Hu and Zhu (arXiv:2510.27159, 2025) constructed the first complete instance beyond the polynomial ring Fq[t]\mathbb{F}_q[t] of the explicit Drinfeld-modular method for optimal function-field towers: for the Dedekind domain A\mathcal{A} of functions on PFq1\mathbb{P}^1_{\mathbb{F}_q} regular outside a degree-two place (a class-number-two ring), they wrote closed-form defining equations for the normalized and minimal Drinfeld modular towers x0(I∞k)\mathbf{x}_0(I_\infty^k), and proved that reduction at any degree-two place IηI_\eta of A\mathcal{A} yields a tower over Fq4\mathbb{F}_{q^4} attaining the Drinfeld-Vladut bound λ=q2−1\lambda=q^2-1. Their abstract and Theorem C, however, claim reduction at an arbitrary non-zero principal A\mathcal{A}-ideal IηI_\eta of degree two, while the proof treats exactly the prime ones; the open problem left by that work is therefore to complete the construction for the degree-two principal ideals of A\mathcal{A} that are not prime, i.e. products of two degree-one places, where the current reduction framework does not apply.

Background

A function field over Fq\mathbb{F}_q is a finitely generated extension F/FqF/\mathbb{F}_q of transcendence degree one; its genus g(F)g(F) measures its complexity and N(F)N(F) counts its Fq\mathbb{F}_q-rational places. The Hasse-Weil and Serre bounds cap N(F)N(F) at roughly 2q g(F)2\sqrt{q}\,g(F). For the asymptotic regime Ihara introduced the quantity Ihara(q)=lim sup⁡g(F)→∞N(F)/g(F)\mathrm{Ihara}(q)=\limsup_{g(F)\to\infty}N(F)/g(F), and the Drinfeld-Vladut bound states Ihara(q)≤q−1\mathrm{Ihara}(q)\le\sqrt{q}-1. The bound is sharp exactly when qq is a square: Ihara (via Shimura modular curves) and Tsfasman-Vladut-Zink (via classical modular curves) proved Ihara(q2)=q−1\mathrm{Ihara}(q^2)=q-1, a result that also yields algebraic-geometry codes beating the Gilbert-Varshamov bound. A function field tower is a sequence of extensions F0⊂F1⊂F2⊂⋯F_0\subset F_1\subset F_2\subset\cdots over Fq\mathbb{F}_q with g(Fk)→∞g(F_k)\to\infty; its limit λ=lim⁡kN(Fk)/g(Fk)\lambda=\lim_k N(F_k)/g(F_k) lower-bounds Ihara(q)\mathrm{Ihara}(q), and the tower is called asymptotically optimal when λ\lambda reaches the Drinfeld-Vladut bound. Garcia and Stichtenoth gave the first fully explicit optimal towers over square fields, defined by simple two-variable recursions; Elkies then identified those towers as reductions of Drinfeld modular curves, and conjectured more broadly that optimal recursive towers must have modular origin. Gekeler confirmed that certain towers of Drinfeld modular curves achieve the bound, and Bassa and Beelen showed how explicit equations for the curves x0(Pn)\mathbf{x}_0(P^n) over Fq[t]\mathbb{F}_q[t] can be read off Drinfeld modular polynomials.

For a Dedekind domain A\mathcal{A} that is the coordinate ring of a smooth curve over Fq\mathbb{F}_q with one marked place at infinity, a Drinfeld A\mathcal{A}-module of rank rr over an Fq\mathbb{F}_q-field LL (equipped with a structural map ι:A→L\iota:\mathcal{A}\to L) is an Fq\mathbb{F}_q-algebra homomorphism ϕ:A→L{τ}\phi:\mathcal{A}\to L\{\tau\} into the twisted polynomial ring, where τ\tau is the qq-Frobenius satisfying τa=aqτ\tau a=a^q\tau, with constant term ι(a)\iota(a) and a growth condition governed by rr; it is the function-field analogue of an elliptic curve with multiplication by A\mathcal{A}. The Drinfeld modular curve x0(n)\mathbf{x}_0(\mathfrak{n}) parametrizes rank-two Drinfeld A\mathcal{A}-modules together with a cyclic n\mathfrak{n}-torsion subgroup (an isogeny level structure); normalized (Hayes) variants quotient by an action coming from the ideal class group. Bassa, Beelen, and Nguyen developed the general theory of good families of such modular curves, and over non-square fields the BBGS towers of Bassa-Beelen-Garcia-Stichtenoth, with modular interpretations by Chen-Chen-Hu and related work, give the best known lower bounds for Ihara(p2m+1)\mathrm{Ihara}(p^{2m+1}). Until 2025, however, the construction had been carried out end-to-end with explicit equations only for the polynomial ring Fq[t]\mathbb{F}_q[t], which has class number one and infinity of degree one.

Hu and Zhu, building on Hu and Huang's explicit rank-one and rank-two formulas for the same ring, treated the next case: A=Fq[x,y]/⟨y2−(ζ+ζq)xy+ζq+1x2−x⟩\mathcal{A}=\mathbb{F}_q[x,y]/\langle y^2-(\zeta+\zeta^q)xy+\zeta^{q+1}x^2-x\rangle with ζ∈Fq2∖Fq\zeta\in\mathbb{F}_{q^2}\setminus\mathbb{F}_q, the ring of functions on PFq1\mathbb{P}^1_{\mathbb{F}_q} regular outside the degree-two place PρP_\rho at the zeros of ρ(t)=(t−ζ)(t−ζq)\rho(t)=(t-\zeta)(t-\zeta^q); this ring has class number two. Its Hilbert class field is H=Fq(t,ζ)H=\mathbb{F}_q(t,\zeta) and its narrow class field is H+=Fq(t,ζ,ν)H^+=\mathbb{F}_q(t,\zeta,\nu) with νq+1=−1/((t−ζ)(tq−ζ))\nu^{q+1}=-1/((t-\zeta)(t^q-\zeta)); the ideal I∞=(x,y)I_\infty=(x,y), the degree-one place at t=∞t=\infty with I∞2=(x)I_\infty^2=(x), plays the role that a linear-polynomial ideal plays classically. Hu and Zhu first write closed-form defining equations for the towers x˙0(I∞k)\dot{\mathbf{x}}_0(I_\infty^k) (normalized) and x0(I∞k)\mathbf{x}_0(I_\infty^k) (minimal) over H+H^+, the technical heart being explicit formulas for the I∞kI_\infty^k-annihilator of a normalized Drinfeld module and for isogenies between Frobenius twists. They then reduce the minimal tower modulo a degree-two principal ideal: for η∈Fq2∖({ζ,ζq}∪Fq)\eta\in\mathbb{F}_{q^2}\setminus(\{\zeta,\zeta^q\}\cup\mathbb{F}_q) the element zη=(t−η)(t−ηq)/ρ(t)z_\eta=(t-\eta)(t-\eta^q)/\rho(t) generates a degree-two prime ideal IηI_\eta (a degree-two place of A\mathcal{A}, with residue field Fq2\mathbb{F}_{q^2}), and their Theorem C proves that the genus of each reduced curve equals

g(x0(I∞k)/Iη)=−1+qk−1(q+1)q−1−2q−1(q⌊k/2⌋+qk−⌊k/2⌋−1−1),g\Big(\mathbf{x}_0(I_\infty^k)/I_\eta\Big)=-1+\frac{q^{k-1}(q+1)}{q-1}-\frac{2}{q-1}\Big(q^{\lfloor k/2\rfloor}+q^{k-\lfloor k/2\rfloor-1}-1\Big),

that all supersingular points (points where the underlying Drinfeld module is supersingular, meaning its endomorphism ring is a maximal order in a division algebra; concretely a large explicitly bounded family) are Fq4\mathbb{F}_{q^4}-rational and number (q+1)2qk−1(q+1)^2q^{k-1}, and hence that the reduced tower over Fq4\mathbb{F}_{q^4} is asymptotically optimal with λ=q2−1\lambda=q^2-1. This is the first complete instance of the explicit Drinfeld-modular tower construction for a coefficient ring other than Fq[t]\mathbb{F}_q[t]. What it does not do defines the problem: the reduction analysis covers only the degree-two principal ideals that are prime. Since A\mathcal{A} is a Dedekind domain, for a product I=PaPbI=P_aP_b of two degree-one places of A\mathcal{A} the Artinian quotient A/I\mathcal{A}/I is isomorphic to Fq×Fq\mathbb{F}_q\times\mathbb{F}_q when a≠ba\ne b and to the dual-number ring Fq[u]/u2\mathbb{F}_q[u]/u^2 when a=ba=b.

Problem Statement

Let qq be a prime power, let ζ∈Fq2∖Fq\zeta\in\mathbb{F}_{q^2}\setminus\mathbb{F}_q, and let A=Fq[x,y]/⟨y2−(ζ+ζq)xy+ζq+1x2−x⟩\mathcal{A}=\mathbb{F}_q[x,y]/\langle y^2-(\zeta+\zeta^q)xy+\zeta^{q+1}x^2-x\rangle be the coordinate ring of PFq1\mathbb{P}^1_{\mathbb{F}_q} with the degree-two place PρP_\rho (zeros of ρ(t)=(t−ζ)(t−ζq)\rho(t)=(t-\zeta)(t-\zeta^q)) removed; let I∞=(x,y)I_\infty=(x,y) be its degree-one place at t=∞t=\infty. The problem left open by Hu-Zhu's construction of the explicit Drinfeld-modular towers x0(I∞k)\mathbf{x}_0(I_\infty^k) and their IηI_\eta-reductions is to complete the known ring's picture for the degree-two principal ideals their reduction theorem claims but does not treat.

The optimality theorem covers exactly the degree-two principal ideals that are prime (those generated by zη=(t−η)(t−ηq)/ρ(t)z_\eta=(t-\eta)(t-\eta^q)/\rho(t) for η∈Fq2∖({ζ,ζq}∪Fq)\eta\in\mathbb{F}_{q^2}\setminus(\{\zeta,\zeta^q\}\cup\mathbb{F}_q)), whereas the paper's abstract and Theorem C speak of an arbitrary non-zero principal A\mathcal{A}-ideal of degree two. For every degree-two principal ideal that is not prime --- equivalently, for every product I=PaPbI=P_aP_b of two degree-one places of A\mathcal{A}, where PaP_a and PbP_b range over all degree-one places of A\mathcal{A} (a complete target: the factors include the place I∞I_\infty at infinity as well as the places at t=at=a, and the ideals generated by (t−a)(t−b)/ρ(t)(t-a)(t-b)/\rho(t), with a=ba=b allowed, are only examples of the class, not the whole of it) --- determine the following: (i) whether an II-reduction of the towers x0(I∞k)\mathbf{x}_0(I_\infty^k) in the sense of Hu-Zhu's reduction framework exists; (ii) the genera of the resulting curves and the constant fields they are defined over; and (iii) whether any resulting tower is asymptotically optimal over some explicit finite field. All three parts are required, and either a uniform answer over the whole target class or a per-ideal answer for each such II resolves the problem; alternatively, a single proof that no extension of the reduction framework to such ideals exists resolves it negatively, so that the claimed generality fails.

References

  1. Chuangqiang Hu and Xiuwu Zhu, New optimal function field towers over finite fields of quartic power, arXiv:2510.27159 (2025). DOI: 10.48550/arXiv.2510.27159, https://arxiv.org/abs/2510.27159
  2. 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), 721-724. Zbl 0509.14019, https://zbmath.org/?q=an%3A0509.14019
  3. V. G. Drinfel'd and S. G. Vladut, Number of points of an algebraic curve, Functional Analysis and Its Applications 17 (1983), no. 1, 53-54. DOI: 10.1007/BF01083182, https://doi.org/10.1007/BF01083182 (Zbl 0522.14011)
  4. M. A. Tsfasman, S. G. Vladut, and Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound, Mathematische Nachrichten 109 (1982), 21-28. DOI: 10.1002/mana.19821090103, https://doi.org/10.1002/mana.19821090103
  5. Arnaldo Garcia and Henning Stichtenoth, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound, Inventiones Mathematicae 121 (1995), 211-222. DOI: 10.1007/BF01884295, https://doi.org/10.1007/BF01884295
  6. Noam D. Elkies, Explicit towers of Drinfeld modular curves, in: 3rd European Congress of Mathematics (Barcelona, 2000), Vol. II, Progress in Mathematics 202, Birkhauser, Basel, 2001, pp. 189-198. DOI: 10.1007/978-3-0348-8266-8_14, https://doi.org/10.1007/978-3-0348-8266-8_14 (Zbl 1031.11035)
  7. Ernst-Ulrich Gekeler, Asymptotically optimal towers of curves over finite fields, in: Algebra, Arithmetic and Geometry with Applications (papers from Shreeram S. Abhyankar's 70th birthday conference), Springer, 2004, pp. 325-336. DOI: 10.1007/978-3-642-18487-1_21, https://doi.org/10.1007/978-3-642-18487-1_21 (Zbl 1151.11326)
  8. Alp Bassa, Peter Beelen, Arnaldo Garcia, and 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 (arXiv:1202.5922)
  9. Alp Bassa and Peter Beelen, Explicit equations for Drinfeld modular towers, arXiv:1110.6076 (2011). https://arxiv.org/abs/1110.6076
  10. Alp Bassa and Peter Beelen, The Galois closure of Drinfeld modular towers, Journal of Number Theory 131 (2011), no. 3, 561-577. DOI: 10.1016/j.jnt.2010.10.006, https://doi.org/10.1016/j.jnt.2010.10.006 (Zbl 1219.11073)
  11. Alp Bassa, Peter Beelen, and Nhut Nguyen, Good families of Drinfeld modular curves, LMS Journal of Computation and Mathematics 18 (2015), no. 1, 699-712. DOI: 10.1112/s146115701500025x, https://doi.org/10.1112/s146115701500025x (arXiv:1608.07422)
  12. Rui Chen, Zhuo Chen, and Chuangqiang Hu, A modular interpretation of BBGS towers, Journal of Number Theory 221 (2021), 143-173. DOI: 10.1016/j.jnt.2020.06.006, https://doi.org/10.1016/j.jnt.2020.06.006 (arXiv:1912.02668)
  13. Chuangqiang Hu and Xiao-Min Huang, Drinfeld module and Weil pairing over Dedekind domain of class number two, Finite Fields and Their Applications 100 (2024), Paper No. 102516. DOI: 10.1016/j.ffa.2024.102516, https://doi.org/10.1016/j.ffa.2024.102516 (arXiv:2312.16919)
  14. Peter Beelen, A survey on recursive towers and Ihara's constant, arXiv:2203.03310 (2022). https://arxiv.org/abs/2203.03310
  15. Chuangqiang Hu, Xiao-Min Huang, and Stephen S.-T. Yau, Anderson generating function of rank-one Drinfeld Module over rational function fields, arXiv:2605.07484 (2026). https://arxiv.org/abs/2605.07484
  16. Chuangqiang Hu, Ze-Dong Huang, and Chang-An Zhao, Rank-Two Drinfeld Module over Elliptic Curves, arXiv:2606.26851 (2026). https://arxiv.org/abs/2606.26851
  17. Kohei Aoyama, Youhei Morita, and Yasuhiro Wakabayashi, Genus formulas for dormant modular curves and asymptotic behavior of their function fields, arXiv:2605.17973 (2026). https://arxiv.org/abs/2605.17973