Improving the lower bounds for the Ihara constants $A(2)$ and $A(3)$ (toward determining $A(q)$ for nonsquare finite fields)

For a finite field $\mathbb{F}_q$, the Ihara constant $A(q)$ measures the asymptotically maximal ratio of rational points to genus on algebraic curves over $\mathbb{F}_q$. It equals $\sqrt{q}-1$ when $q$ is a square, but for nonsquare $q$ its exact value is unknown. The current best lower bounds for the two smallest nonsquare cases, $A(2)\geq 0.316999\ldots$ and $A(3)\geq 0.492876\ldots$, were obtained by Duursma and Mak (2013) via a variant of Serre's class field tower method with explicit ray class field conductor control, and the authors explicitly flag the bottlenecks of their technique: exact conductor-degree data for the characters of the ray class field extension, the search for better base function fields, and the combinatorial optimization of conductor sets, conductor exponents, and splitting sets. The records remain the best known and the flagged bottlenecks remain open, so the problem of proving a strictly larger lower bound for $A(2)$ or $A(3)$ — or determining either constant exactly, anywhere up to the Drinfeld–Vlăduţ ceilings $\sqrt{2}-1\approx 0.414214$ and $\sqrt{3}-1\approx 0.732051$ — is open.

Background

Let $\mathbb{F}_q$ be the finite field with $q$ elements and let $\mathcal{X}/\mathbb{F}_q$ be a projective, nonsingular, geometrically irreducible curve of genus $g(\mathcal{X})$ (the genus is the dimension of the space of holomorphic differentials, a birational invariant measuring arithmetic complexity). Write $N(\mathcal{X})$ for the number of $\mathbb{F}_q$-rational points of $\mathcal{X}$. The Hasse–Weil bound $N(\mathcal{X})\leq q+1+2g\sqrt{q}$ is far from sharp when $g$ is large relative to $q$, so one studies $N_q(g)=\max N(\mathcal{X})$ over genus-$g$ curves and its asymptotic slope, the Ihara constant

$$A(q)=\limsup_{g\to\infty}\frac{N_q(g)}{g}.$$

This quantity governs, among other things, the asymptotic performance of algebraic-geometry (Goppa) error-correcting codes and of arithmetic secret-sharing schemes built from curves. A theorem of Drinfeld and Vlăduţ gives $A(q)\leq\sqrt{q}-1$ for every $q$, and work of Ihara and of Tsfasman–Vlăduţ–Zink on modular and Shimura curves shows $A(q)=\sqrt{q}-1$ whenever $q$ is a square. For nonsquare $q$ — starting with $q=2$ and $q=3$ — the exact value of $A(q)$ is unknown, and the Drinfeld–Vlăduţ bound remains the best known upper bound.

The leading technique for lower bounds when $q$ is prime is Serre's class field tower method, transported to function fields via the curve–function field dictionary (rational points of the curve correspond to degree-one places of its function field). One starts from a base function field $k/\mathbb{F}_q$ of genus $g$ with $t$ rational places, forms a suitable class field tower above it, and proves the tower is infinite using the Golod–Shafarevich criterion — a group-theoretic inequality saying that a nontrivial finite $p$-group $G$ must satisfy $r(G)>d(G)^2/4$, where $d(G)$ is the minimal number of generators and $r(G)$ the minimal number of defining relations; if the generator and relation ranks established for the Galois group of the tower contradict this inequality, the group cannot be finite and the tower is infinite. An infinite tower in which the $t$ rational places split completely yields, by the Hurwitz genus formula (which relates the genus of an extension to the degree and ramification divisor), a lower bound $A(q)\geq t/(g-1)$, refined by controlled ramification. Temkine and others generalized the ramification analysis; Kuhnt's thesis (as documented by Duursma and Mak) considered towers over a finite Galois $p$-extension $K/k$ with wild ramification concentrated in a chosen finite set $S$ of places.

Duursma and Mak (Compositio Mathematica 149 (2013) 1108–1128) developed this into a practical criterion. Given a conductor $\mathfrak{m}=\sum_{\mathfrak{p}\in S}\nu_{\mathfrak{p}}\mathfrak{p}$ (a formal sum of places with positive integer exponents directing allowed ramification) and a set $T$ of $t$ rational places required to split completely, they form the ray class field $K=k^\mathfrak{m}_T$ — the maximal abelian extension of conductor dividing $\mathfrak{m}$ in which all places of $T$ split — and prove (Theorem 1.1) that if the splitting set and conductor exponents satisfy an explicit numerical inequality of Golod–Shafarevich type, then the $(T,p)$-class field tower over $K$ is infinite and

$$A(q)\ \geq\ \frac{t}{g-1+\tfrac{1}{2[K:k]}\sum_{\chi}\deg\mathfrak{f}_\chi},$$

where $\mathfrak{f}_\chi$ is the Artin conductor of the character $\chi$ of $\mathrm{Gal}(K/k)$ (a divisor measuring the ramification of $\chi$), and the sum runs over the characters of that group. Their records $A(2)\geq 0.316999\ldots$ and $A(3)\geq 0.492876\ldots$ use explicit Artin–Schreier models (equations of the form $y^p-y=f$ for a rational function $f$, which define the cyclic degree-$p$ covers in characteristic $p$) of elementary abelian extensions of small-genus function fields over $\mathbb{F}_2$ and $\mathbb{F}_3$: a genus-2 hyperelliptic function field $H=\mathbb{F}_2(x,y)$, $y^2+(x^3+x+1)y=x^2+x$, underlies the $A(2)$ record, while the $A(3)$ record is built over the elliptic curve $y^2=x^3-x+1$ over $\mathbb{F}_3$. The authors state the acknowledged limitations of the method verbatim: 'In general it may be difficult to determine exactly the degrees of the conductors for the characters of the group $\mathrm{Gal}(K/k)$' — forcing either the crude bound $\deg\mathfrak{f}_\chi\leq\deg\mathfrak{m}$ (their Corollary 1.2) or average-conductor factors $(1-p^{-d})$ per degree-$d$ place, which the authors use as rigorous upper bounds on the average conductor — and the systematic search for better base fields and the optimization of $S$, the exponents $\nu_{\mathfrak{p}}$, and $T$ are left open. Determining $A(2)$ or $A(3)$ exactly would additionally require a genuinely new upper-bound technique at or below the Drinfeld–Vlăduţ ceiling for nonsquare $q$, none being currently known.

Problem Statement

Improve the best known lower bounds for the Ihara constants $A(2)$ and $A(3)$ — or determine either constant exactly. Concretely: give a rigorous proof that $A(2)\geq c_2$ for an explicit constant $c_2>0.316999\ldots$ (the 2013 Duursma–Mak record), or that $A(3)\geq c_3$ for an explicit $c_3>0.492876\ldots$, or determine the exact value of $A(2)$ or $A(3)$ (either of which would require a matching upper bound at or below the Drinfeld–Vlăduţ ceilings $\sqrt{2}-1\approx 0.414214$ and $\sqrt{3}-1\approx 0.732051$). This is the derived small-field instance of the general open problem of determining $A(q)$ for nonsquare $q$, posed in the exact form the source paper leaves open. Any method is admissible.

References

1. Iwan Duursma, Kit-Ho Mak, On lower bounds for the Ihara constants $A(2)$ and $A(3)$, Compositio Mathematica 149 (2013), no. 7, 1108–1128. DOI: 10.1112/s0010437x12000796, https://doi.org/10.1112/s0010437x12000796 (author preprint arXiv:1102.4127, https://arxiv.org/abs/1102.4127)
2. Peter Beelen, A survey on recursive towers and Ihara's constant, 2022, arXiv:2203.03310, https://arxiv.org/abs/2203.03310
3. Peter Beelen, Maria Montanucci, Lara Vicino, On the constant $D(q)$ defined by Homma, in: Arithmetic, Geometry, Cryptography, and Coding Theory 2021, Contemporary Mathematics 779, American Mathematical Society, 2022. DOI: 10.1090/conm/779/15669, https://doi.org/10.1090/conm/779/15669 (author preprint arXiv:2201.00602, https://arxiv.org/abs/2201.00602)
4. L. L. Hall-Seelig, New lower bounds for the Ihara function $A(q)$ for small primes, Journal of Number Theory 133 (2013), no. 10, 3319–3324. DOI: 10.1016/j.jnt.2013.04.002, https://doi.org/10.1016/j.jnt.2013.04.002
5. Chung Pang Mok, Xiaoyu Wang, On an analogue of the Ihara constant for exponential sums, Finite Fields and Their Applications 115 (2026), article 102866. DOI: 10.1016/j.ffa.2026.102866, https://doi.org/10.1016/j.ffa.2026.102866
6. Alexandre Temkine, Hilbert class field towers of function fields over finite fields and lower bounds for $A(q)$, Journal of Number Theory 87 (2001), no. 2, 189–210. DOI: 10.1006/jnth.2000.2596, https://doi.org/10.1006/jnth.2000.2596
7. Farshid Hajir, Christian Maire, Asymptotically good towers of global fields, in: European Congress of Mathematics, Vol. II (Barcelona, 2000), Progress in Mathematics 202, Birkhäuser, Basel, 2001, pp. 207–218. DOI: 10.1007/978-3-0348-8266-8_16, https://doi.org/10.1007/978-3-0348-8266-8_16
8. Chaoping Xing, Sze Ling Yeo, Algebraic curves with many points over the binary field, Journal of Algebra 311 (2007), no. 2, 775–780. DOI: 10.1016/j.jalgebra.2006.12.029, https://doi.org/10.1016/j.jalgebra.2006.12.029
9. Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth, Towers of function fields over non-prime finite fields, Moscow Mathematical Journal 15 (2015), no. 1, 1–29. DOI: 10.17323/1609-4514-2015-15-1-1-29, https://doi.org/10.17323/1609-4514-2015-15-1-1-29 (author preprint arXiv:1202.5922)
10. M. A. Tsfasman, S. G. Vlăduţ, Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than Varshamov–Gilbert bound, Mathematische Nachrichten 109 (1982), no. 1, 21–28. DOI: 10.1002/mana.19821090103, https://doi.org/10.1002/mana.19821090103
11. S. G. Vlădut, V. G. Drinfel'd, Number of points of an algebraic curve, Functional Analysis and Its Applications 17 (1983), no. 1, 53–54 (English translation of Funktsional. Anal. i Prilozhen. 17, no. 1, 68–69). DOI: 10.1007/BF01083182, https://doi.org/10.1007/BF01083182
12. Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, Journal of the Faculty of Science, University of Tokyo, Section IA Mathematics 28 (1981), no. 3, 721–724. zbMATH: 0509.14019, https://zbmath.org/?q=an%3A0509.14019