresponding curves.NEWLINENEWLINEThe key lemma is that a finite subgroup \(G\) of the automorphism group of the associated elliptic curve is a Galois group at a Galois point if and only if \(|G| \geq 3\) or it contains non-trivial automorphisms of the elliptic curve fixing the neutral element.NEWLINENEWLINEThe authors mention that the positive characteristic cases present many new phenomenas (see \textit{S. Fukasawa}, [Geom. Dedicata 139, 211--218 (2009; Zbl 1160.14304)]) and state as an open problem in characteristic zero to prove that the number of Galois points is at most three.\end{itemize}