Explicit constants, optimal slope, and limit points for linear realizability of prescribed rational-point counts on curves of every genus over finite fields

For a prime power $q$ and genus $g$, let $\mathcal{M}_q(g)$ be the set of integers $N$ that occur as the number of $\mathbb{F}_q$-rational points of a genus-$g$ curve, and let $N_q(g) = \max \mathcal{M}_q(g)$. Anbar and Stichtenoth (2013) proved that there are constants $a_q, b_q > 0$ such that every pair $(N, g)$ with $g \ge a_q N + b_q$ is realized - equivalently, $[0, \alpha_q g - \beta_q]$ intersected with the nonnegative integers lies in $\mathcal{M}_q(g)$ for every $g \ge 0$ - strengthening the theorem of Elkies, Howe, Kresch, Poonen, Wetherell and Zieve that $N_q(g) \ge c \log q \cdot g$ for every genus. Their constants are inexplicit (they inherit an ineffective absolute constant $c$), and explicit constants are known only when $q$ is a square and for $q = 2$. The problem left open there and in the authors' companion note has three facets: (i) determine explicit numerical values of $a_q, b_q$ for at least one non-square $q \ne 2$; (ii) determine the optimal slope $\alpha^*_q$, the supremum of slopes $\alpha$ for which $[0, \alpha g - \beta]$ is contained in $\mathcal{M}_q(g)$ for all $g$ with some offset $\beta$, in particular whether $\alpha^*_q$ equals $A^-(q) = \liminf N_q(g)/g$; (iii) determine the set $L_q$ of limit points of the point-to-genus ratio $N(C)/g(C)$, in particular whether $L_q = [0, A(q)]$ where $A(q)$ is Ihara's constant.

Background

A fundamental object in the arithmetic of curves over finite fields is the set $\mathcal{M}_q(g) := \{N \in \mathbb{Z}_{\ge 0} \mid \text{there is a curve } C/\mathbb{F}_q \text{ of genus } g \text{ with exactly } N \text{ rational points}\},$ where a curve over the finite field $\mathbb{F}_q$ (q a prime power) means a smooth, projective, geometrically irreducible algebraic curve, its rational points are its $\mathbb{F}_q$-points, and its genus g is the nonnegative integer measuring its complexity (the number of independent holomorphic differentials). The Hasse-Weil-Serre bound confines $\mathcal{M}_q(g)$ to the interval $[0, q+1+g\lfloor 2\sqrt{q}\rfloor]$, but a complete description of $\mathcal{M}_q(g)$ is considered out of reach, and $\mathcal{M}_q(g)$ is in general not an interval: for some (q, g) gaps occur. Research has therefore concentrated on the maximum $N_q(g) := \max \mathcal{M}_q(g)$ and its asymptotics. Ihara's constant $A(q) := \limsup_{g\to\infty} N_q(g)/g$ satisfies the Drinfeld-Vladut bound $A(q) \le \sqrt{q}-1$, with equality when q is a square, attained by the explicit Artin-Schreier tower of Garcia and Stichtenoth (a nested family of cyclic degree-$p$ covers of the kind described below); for non-square q the exact value of $A(q)$ is unknown, and the best general lower bound, $A(q) \ge c\log q$ from Serre's class-field-tower method (iterated unramified class field extensions producing infinite families of covers with many points), involves an ineffective constant c (the proof establishes that such a c exists but yields no explicit numerical value for it).

Two every-genus refinements of this asymptotic picture drive the present problem. First, a question of Serre from 1983 (as recorded in the work of Elkies, Howe, Kresch, Poonen, Wetherell and Zieve) asked whether curves with many points exist at every genus, that is, whether $A^-(q) := \liminf_{g\to\infty} N_q(g)/g$ is positive; those authors proved that for every prime power q there is an absolute constant $c > 0$ with $N_q(g) \ge c\log q \cdot g$ for every $g \ge 0$, and their constant c is ineffective. Second, Stichtenoth (2011) showed that for every fixed N there is a genus threshold $g_0(N)$ such that every genus $g \ge g_0(N)$ carries a curve with exactly N rational points, but the realized ratios N/g tend to 0.

Anbar and Stichtenoth (2013) unified and substantially strengthened both: their main theorem states that for every q there are constants $a_q, b_q > 0$ with $\{(N,g) \mid g \ge a_q N + b_q\} \subseteq \mathcal{M}_q,$ that is, for every genus $g \ge 0$, every integer N with $0 \le N \le \alpha_q g - \beta_q$ (where $\alpha_q = 1/a_q$ and $\beta_q = b_q/a_q$) occurs as an exact rational-point count. The constants take the form $a_q = \gamma_q^{-1}(3p-2+2(p-1)\sqrt{q})$ and $b_q = (p-1)(q+2(p-1)\sqrt{q}+3p-1)$, where $p = \operatorname{char}\mathbb{F}_q$ and $\gamma_q = c\log q$ comes from the ineffective constant of Elkies et al. The proof constructs Artin-Schreier extensions (cyclic degree-p covers of function fields given by equations $y^p - y = f(x)$, whose genera are controlled by the Hurwitz genus formula, the standard formula computing the genus of a cover from its ramification data) over a base curve with many points, eliminating surplus rational points one at a time while raising the genus to any prescribed value in the right residue class modulo $p-1$. Two consequences: an interval $[0, \varepsilon_q g] \subseteq \mathcal{M}_q(g)$ for all sufficiently large g, and containment of an interval in the set $L_q$ of limit points of the point-to-genus ratio $N(C)/g(C)$, defined as the set of real $\alpha$ for which there exist curves $C_i/\mathbb{F}_q$ with $g(C_i) \to \infty$ and $N(C_i)/g(C_i) \to \alpha$; concretely $[0, \delta_q] \subseteq L_q$ with $\delta_q = 1/a_q$. Since always $L_q \subseteq [0, A(q)]$ and $A(q) \in L_q$, the sharp question is whether $L_q$ is the entire interval $[0, A(q)]$. Explicit constants are obtained in the paper only for square q (roughly $g \ge \frac{2q}{q-1}p(p+4)N$ for $N \ge q-1$, with small N handled by a separate bound) and for $q = 2$ ($g \ge 27.237N$ for $N \ge 3$, plus explicit equations realizing $N = 0, 1, 2$ at every genus). The same paper also proves, as logically separate companion results, that prescribed place counts of degrees $1, \dots, m$ and prescribed initial coefficients of the L-polynomial (the numerator of the zeta function of the function field, which encodes the numbers of places of every degree) can be realized for all sufficiently large genera; those refinements are not part of the problem formulated here.

Problem Statement

Fix a prime power $q$ of characteristic $p$. For $g \ge 0$ let $\mathcal{M}_q(g)$ denote the set of nonnegative integers $N$ such that some curve over $\mathbb{F}_q$ of genus $g$ has exactly $N$ rational points, and let $N_q(g) = \max \mathcal{M}_q(g)$, $A(q) = \limsup_{g\to\infty} N_q(g)/g$, $A^-(q) = \liminf_{g\to\infty} N_q(g)/g$. Let $L_q$ be the set of limit points of the point-to-genus ratio, $L_q := \{\alpha \in \mathbb{R} \mid \exists\ \text{curves } C_i/\mathbb{F}_q \text{ with } g(C_i) \to \infty \text{ and } N(C_i)/g(C_i) \to \alpha\},$ where $N(C)$ denotes the number of $\mathbb{F}_q$-rational points of the curve $C$, and define the optimal realizability slope $\alpha^*_q := \sup\{\alpha \ge 0 \mid \exists \beta \in \mathbb{R} \text{ such that } [0, \alpha g - \beta] \cap \mathbb{Z}_{\ge 0} \subseteq \mathcal{M}_q(g) \text{ for every } g \ge 0\}.$ Anbar and Stichtenoth proved $\alpha^*_q \ge \delta_q > 0$ with $\delta_q = \gamma_q/(3p-2+2(p-1)\sqrt{q})$ for an ineffective $\gamma_q = c\log q$, and $[0, \delta_q] \subseteq L_q$; plainly $\alpha^*_q \le A^-(q)$ and $L_q \subseteq [0, A(q)]$ with $A(q) \in L_q$. The problem - one scientific objective, namely to determine how large, how sharp, and how explicit the guaranteed linear realizability region for prescribed exact point counts is - has three linked facets:

(a) Effectivity. For at least one non-square prime power $q \ne 2$ (for instance $q = 3$ or any odd prime), determine explicit numerical constants $a_q, b_q$ with $\{(N,g) \mid g \ge a_q N + b_q\} \subseteq \mathcal{M}_q$, together with an explicit construction of the realizing curves. Alternatively, or more generally, give an effective value of, or an effective substitute for, the constant $\gamma_q$ in the bound $N_q(g) \ge \gamma_q g$ valid for all g (an effective value or substitute being an explicit, computable constant, not necessarily equal to the ineffective $\gamma_q$, for which the inequality is proved by an argument that yields its numerical value). Either form of deliverable, on its own, resolves facet (a).

(b) Optimal slope. Determine $\alpha^*_q$. In particular, decide whether $\alpha^*_q = A^-(q)$, or whether $\alpha^*_q < A^-(q)$; in the latter case identify the obstruction, that is, give a quantitative account of the gap between realizing the maximal count and realizing every smaller count at every genus.

(c) Limit points. Determine $L_q$; in particular, decide whether $L_q = [0, A(q)]$. For square q, where $A(q) = \sqrt{q}-1$ is known, this is the already-concrete question of whether every ratio in $[0, \sqrt{q}-1]$ is a limit of point-to-genus ratios of curves over $\mathbb{F}_q$.

The three facets quantify the same realizability region from the effective, the uniform, and the asymptotic side; they are independent, self-contained deliverables, so an answer is judged against the stated target of whichever facet it addresses, a complete resolution of one facet stands as a resolution of that facet alone, and partial progress on one facet resolves neither it nor the others.

References

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