Genus-five hyperelliptic or trigonal curves with many rational points in characteristic three

2022/04/14 by Kudo, Momonari, Harashita, Shushi
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2204.06805

Abstract

The number N₉(5), the maximal number of \mathbbF₉-rational points on curves over \mathbbF₉ of genus 5 is unknown, but it is known that 32 ≤ N₉(5)≤ 35. In this paper, we enumerate hyperelliptic curves and trigonal curves over \mathbbF₃ which have many \mathbbF₉-rational points (and \mathbbF₃-rational points), especially the maximal number of \mathbbF₉-rational points of those curves is 30. Kudo-Harashita studied the nonhyperelliptic and nontrigonal case,where they found a new example of curves (over \mathbbF₃) of genus five which attains 32 and proved that there is no example attaining more than 32, among sextic plane curves with mild singularities. We conclude from the main results in this paper that we need to search sextic models (i.e., nonhyperelliptic and nontrigonal) with bad singularities, in order to find a genus-five curve over \mathbbF₃ with at least 33 \mathbbF₉-rational points.

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