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Configurations on elliptic curves

Andrin Halbeisen, Lorenz Halbeisen and Norbert Hungerbühler

Vol. 19 (2022), No. 3, 111–135
Abstract

An elliptic configuration is a configuration with all its points on a cubic curve, or more precisely, where all points are in the torsion group of an elliptic curve. We investigate the existence of elliptic (3r4,4r3) configurations for r 5. In particular, we construct elliptic ((p 1)3) configurations for every prime p > 7 and show that there are (3r4,4r3) configurations whenever 3r = p 1 for some prime p > 7. Furthermore, we show that for every k 2 there is an elliptic (9k4,12k3) configuration with a rotational symmetry of order 3, where we introduce a new normal form for D3-symmetric elliptic curves.

Dedicated to the memory of Branko Grünbaum

Keywords
configurations, elliptic curves
Mathematical Subject Classification
Primary: 51A20
Secondary: 51A05
Milestones
Received: 5 November 2021
Revised: 15 July 2022
Accepted: 7 August 2022
Published: 10 October 2022
Authors
Andrin Halbeisen
Winterthur
Switzerland
Lorenz Halbeisen
Department of Mathematics
ETH Zentrum
Zürich
Switzerland
Norbert Hungerbühler
Department of Mathematics
ETH Zentrum
Zürich
Switzerland