Closed-form Serre-type arithmetic refinements of Weil–Oesterlé bounds of order $n\ge 3$

The maximum number $N_q(g)$ of rational points on a curve of genus $g$ over the finite field $\mathbb F_q$ is controlled by the Weil–Oesterlé hierarchy: a sequence of semidefinite bounds of Hallouin and Perret whose first steps recover Weil's and Ihara's bounds. Hallouin, Moustrou, and Perret (arXiv:2506.05212) combined this hierarchy with Serre's arithmetic-integrality argument — the observation that a Galois-invariant product of positive algebraic integers is a positive integer — and obtained, for order 2, a closed-form and provably optimal 'Ihara–Serre' improvement of Ihara's bound, while for order 3 they obtained only experimental improvements via computer search over admissible test matrices. The problem left open by that work is: extend the arithmetic refinement theory to orders $n\ge 3$. Concretely: find a closed-form, provably optimal arithmetic test matrix together with an analytic gain formula for the order-$n$ Weil–Oesterlé bound; give a systematic analytic description of the floor/ceiling (fractional-part) phenomena that govern when the gain is nonzero; and provide a quantitative theory of how the refinement's efficacy decays as the order grows. The problem remains open.

Background

Let $q$ be a prime power and $X$ an absolutely irreducible smooth projective curve of genus $g$ over the finite field $\mathbb F_q$. A basic quantitative problem of the field, with direct consequences for algebraic-geometry error-correcting codes, is to estimate $N_q(g)$, the maximum of $\#X(\mathbb F_q)$ over such curves; the record bounds are curated in the manYPoints database. Weil proved $|N_1-(q+1)|\le 2g\sqrt q$, where $N_1=\#X(\mathbb F_q)$. The estimate flows from the Frobenius eigenvalues: there are algebraic integers $\omega_1,\dots,\omega_g$ of modulus $\sqrt q$, stable under the absolute Galois group of $\mathbb Q$, such that with $N_k=\#X(\mathbb F_{q^k})$ the traces $t_k=1+q^k-N_k=\sum_j(\omega_j^k+\bar\omega_j^k)$ convert any lower bound on $t_1$ into an upper bound on $N_1$. Serre sharpened Weil's bound to $N_1\le(q+1)+g\lfloor 2\sqrt q\rfloor$ by an arithmetic argument: each $\tau_1(\omega)=\omega+\bar\omega$ satisfies $\tau_1(\omega)+\lfloor 2\sqrt q\rfloor+1>0$, and by the arithmetic–geometric mean (AM–GM) inequality the average $t_1/g+\lfloor 2\sqrt q\rfloor+1$ of these $g$ positive numbers is at least their geometric mean, which is at least $1$ because their product is rational, an algebraic integer, and positive — hence a positive integer; therefore $t_1\ge -g\lfloor 2\sqrt q\rfloor$. Independently, Ihara showed that Weil's bound cannot be optimal for large genus; the method of explicit formulae — optimizing over trigonometric polynomials in the Frobenius angles that are nonnegative on the whole angle range — solved explicitly by Oesterlé and systematized in Serre's published Harvard course notes, yields the stronger Oesterlé bounds. Hallouin and Perret reformulated these bounds geometrically: by the Hodge index theorem (the classical result fixing the signature of the intersection pairing on divisors of a smooth projective surface) applied to the surface $X\times X$, the Gram matrix of the projections of the diagonal correspondences $\Gamma_0,\dots,\Gamma_n$ (orthogonal to the horizontal and vertical divisors) is a positive semidefinite (PSD) Toeplitz-type matrix in $t_1,\dots,t_n$; together with the affine inequalities $t_k\le t_1+q^k-q$ (valid since $N_k\ge N_1$), the vector $(t_1,\dots,t_n)$ of any curve lies in a spectrahedron — a convex set defined by PSD constraints on a matrix whose entries depend affinely on the coordinates — and minimizing $t_1$ over this spectrahedron gives the order-$n$ Weil–Oesterlé bound $N^\star_n(g)$. The hierarchy starts with Weil's bound ($n=1$) and Ihara's bound ($n=2$); $N^\star_{n+1}$ improves on $N^\star_n$ only beyond an explicit genus threshold $g_{n+1}$, and as $n$ grows the hierarchy approaches the Drinfeld–Vlăduț asymptotic bound on Ihara's constant $A(q)=\limsup_{g\to\infty}N_q(g)/g$, the supremal limiting ratio of rational points to genus. The source combines the two strategies. A positive (semi)definite 'test matrix' $A$ with arithmetic coefficients produces, by trace duality with the rank-one Gram block $M(\omega)$ attached to a single eigenvalue, an integer-affine inequality $\sum_k a_k\tau_k(\omega)+a_0>0$ valid for every $|\omega|=\sqrt q$; the source's Lemma 2.1 then replays Serre's AM–GM/Galois-integrality argument to give $\sum_k a_k t_k+g\,a_0\ge g$, and substituting the constraints $t_k\le t_1+q^k-q$ (its Theorem 2.4) yields an arithmetic lower bound on $t_1$. For $n=2$ the authors solve the optimization completely: their Theorem 2.5 exhibits, in closed form, the optimal matrix under the normalization fixing the top-left entry to $1$, and proves that the resulting 'Ihara–Serre' bound improves Ihara's bound by $g\,(\alpha-\lfloor\alpha\rfloor)(\lceil\alpha\rceil-\alpha)/(2\lceil\alpha\rceil)$, where $\alpha=-t_I/g$ and $t_I$ is Ihara's lower bound on $t_1$. They analyze this gain — for fixed $q\ne 3$ it grows linearly in $g$; it improves the fixed-order asymptotic constant; it is optimal on an explicit infinite family of pairs $(q,g)$ in the Ihara range $[g_2,g_3]$, the range where Ihara's bound is not improved by higher orders of the hierarchy — and it improves more than 150 tabulated entries, setting 20 new records for $q\le 100$, $g\le 50$. For $n=3$ the analogue (their Theorem 3.1) requires the coefficient combinations $d$, $2a$, and $2a\sqrt q+b$ to be natural integers, and a computer search over admissible matrices yields four new records (their Theorem 3.2), but with no closed form and no optimality proof. Section 3.2 of the source details the obstacles: the kernel-vector construction of a good starting matrix is defined only up to scale; the top-left normalization that was optimal for $n=2$ appears suboptimal for $n=3$; the optimal point $t_W$ of the order-$n$ program is cut out by polynomial equations of degree growing with $n$; and irrational quantities (powers of $\sqrt q$) enter the coefficients, so no canonical candidate for the optimal $A$ is known for $n\ge 3$.

Problem Statement

The problem left open by Hallouin–Moustrou–Perret (arXiv:2506.05212, Section 3, 'Obstacles and limits'): extend the arithmetic (Serre-type) refinement of the Weil–Oesterlé hierarchy to orders $n\ge 3$, replacing the current experimental, search-based order-3 results by a theory comparable to the order-2 theory of Theorem 2.5 of the source. Concretely, for each order $n\ge 3$ (or uniformly in $n$):

(1) Closed-form optimal certificate. Exhibit in closed form a positive (semi)definite test matrix $A$ with arithmetic coefficients — in the source these are $2\times2$ matrices $A=\begin{pmatrix}d & a\\ a & b\end{pmatrix}$ subject, for $n=2$, to $d,b\in\mathbb N$ and $2a\in\mathbb N$ (its Theorem 2.4) and, for $n=3$, to $d,2a,2a\sqrt q+b\in\mathbb N$ (its Theorem 3.1), the general admissible class being the positive (semi)definite test matrices $A$ of the order-$n$ block whose trace pairing $\langle M(\omega),A\rangle=\sum_{k=1}^n a_k\tau_k(\omega)+a_0$ is an integer-affine certificate with integer coefficients $a_0,\dots,a_n$ and $a_k\ge0$ for $1\le k\le n$ — together with a closed-form expression for the resulting Serre-type improvement of the order-$n$ Weil–Oesterlé bound $N^\star_n(g)$, valid for all prime powers $q$ and all genera $g$ in the range where the order-$n$ bound is not improved by higher orders of the hierarchy; and prove that the exhibited $A$ is optimal within the class of admissible arithmetic certificates (optimality is required in the full class, with no normalization imposed; Theorem 2.5 proves the order-$2$ analogue under the normalization fixing the top-left entry to $1$).

(2) Rounding analysis. Give a systematic analytic description — beyond case-by-case or computational checks — of the set of parameters $(q,g,n)$ for which the floor/ceiling (fractional-part) phenomena governing the gain produce a strict improvement of the order-$n$ Weil–Oesterlé bound.

(3) Decay of efficacy. Provide a conceptual and quantitative account of how the improvement achievable by optimal integer-affine truncations decays as the order $n$ (and as $q,g$) vary, explaining the numerically observed weakening from Serre's gain on Weil's bound, to the order-2 gain on Ihara's bound, to the order-3 gain on Oesterlé's bound.

These items elaborate obstacles discussed in the source (Sections 3 and 3.2). Together they decompose a single scientific objective — an arithmetic-refinement theory of the higher-order Weil–Oesterlé hierarchy — of which the closed-form order-2 theory and the four order-3 computational records currently mark the boundary of knowledge; this objective is to be established for every order $n\ge 3$ or by a single treatment uniform in $n$.

References

1. Emmanuel Hallouin, Philippe Moustrou, Marc Perret, Refinements on higher order Weil-Oesterlé bounds via a Serre type argument, arXiv:2506.05212 (2025), https://arxiv.org/abs/2506.05212
2. Emmanuel Hallouin, Marc Perret, A unified viewpoint for upper bounds for the number of points of curves over finite fields via Euclidean geometry and semi-definite symmetric Toeplitz matrices, Transactions of the American Mathematical Society 372 (2019), no. 8, 5409–5451, DOI 10.1090/tran/7813, https://doi.org/10.1090/tran/7813
3. Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 9, 397–402 (reprinted in Œuvres – Collected Papers III, Springer), DOI 10.1007/978-3-642-39816-2_128, https://doi.org/10.1007/978-3-642-39816-2_128
4. Jean-Pierre Serre, Rational Points on Curves over Finite Fields, with contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler, Documents Mathématiques 18, Société Mathématique de France, Paris, 2020, ISBN 978-2-85629-923-4, https://smf.emath.fr/publications/rational-points-curves-over-finite-fields
5. Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724, MR 656048, Zbl 0509.14019, https://zbmath.org/0509.14019
6. Sergei G. Vlăduț, Vladimir G. Drinfel'd, Number of points of an algebraic curve, Functional Analysis and Its Applications 17 (1983), 53–54, DOI 10.1007/BF01083182, https://doi.org/10.1007/BF01083182
7. manYPoints — Tables of curves with many points (online database of upper and lower bounds for $N_q(g)$), http://manypoints.org