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 $(3r_4,4r_3)$ configurations for $r\ge 5$. In particular, we construct elliptic $({(p-1)}_3)$ configurations for every prime $p>7$ and show that there are $(3r_4,4r_3)$ configurations whenever $3r=p-1$ for some prime $p>7$. Furthermore, we show that for every $k\ge 2$ there is an elliptic $(9k_4,12k_3)$ configuration with a rotational symmetry of order $3$, where we introduce a new normal form for $D_3$-symmetric elliptic curves.

Keywords

Mathematical Subject Classification

Milestones

Authors