From rank-two Drinfeld modules over the coordinate ring A=F2[S,T]/⟨S2+S+T3+T⟩A=\mathbb{F}_2[S,T]/\langle S^2+S+T^3+T\rangle of an elliptic curve with five rational points, Bassa, Beelen and Nguyen (2013) constructed an explicitly defined tower F=(Fn)\mathcal{F}=(\mathcal{F}_n) of function fields over F210\mathbb{F}_{2^{10}}, defined recursively by Φ(α8n,un,un+1)=0\Phi(\alpha^{8^n},u_n,u_{n+1})=0, and proved only that its limit satisfies λ(F)≥1\lambda(\mathcal{F})\ge 1 (via N1(Fn)≥13⋅2n−1N_1(\mathcal{F}_n)\ge 13\cdot 2^{n-1} and g(Fn)≤13⋅2n−1+1g(\mathcal{F}_n)\le 13\cdot 2^{n-1}+1). The Drinfeld–Vlăduţ bound gives λ(F)≤210−1=31\lambda(\mathcal{F})\le\sqrt{2^{10}}-1=31, and neither the source nor the later literature determines the exact value. The general question of which this is the emblematic instance is also open: for families of reductions of Drinfeld modular curves over Fq2de\mathbb{F}_{q^{2de}} at a non-principal AA-characteristic (order e>1e>1), the general theory of Bassa–Beelen–Nguyen (2015) establishes only the lower bound qd−1q^d-1 on the limit, and the true asymptotic behaviour of these families is unknown. The problem asks for the exact limit of this tower; the general question of the actual limits of Drinfeld-modular towers over more general base rings is posed in part (2) below as the broader setting.

Background

For an algebraic curve over a finite field Fq\mathbb{F}_q, the Hasse–Weil bound limits its number of Fq\mathbb{F}_q-rational points in terms of the genus. Asymptotically, one studies A(q)=lim sup⁡N1(X)/g(X)A(q)=\limsup N_1(X)/g(X) over curves of growing genus gg; Drinfeld and Vlăduţ proved A(q)≤q−1A(q)\le\sqrt{q}-1, and this is attained when qq is a square. A standard way to exhibit large A(q)A(q) is a tower of function fields: a chain F=(Fn)\mathcal{F}=(\mathcal{F}_n) of function fields over Fq\mathbb{F}_q with full constant field Fq\mathbb{F}_q, finite separable steps, and g(Fn)→∞g(\mathcal{F}_n)\to\infty, whose limit is

λ(F)=lim⁡n→∞N1(Fn)g(Fn),\lambda(\mathcal{F})=\lim_{n\to\infty}\frac{N_1(\mathcal{F}_n)}{g(\mathcal{F}_n)},

where N1N_1 counts rational places. A tower is good if λ(F)>0\lambda(\mathcal{F})>0 and optimal if λ(F)=q−1\lambda(\mathcal{F})=\sqrt{q}-1; then A(q)≥λ(F)A(q)\ge\lambda(\mathcal{F}). Garcia and Stichtenoth (1995) gave the first completely explicit optimal tower, and most explicit towers found since admit a modular explanation, due to Elkies: they arise as reductions of Drinfeld modular curves. A rank-two Drinfeld module over Fq[T]\mathbb{F}_q[T] is the function-field analogue of an elliptic curve: a ring homomorphism sending TT to an additive (Frobenius-twisted) polynomial such that the torsion points form a rank-two Fq[T]\mathbb{F}_q[T]-module. Isomorphism classes are classified by a jj-invariant, and pairs of isogenous Drinfeld modules are parametrized by Drinfeld modular curves X0(N)X_0(N), related by the Drinfeld modular polynomial ΦN(X,Y)\Phi_N(X,Y), the analogue of the classical modular polynomial: it vanishes on the jj-invariants of NN-isogenous pairs. Gekeler showed that reductions of the tower (X0(Pn))n(X_0(P^n))_n at a prime LL are asymptotically optimal over a quadratic extension of Fq[T]/⟨L⟩\mathbb{F}_q[T]/\langle L\rangle. Bassa, Beelen and Nguyen showed how the recursive defining equations of such towers are read off directly from ΦP\Phi_P, the polynomial ΦN\Phi_N itself being known explicitly only at low level (a closed form for ΦT\Phi_T is due to Bassa–Beelen, 2012). The framework generalizes: replace Fq[T]\mathbb{F}_q[T] by the ring AA of functions on a function field F/FqF/\mathbb{F}_q that are regular outside a fixed place ∞\infty of degree δ\delta, with prime ideals of AA playing the role of levels and of characteristics. In this generality Bassa–Beelen–Nguyen (2015) computed the genus of the components x0(n)x_0(\mathfrak{n}) of X0(n)X_0(\mathfrak{n}) and proved: for a reduction at an AA-characteristic PP of degree dd and order ee (the order of PP in the ideal class group of AA), the family (x0(nk))k(x_0(\mathfrak{n}_k))_k over Fq2de\mathbb{F}_{q^{2de}} satisfies lim⁡kN1/g≥qd−1\lim_k N_1/g\ge q^d-1, sharp when PP is principal; for non-principal PP only this lower bound is known. The emblematic worked example — the source of this problem — takes A=F2[S,T]/⟨S2+S+T3+T⟩A=\mathbb{F}_2[S,T]/\langle S^2+S+T^3+T\rangle, the coordinate ring of the elliptic curve y2+y=x3+xy^2+y=x^3+x over F2\mathbb{F}_2 (an affine model of S2+S=T3+TS^2+S=T^3+T), whose five rational points form a cyclic group, so δ=1\delta=1; in the characteristic P=⟨S,T⟩P=\langle S,T\rangle (degree d=1d=1, order e=5e=5: the ideal class group of AA is isomorphic to the group of rational points of the curve, which is cyclic of order five, and PP is a generator) the isomorphism classes of rank-two AA-Drinfeld modules form five components defined over F32\mathbb{F}_{32}, permuted transitively by the Frobenius of F32/F2\mathbb{F}_{32}/\mathbb{F}_2. Computing an explicit ⟨S+1,T+1⟩\langle S+1,T+1\rangle-isogeny yields a trivariate relation Φ(α,X,Y)\Phi(\alpha,X,Y) between the parameters of isogenous modules (Eq. (14) of the source, with α5+α2+1=0\alpha^5+\alpha^2+1=0), and iterating it with the Frobenius twists α↦α8n\alpha\mapsto\alpha^{8^n} defines the tower F\mathcal{F} over F210\mathbb{F}_{2^{10}} by F0=F210(u0)\mathcal{F}_0=\mathbb{F}_{2^{10}}(u_0) and Fn+1=Fn(un+1)\mathcal{F}_{n+1}=\mathcal{F}_n(u_{n+1}) with Φ(α8n,un,un+1)=0\Phi(\alpha^{8^n},u_n,u_{n+1})=0, taking the degree-two factor for n≥1n\ge 1. The first step has degree 33 and genus g(F1)=4g(\mathcal{F}_1)=4; all later steps are Artin–Schreier extensions (degree 22, wildly ramified in characteristic 22). The source proves N1(Fn)≥13⋅2n−1N_1(\mathcal{F}_n)\ge 13\cdot 2^{n-1} and g(Fn)≤13⋅2n−1+1g(\mathcal{F}_n)\le 13\cdot 2^{n-1}+1, hence λ(F)≥1\lambda(\mathcal{F})\ge 1; the genus bound uses only an estimate of the ramification (2-boundedness: every place QQ of Fn\mathcal{F}_n lying above a place PP satisfies d(Q∣P)=2e(Q∣P)−2d(Q|P)=2e(Q|P)-2, where dd is the different exponent and ee the ramification index), and no exact asymptotic — limit value, complete splitting locus, or exact different growth — is determined, by the source or by subsequent work.

Problem Statement

Let A=F2[S,T]/⟨S2+S+T3+T⟩A=\mathbb{F}_2[S,T]/\langle S^2+S+T^3+T\rangle, the coordinate ring of the elliptic curve over F2\mathbb{F}_2 with five rational points, with characteristic prime P=⟨S,T⟩P=\langle S,T\rangle (degree d=1d=1, order e=5e=5). Let F=(Fn)\mathcal{F}=(\mathcal{F}_n) be the tower over F210\mathbb{F}_{2^{10}} defined by Bassa–Beelen–Nguyen: F0=F210(u0)\mathcal{F}_0=\mathbb{F}_{2^{10}}(u_0), Fn+1=Fn(un+1)\mathcal{F}_{n+1}=\mathcal{F}_n(u_{n+1}) with Φ(α8n,un,un+1)=0\Phi(\alpha^{8^n},u_n,u_{n+1})=0, where Φ(α,X,Y)\Phi(\alpha,X,Y) is the trivariate relation of Eq. (14) of arXiv:1309.4951 (with α5+α2+1=0\alpha^5+\alpha^2+1=0); the first step Φ(α,u0,u1)=0\Phi(\alpha,u_0,u_1)=0 is irreducible of degree 33, while for n≥1n\ge 1 the polynomial Φ(α8n,un,T)\Phi(\alpha^{8^n},u_n,T) factors and the degree-two factor is taken. It is known that 1≤λ(F)≤210−1=311\le\lambda(\mathcal{F})\le\sqrt{2^{10}}-1=31; moreover, since PP is non-principal, Remark 4.3 of Bassa–Beelen–Nguyen (2015) asserts that the tower is asymptotically good but not optimal, i.e. λ(F)<31\lambda(\mathcal{F})<31, although no explicit upper bound below 3131 is proved.

(1) Determine the exact limit: prove that λ(F)=lim⁡nN1(Fn)/g(Fn)\lambda(\mathcal{F})=\lim_n N_1(\mathcal{F}_n)/g(\mathcal{F}_n) exists and compute its value. In particular, decide whether λ(F)=1\lambda(\mathcal{F})=1 or strictly larger.

(2) The general question of which (1) is the emblematic instance: for a function field F/FqF/\mathbb{F}_q with a fixed place ∞\infty of degree δ\delta, the ring AA of functions regular outside ∞\infty, and reduction at an AA-characteristic PP of degree dd and order e>1e>1 (non-principal), consider sequences of ideals (nk)k≥1(\mathfrak{n}_k)_{k\ge 1} of AA, coprime to PP, with deg⁡nk→∞\deg\mathfrak{n}_k\to\infty; Theorem 4.2 of Bassa–Beelen–Nguyen (2015) establishes, for every such sequence, the lower bound lim⁡kN1(x0(nk))/g(x0(nk))≥qd−1\lim_k N_1(x_0(\mathfrak{n}_k))/g(x_0(\mathfrak{n}_k))\ge q^d-1 for the family of components of Drinfeld modular curves over Fq2de\mathbb{F}_{q^{2de}}. Determine the actual limit for every such sequence: decide whether the limit exists, whether its value can depend on the chosen sequence, and whether the lower bound qd−1q^d-1 is attained by every admissible sequence or only by some; if the bound is not attained, characterize how large the limit can be.

Resolving (1) is by itself a complete resolution of the concrete problem about the tower F\mathcal{F}; a full resolution of the general question (2) for all admissible sequences is not required.

References

  1. Alp Bassa, Peter Beelen, Nhut Nguyen, Good Towers of Function Fields, in: Algebraic Curves and Finite Fields (H. Niederreiter, A. Ostafe, D. Panario, A. Winterhof, eds.), de Gruyter, 2014, pp. 23–40. DOI: 10.1515/9783110317916.23; arXiv:1309.4951; https://arxiv.org/abs/1309.4951
  2. Alp Bassa, Peter Beelen, Nhut Nguyen, Good families of Drinfeld modular curves, LMS Journal of Computation and Mathematics 18 (2015), no. 1, pp. 699–712. DOI: 10.1112/S146115701500025X; arXiv:1608.07422; https://arxiv.org/abs/1608.07422
  3. Peter Beelen, A survey on recursive towers and Ihara's constant, arXiv:2203.03310 (2022); https://arxiv.org/abs/2203.03310
  4. Alp Bassa, Peter Beelen, A closed-form expression for the Drinfeld modular polynomial ΦT(X,Y)\Phi_T(X,Y), Archiv der Mathematik 99 (2012), pp. 237–245. DOI: 10.1007/s00013-012-0423-x; arXiv:1203.6288; https://arxiv.org/abs/1203.6288
  5. Florian Breuer, Fabien Pazuki, Zhenlin Ran, Heights of Drinfeld modular polynomials and Hecke images, arXiv:2410.11132 (2024); https://arxiv.org/abs/2410.11132
  6. Florian Breuer, Mahefason Heriniaina Razafinjatovo, Drinfeld modular polynomials of level TT, arXiv:2412.11324 (2024); https://arxiv.org/abs/2412.11324
  7. Chuangqiang Hu, Xiuwu Zhu, New optimal function field towers over finite fields of quartic power, arXiv:2510.27159 (2025); https://arxiv.org/abs/2510.27159
  8. Arnaldo Garcia, Henning Stichtenoth, A tower of Artin–Schreier extensions of function fields attaining the Drinfeld–Vladut bound, Inventiones Mathematicae 121 (1995), pp. 211–222. DOI: 10.1007/BF01884295; https://doi.org/10.1007/BF01884295