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arXiv · 2003.07172

Orchards in elliptic curves over finite fields

R. Padmanabhan·Alok Shukla
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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, Gr\"unbaum 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.

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R. Padmanabhan, Alok Shukla. 2020-03-03. Orchards in elliptic curves over finite fields. https://doi.org/10.1016/j.ffa.2020.101756

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