Orchards in elliptic curves over finite fields

Abstract

Consider a set of $n$ points on a plane. A line containing exactly $3$ out of the $n$ points is called a $3$-rich line. The classical orchard problem asks for a configuration of the $n$ points on the plane that maximizes the number of $3$-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of $3$-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of $3$-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of $3$-rich lines agrees with the Green-Tao formula.

arXiv comment15 pages