Attainable spectra of Tsfasman–Vlăduţ invariants of towers of global fields: infinite supports, order-$r$ optimality over non-square finite fields, and minimal spectra

Hess, Stichtenoth and Tutdere (J. Algebra Appl. 2013) initiated the study of which spectra of Tsfasman–Vlăduţ invariants $\beta_r$ (asymptotic densities of degree-$r$ places per unit genus) are realized by towers of function fields over finite fields, and left open a precise cluster of problems. The square-field cases were settled by the source paper itself (recursive towers attaining the Drinfeld–Vladut bound of order $r$ whenever $q^r$ is a square) and by Ballet–Rolland ($r=2$, any $q$; and $q=2$, $r=4$); Lebacque (2015) constructed infinite global fields with prescribed finite positive and prescribed zero invariants in both the number-field and function-field settings; and Hajir–Maire–Ramakrishna and Hamza–Maire (2019–2021) produced number-field towers in which infinitely many primes split completely. Four questions remain open: (1) does any asymptotically exact tower of function fields over a finite field have infinitely many positive $\beta_r$; (2) does any asymptotically exact sequence over $\mathbb{F}_q$ attain the generalized Drinfeld–Vladut bound of order $r\ge 2$ when $q^r$ is not a square (a related existential form posed in the source asks whether $A(q^r)=rA_r(q)$ is ever true for non-square $q^r$; order-$r$ attainment implies this equality, but the converse need not hold); (3) do towers with a single positive invariant, or with at least one positive invariant and a prescribed zero invariant set, exist over the remaining non-square fields such as odd prime fields and $\mathbb{F}_8$; (4) do infinite number fields with all but one invariant zero exist.

Background

Let $\mathbb{F}_q$ be the finite field with $q$ elements. An algebraic function field $F/\mathbb{F}_q$ (a finitely generated extension of transcendence degree one whose full constant field is $\mathbb{F}_q$) carries a genus $g(F)$ — the genus of the corresponding algebraic curve — and a set of places (equivalence classes of discrete valuations, corresponding to closed points of the curve), each with a degree $\deg P$ equal to the degree of its residue field over $\mathbb{F}_q$. Write $B_r(F)$ for the number of places of $F$ of degree $r$, so $B_1(F)=N(F)$ counts $\mathbb{F}_q$-rational points.

A tower of function fields over $\mathbb{F}_q$ is a sequence $\mathcal{F}=(F_n)_{n\ge 0}$ with $F_n\subset F_{n+1}$ finite separable extensions, constant field $\mathbb{F}_q$ at every step, and $g(F_n)\to\infty$. Motivated by algebraic-geometry codes (the Tsfasman–Vlăduţ–Zink bound) and by asymptotic class-number questions, Tsfasman (1992) called such a sequence asymptotically exact if all limits

$\beta_r(\mathcal{F})=\lim_{n\to\infty}\frac{B_r(F_n)}{g(F_n)},\qquad r\ge 1,$

exist, and proved the generalized Drinfeld-Vladut bound (extending the 1983 bound of Drinfeld and Vlăduţ on rational points)

$\sum_{r\ge 1}\frac{r\,\beta_r}{q^{r/2}-1}\le 1.$

The case $r=1$ recovers $\beta_1\le\sqrt{q}-1$ for the Ihara constant $A(q)=\limsup N(F)/g(F)$; the extremal value $A(q)=\sqrt{q}-1$ holds when $q$ is a square (realized by the modular-curve towers of Tsfasman–Vlăduţ–Zink and by the explicit Artin–Schreier tower of Garcia–Stichtenoth), and the exact value of $A(q)$ remains unknown for every non-square $q$ (confirmed by the reviews of Anbar–Beelen 2017 and Chara–Navarro–Toledano 2023). Analogously one defines the generalized Ihara constants $A_r(q)=\limsup B_r(F)/g(F)$; a sequence attains the Drinfeld-Vladut bound of order $r$ if $\beta_r=(q^{r/2}-1)/r$, which, as Hess–Stichtenoth–Tutdere observe, forces $A(q^r)=q^{r/2}-1=rA_r(q)$. The deficiency $\delta=1-\sum_r r\beta_r/(q^{r/2}-1)$ measures failure of optimality and governs asymptotic class numbers through the generalized Brauer–Siegel theorem of Tsfasman–Vlăduţ (2002), which extends the framework to number fields: there $g(K)=\log\sqrt{|D_K|}$ with $D_K$ the absolute discriminant, $\Phi_\alpha(K)$ counts places of norm $\alpha$ (plus real and complex places), and an infinite number field (a tower $K_0\subset K_1\subset\cdots$) has invariants $\phi_\alpha=\lim\Phi_\alpha(K_n)/g(K_n)$ when all limits exist.

Hess–Stichtenoth–Tutdere (2013) developed a method, based on a finite explicit extension $E/F_0$ and the compositum tower $\mathcal{E}=E\cdot\mathcal{F}$, that constructs: towers with a prescribed finite set of positive invariants over any $\mathbb{F}_q$ (from any base tower of finite support); towers over square fields with at least one positive invariant and any prescribed set of invariants zero (using the Garcia–Stichtenoth base tower, whose only positive invariant is $\beta_1$); recursive towers attaining the order-$r$ bound whenever $q^r$ is a square (their Corollary 4.10, settling a problem of Ballet–Rolland in that case); and — via Möbius inversion between $\beta_r(\mathcal{F}/\mathbb{F}_q)$ and $\beta_1$ of the constant-field extensions $\mathcal{F}\mathbb{F}_{q^d}$, a technique from Beelen's graph analysis of recursive towers — non-optimal towers with all but one invariant zero over $\mathbb{F}_{2^e}$ with $3\nmid e$ (over $\mathbb{F}_2$ one has $\beta_3=1/2$), over $\mathbb{F}_9$ and $\mathbb{F}_{81}$, and over $\mathbb{F}_{p^e}$ for primes $p\ge 3$ and odd $e>2$. Ballet–Rolland had earlier built order-2 attaining sequences over every $\mathbb{F}_q$ (note $q^2$ is automatically a square) and one over $\mathbb{F}_2$ for $r=4$. On the number-field side, Hasegawa and Lebacque had used class field theory to realize finitely many prescribed positive invariants, and Lebacque posed the question of infinite number fields with all but one invariant zero; his 2015 paper later combined Schmidt's theory of $S$-ramified, $T$-split pro-$\ell$ extensions with an effective weak Grunwald–Wang theorem to construct infinite global fields (totally real number fields, and function fields over any constant field) with a prescribed finite set of positive invariants, a prescribed finite set of zero invariants, and all higher-norm splitting invariants $\phi_{\mathfrak{p},(N\mathfrak{p})^m}$ with $m\ge 2$ equal to zero — but with the degree-one support outside the prescribed sets left uncontrolled. The unresolved residue of this program is the problem stated below.

Problem Statement

The problem is the cluster of four questions left open by Hess–Stichtenoth–Tutdere (stated in their introduction, in Section 3, and at the end of Section 5). Their common objective is to determine which configurations of Tsfasman–Vlăduţ invariants are realized by towers of global fields.

Q1 (infinite spectra). Does there exist an asymptotically exact tower $\mathcal{F}=(F_n)$ of function fields over a finite field $\mathbb{F}_q$ whose set of positive invariants $\{r\ge 1:\beta_r(\mathcal{F})>0\}$ is infinite? No example is known for global fields of either characteristic.

Q2 (order-$r$ optimality over non-square fields). Let $r\ge 2$ and let $q$ be a prime power such that $q^r$ is not a square. Does there exist an asymptotically exact sequence of function fields over $\mathbb{F}_q$ attaining the Drinfeld–Vladut bound of order $r$, i.e. with $r\,\beta_r/(q^{r/2}-1)$ equal to $1$ in the generalized bound? The source also poses, at the end of Section 5, the related existential form: do there exist $r\ge 2$ and $q$ with $q^r$ non-square such that $A(q^r)=r\,A_r(q)$? (For $q^r$ a square the answer is yes by Corollary 4.10 of the source, and for $r=2$ with any $q$ by Ballet–Rolland — and $q^2$ is always a square — so the non-square cases are exactly the open ones; order-$r$ attainment over $\mathbb{F}_q$ forces $A(q^r)=q^{r/2}-1$, so a positive answer would exhibit an optimal sequence over the non-square field $\mathbb{F}_{q^r}$.)

Q3 (minimal spectra over the remaining non-square fields). For every non-square constant field not covered by the known examples — the odd prime fields $\mathbb{F}_p$ and fields such as $\mathbb{F}_8$ are instances of this class — does there exist a tower with a single positive invariant? A stronger form of this question asks whether there exists a tower with at least one positive invariant and a prescribed set of invariants equal to zero (the analogue for non-square $q$ of Corollary 3.12 of the source); this stronger form implies the single-invariant form.

Q4 (number-field analogue). Do there exist infinite number fields — asymptotically exact towers $K_0\subset K_1\subset\cdots$ of number fields with $[K_n:\mathbb{Q}]\to\infty$, so that all limits $\phi_\alpha$ exist — whose Tsfasman–Vlăduţ invariants $\phi_\alpha$ vanish for all but exactly one value of $\alpha$, the remaining invariant being positive (Lebacque's question)?

The four questions are independent sub-claims of this single objective: each can be settled on its own, in either direction (a proof of existence or a proof of non-existence).

References

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