Refinements on higher order Weil-Oesterl\'e bounds via a Serre type argument

TL;DR

This paper combines Serre's arithmetic constraints with recent semi-definite programming approaches to refine bounds on the number of points of algebraic curves over finite fields, improving upon existing bounds.

Contribution

It introduces a new method that merges Serre's refinement with semi-definite programming to strengthen Ihara's bound on curve points over finite fields.

Findings

The new bound generally improves Ihara's bound.

The approach recovers Weil's and Ihara's bounds at initial steps.

Potential extensions to higher order bounds are discussed.

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 Oesterl\'e 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-Oesterl\'e bounds.