Refinements on higher order Weil-Oesterlé bounds via a Serre type argument

2025/06/05 by Hallouin, Emmanuel, Moustrou, Philippe, Perret, Marc · 1 citation
#11G20 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.05212

Abstract

Weil's theorem gives the most standard bound on the number of points of a curve over a finite field. This bound was improved by Ihara and Oesterlé for larger genus. Recently, Hallouin and Perret gave a new point of view on these bounds, that can be obtained by solving a sequence of semi-definite programs, and the two first steps of this hierarchy recover Weil's and Ihara's bounds. On the other hand, by taking into account arithmetic constraints, Serre obtained a refinement on Weil's bound. In this article, we combine these two approaches and propose a strengthening of Ihara's bound, based on an argument similar to Serre's refinement. We show that this generically improves upon Ihara's bound, even in the range where it was the best bound so far. Finally we discuss possible extensions to higher order Weil-Oesterlé bounds.

Citations

• Refinements of Katz-Sarnak theory for the number of points on curves over finite fields
• The maximum number of points on a curve of genus eight over the field of\n four elements
• On the maximum number of rational points on singular curves over finite fields
• New bounds on the maximum number of points on genus-4 curves over small finite fields

Cited by

• A closed formula for the Geil-Matsumoto bound on semigroups and curves with many points

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• The maximum number of points on a curve of genus eight over the field of\n four elements