Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.05212

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

Emmanuel Hallouin·Philippe Moustrou·Marc Perret
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emmanuel Hallouin, Philippe Moustrou, Marc Perret. 2025-06-05. Refinements on higher order Weil-Oesterlé bounds via a Serre type argument. https://arxiv.org/abs/2506.05212

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Documentation for the ratpoints program

This note explains how to obtain, install, and use the ratpoints program. The program finds rational points up to a specified height on hyperelliptic curves using a highly optimized quadratic sieving algorithm.

math.NT↗
arXiv2026

Transfer operator for the Gauss' continued fraction map. I. Structure of the eigenvalues and trace formulas

Let L be the transfer operator associated with the Gauss' continued fraction map, known also as the Gauss-Kuzmin-Wirsing operator, acting on the Banach space. In this work we prove a two-term asymptotic formula for the eigenvalues of L, show their algebraic simplicity, sign alternation pattern, and decrease in absolute value. This settles, in a stronger form, the conjectures of D. Mayer and G. Roepstorff (1988), A.J. MacLeod (1992), Ph. Flajolet and B. Vallee (1995), also supported by several other authors. Further, we find an exact series for the eigenvalues, which also gives the canonical decomposition of trace formulas due to D. Mayer (1976) and K.I. Babenko (1978). This crystallizes the contribution of each individual eigenvalue in the trace formulas.

math.NT↗
arXiv2026

Arithmetic Sparsity and Obstructions in Weighted Projective Spaces

Let $\mathbb{WP}^n_{\mathbf{q}}$ be a weighted projective space with weights $\mathbf{q} = (q_0, \dots, q_n)$, $q = \operatorname{lcm}(q_i)$, and let $ϕ\colon \mathbb{WP}^n_{\mathbf{q}} \to \mathbb{P}^n$, $[x_i] \mapsto [x_i^{q/q_i}]$, be the Veronese morphism. A point of $\mathbb{P}^n(\mathbb{Q})$ is the image of a rational point of $\mathbb{WP}^n_{\mathbf{q}}$ only if its valuation vector at every prime satisfies a Kummer congruence. We count the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ of bounded weighted height $\mathfrak{h} = H(ϕ(\,\cdot\,))^{1/q}$ and prove that, on the locus where all coordinates are nonzero, $$ Z^{\circ}_{\mathfrak{h}}\big( \mathbb{WP}^n_{\mathbf{q}}(\mathbb{Q}), X \big) = X^{q\,a(\mathbf{q})} P_{\mathbf{q}}(\log X) + O\big( X^{q\,a(\mathbf{q}) - θ} \big), \qquad θ> 0, $$ with $P_{\mathbf{q}}$ of exact degree $β(\mathbf{q})$, where $a(\mathbf{q})$ and $β(\mathbf{q})$ are the value and the dimension of the optimal face of a linear program determined by the Kummer congruences. The exponent satisfies $Q \leq q\,a(\mathbf{q}) \leq q(n+1)$, $Q = \sum q_i$, with equality on the right if and only if the exponents $q/q_i$ are pairwise coprime; the difference $q(n+1) - q\,a(\mathbf{q})$ measures the sparsity of the rational points of $\mathbb{WP}^n_{\mathbf{q}}$ relative to those of its Veronese image. The leading constant is evaluated when the dual optimum is diagonal and for the weights $(2,2,3,3)$. The full counting function follows by stratification, and we formulate the conjecture over an arbitrary number field.

math.NT↗