CURVES OF EVERY GENUS WITH MANY POINTS, II: ASYMPTOTICALLY GOOD FAMILIES

NOAM D. ELKIES, EVERETT W. HOWE, ANDREW KRESCH, BJORN POONEN, JOSEPH L. WETHERELL, AND MICHAEL E. ZIEVE

Abstract. We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_{q} with the following property: for every integer g ≥ 0, there is a genus-g curve over 𝔽_{q} with at least c_{q}g rational points over 𝔽_{q}. Moreover, we show that there exists a positive constant d such that for every q we can choose c_{q} = d log q. We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over 𝔽_{q} that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (ℤ/nℤ)^{r} for some r > cg/n.

1. Introduction

Let q be a power of a prime p and let 𝔽_{q} be the field with q elements. We will study the function

N_{q}(g) := max{#C(𝔽_{q}) : C is a curve of genus g over 𝔽_{q}},

where by a curve over a field k we mean a smooth, projective, geometrically irreducible 1-dimensional variety over k. In particular, we will be interested in upper and lower bounds on the asymptotic quantities

A^{+}(q) := lim sup_{g→∞} N_{q}(g)/g and A^{−}(q) := lim inf_{g→∞} N_{q}(g)/g.

The quantity that we call A^{+}(q) is denoted A(q) in most current literature.

We begin by reviewing the known bounds on A^{+}(q). The upper bound A^{+}(q) ≤ 2\sqrt{q} follows from the well-known inequality

N_{q}(g) ≤ (2\sqrt{q})g + (q + 1),

which was proved by Hasse [9] (for the case g = 1) in the 1930s and by Weil [27, 28, 29, 30] (for general g) in the 1940s. Serre writes [22] that for a long time Weil’s inequality was considered “essentiellement optimale .” The situation changed in 1981, when Manin [17] (following Goppa) and Ihara [13] found bounds on N_{q}(g) that are significantly better than the Weil bound when g is large compared to q. Ihara’s idea was refined by Drinfel’d and Vlăduţ [2] to prove that, for fixed q, we have

(1)  N_{q}(g) ≤ (\sqrt{q} − 1 + o(1))g as g → ∞,

Date: 29 April 2003. 2000 Mathematics Subject Classification. Primary 11G20; Secondary 14G05, 14G15. The authors thank Jordan Ellenberg, Yasutaka Ihara, and Felipe Voloch for helpful conversations. N. E.

and B. P. were supported in part by Packard Fellowships; in addition, B. P. was supported by NSF Grant DMS-9801104 and a Sloan Fellowship. A. K., J. W., and M. Z. were supported in part by NSF Mathematical Sciences Postdoctoral Research Fellowships.

which shows that A^{+}(q) ≤ \sqrt{q} − 1.

The problem of finding lower bounds for A^{+}(q) leads to the problem of producing curves over 𝔽_{q} with many points. Serre [21, 23] used class field towers to construct infinite sequences of such curves; in this way, he showed that A^{+}(q) > 0 for all q. In fact he proved that A^{+}(q) > c log q for some c > 0 not depending on q, and his proof gave better bounds for nonprime q. Ihara [8, 10, 12, 13] used supersingular points on Shimura curves to show that if q is a square we have A^{+}(q) ≥ \sqrt{q} − 1; together with the Drinfel’d-Vlăduţ bound (1), this proves that A^{+}(q) = \sqrt{q} − 1 for every square q. Several other authors proved further results along similar lines. (For a survey, see the introduction to [15].) But qualitatively the situation today is similar to that in 1983:

• There is no nonsquare q for which one knows whether or not A^{+}(q) = \sqrt{q} − 1.

• For square q, every known sequence {C_{i}} of curves over 𝔽_{q} whose genera g_{i} tend to infinity and for which #C_{i}(𝔽_{q})/g_{i} → \sqrt{q} − 1 has the property that C_{i} is modular (of elliptic, Shimura, or Drinfel’d types) for all sufficiently large i (see [3, 4]).

In this paper we consider the problem of finding lower bounds for A^{−}(q). To show that A^{−}(q) is greater than some constant c, one must show that for every sufficiently large g there is a curve of genus g over 𝔽_{q} with more than cg points. Unfortunately, the class field tower and modular curve constructions mentioned above miss infinitely many genera [1], so they do not lead immediately to lower bounds for A^{−}(q). In fact, before now there was no q for which it was known whether A^{−}(q) > 0; furthermore, even the much weaker assertion lim_{g→∞} N_{q}(g) = ∞ was proved only recently [15].

The main result of this paper gives an affirmative answer to a 1983 question of Serre [22]:

Theorem 1.1. For every q, we have A^{−}(q) > 0.

We also discuss quantitative refinements of this result. For instance, at the end of Section 3.2 we show that there is a constant d > 0 such that N_{q}(g) ≥ (d log q)g. In particular, we find that A^{−}(q) ≥ d log q for all q. This can be strengthened for square q:

Theorem 1.2. We have

A^{−}(q) ≥ \begin{cases}\frac{\sqrt{q} − 1}{2 + (log 2/ log q)} & if q is an even square; \\ \frac{\sqrt{q} − 1}{2 + (log 4/ log q)} & if q is an odd square. \end{cases}

For odd square q < 417, Corollary 6.2 in Section 6 gives an improved lower bound.

Theorem 1.1 shows that for every sufficiently large integer g there exist curves of genus g with many points. Theorem 1.3 below shows that we can simultaneously force the rational points of the Jacobian to contain certain groups. Suppose C is a curve over a field k and n > 1 is an integer. We define the n-rank r_{n}(C) of C to be the largest integer r such that the group (ℤ/nℤ)^{r} can be embedded in the group of k-rational points of the Jacobian of C.

Theorem 1.3. There exists a constant c > 0 such that for all integers n > 1 and prime powers q the following statement holds: For all sufficiently large integers g, there is a genus-g curve C over 𝔽_{q} with r_{n}(C) > (c/n)g and #C(𝔽_{q}) > (c/n)g.

Our proofs of these results involve covers of curves. A cover of a curve C over a field k is another curve B over k together with a nonconstant separable map (a covering map) B → C over k. Our strategy for proving Theorems 1.1 and 1.2 is to begin with some sequence of curves with many points that achieve “enough” genera, and then to fill in the missing genera via degree-2 covers. More specifically, we start with a sequence of curves over 𝔽_{q} that have many points and whose genera grow at most exponentially. Then we show that for every curve C in this sequence, and for every integer h greater than some constant multiple of the genus of C, there exists a degree-2 cover B → C over 𝔽_{q} such that B has genus h. Either B or its quadratic twist B′ will have at least as many 𝔽_{q}-rational points as C, because #B(𝔽_{q}) + #B′(𝔽_{q}) = 2#C(𝔽_{q}). To produce the sequence of curves we use class field towers and Shimura curves (see Section 3). To produce the degree-2 covers, we use the following result of independent interest:

Proposition 1.4. Let C be a curve of genus g over a finite field k. For every integer h ≥ 4g, there exists a genus-h curve B over k that admits a degree-2 covering map B → C defined over k.

In the next section we prove Proposition 1.4, as well as a generalization of the proposition to curves over arbitrary fields. We prove Theorem 1.1 and a weak version of Theorem 1.2 in Sections 3.2 and 3.3, respectively. In Section 4 we prove Theorem 1.3 and in Section 5 we prove the full version of Theorem 1.2. We end in Section 6 by discussing limitations on the results obtainable by our methods. Our proof of Theorem 1.1 relies on Serre’s proof [23] that A^{+}(q) > 0 for all q; since Serre’s proof has not been published, we include a version of it as an appendix.

2. Degree-2 covers

In this section we prove Proposition 1.4 and give an analogous result for curves over infinite fields.

2.1. Finite fields. Suppose C is a curve over 𝔽_{q}. Let g(C) denote the genus of C, and let 𝔽_{q}(C) denote the function field of C. For every positive integer d, let n_{d}(C) denote the number of places of C of degree d. We begin with a lemma that gives a lower bound on n_{d}(C) for large enough d.

Lemma 2.1. Let C be a curve over 𝔽_{q} of genus g.

(i) For every d > 0 we have n_{d}(C) > (q^{d} − (6g + 3)q^{d/2})/d.

(ii) If g > 1 and d > 2g we have n_{d}(C) > 0.

(iii) If q is odd and d > 2g then n_{d}(C) ≥ 2^{2g}, with equality if and only if q = d = 3, g = 1,

and #C(𝔽_{3}) = 6. (iv) Assume g ≥ 900. If j is a positive integer and d ≥ log_{q}(j log j + 1) + \sqrt{g}, then

n_{d}(C) > j.

Proof. (i) Clearly #C(𝔽_{q^{d}}) = ∑_{m|d} m · n_{m}(C), so by Möbius inversion we find that

(2)  dn_{d}(C) = ∑_{m|d} μ(d/m)#C(𝔽_{q^{m}}).

Combining the Weil lower bound on #C(𝔽_{q^{d}}) with the Weil upper bounds on the various #C(𝔽_{q^{m}}), we find that

dn_{d}(C) ≥ q^{d} + 1 − 2gq^{d/2} − ∑_{{m|d \atop 1≤m<d}} (q^{m} + 1 + 2gq^{m/2}) .

Using the fact that q + q^{2} + ⋯ + q^{⌊d/2⌋} < 2q^{d/2}, we see that

∑_{{m|d \atop 1≤m<d}} (q^{m} + 1 + 2gq^{m/2}) < ∑_{1≤m≤d/2} ((2g + 1)q^{m} + 1) < (4g + 2)q^{d/2} + d/2.

Thus dn_{d}(C) > q^{d} − (6g + 2)q^{d/2} − d/2, and (i) follows.

(ii) Suppose g ≥ 2 and d ≥ 2g + 1. If q > 2 then q^{d/2} ≥ q^{g+1/2} > 6g + 3, so by (i) we have n_{d}(C) > 0. Suppose q = 2. If g ≥ 5 or if d ≥ g + 6 then once again we have 2^{d/2} > 6g + 3, and (i) shows that n_{d}(C) > 0. We are left to consider six triples (q, d, g). For each one, we can show that n_{d}(C) > 0 by combining equation (2) with the refined Weil bound |#C(𝔽_{q})−q −1| ≤ g⌊2\sqrt{q}⌋ (see [21, Théorème 1]) and the trivial bound #C(𝔽_{q}) ≥ 0.

(iii) We assume g ≥ 1, since (iii) is well-known when g = 0 (it amounts to the existence of irreducible polynomials over 𝔽_{q} of arbitrary positive degree). If q ≥ 5 then (iii) follows from (i), so assume q = 3. In this case (iii) again follows from (i) except when (g, d) is one of (1, 3), (1, 4), (2, 5). When (g, d) is either (1, 4) or (2, 5) then (iii) follows from (2) and the Weil bounds. When (g, d) = (1, 3), the statement can be proved by considering the possible zeta functions of elliptic curves over 𝔽_{3}.

(iv) Define f(x) = 2^{\sqrt{x}/2}/2 − (6x + 3). A short computation shows that f(900) is positive and that f(x) is increasing for x ≥ 900. We conclude that (6g + 3) ≤ 2^{\sqrt{g}/2}/2 ≤ q^{d/2}/2; applying (i), we find that n_{d}(C) ≥ q^{d}/2d.

Note that d ≥ 30 and that q^{d}/2d is an increasing function of d for d in this range. When j = 1, this shows that q^{d}/2d ≥ 2^{30}/60 > j. When j ≥ 2, we find that

\frac{q^{d}}{2d} ≥ \frac{(j log j + 1)q^{30}}{2 log_{q}(j log j + 1) + 60} ≥ \frac{2^{30}j log j}{4 log_{2} j + 60} ≥ \frac{2^{30}j log j}{64 log_{2} j} > j.

In both cases, n_{d}(C) > j, as desired. □

Proof of Proposition 1.4. The assertion is clear when g = 0, so we assume g ≥ 1.

First suppose q is even. If #C(𝔽_{q}) > 0 then let R be a rational point of C. Set m = 2h − 4g + 1; by Riemann-Roch, there exists a function f in 𝔽_{q}(C) having polar divisor mR. Let B → C be the cover of C obtained by adjoining a root of X^{2} + X + f to 𝔽_{q}(C). This cover has degree two; its different is (m + 1)R′, where R′ is the point of B lying over R. By Hurwitz, the genus of B is h, and we are done. We are left with the case where #C(𝔽_{q}) = 0. Since every genus-1 curve over a finite field has a rational point, we must have g ≥ 2. Then Lemma 2.1(ii) guarantees the existence of a place P of degree d := h − 2g + 1. By Riemann-Roch, there exists f ∈ 𝔽_{q}(C) with polar divisor P. If we form B as before, the different of the cover B → C is 2P′, where P′ is the place of B lying above P. Again the genus of B is h, and we are done.

Now suppose q is odd. Let J(𝔽_{q}) be the group of degree-zero divisor classes on C (or equivalently, the group of 𝔽_{q}-rational points on the Jacobian of C). Let d = h − 2g + 1, and let T_{d} denote the set of places on C of degree d. Consider the map T_{d} → J(𝔽_{q})/2J(𝔽_{q}) defined by P ↦ [P − P_{0}], where P_{0} is any fixed degree-d divisor on C. Suppose that #T_{d} > #(J(𝔽_{q})/2J(𝔽_{q})). Then there are distinct places P, P′ ∈ T_{d} with the same image in J(𝔽_{q})/2J(𝔽_{q}), so there is a function f on C having divisor P − P′ + 2D for some divisor D. Adjoining \sqrt{f} to the function field of C produces a degree-2 cover B → C over 𝔽_{q}, with the genus of B equal to 2g + d − 1 = h. Thus we need only show that #T_{d} > #(J(𝔽_{q})/2J(𝔽_{q})). But #(J(𝔽_{q})/2J(𝔽_{q})) is equal to the number of 2-torsion points in J(𝔽_{q}), and this is at most 2^{2g}. Lemma 2.1(iii) supplies the needed inequality so long as we are not in the exceptional case where q = d = 3, g = 1, and #C(𝔽_{q}) = 6. But in that case #J(𝔽_{q}) = 6, so #(J(𝔽_{q})/2J(𝔽_{q})) = 2 and the result follows. □

Remark. The argument given above for even q can be modified to show that every curve over a finite field of characteristic 3 has Artin-Schreier covers of every sufficiently large genus. However, the analogous statement in characteristic p > 3 is not true; the degree of the different of an Artin-Schreier cover B → C is divisible by p − 1, so for any such cover we have g(B) ≡ g(C) mod (p − 1)/2.

2.2. Arbitrary fields. Although we will not need it for our main results, we now discuss a generalization of Proposition 1.4 to arbitrary fields. The index of a curve C is the positive integer I such that the image of the degree map Div(C) → ℤ equals Iℤ; equivalently, I is the greatest common divisor of the degrees of all places of C. A classical result of F. K. Schmidt asserts that curves over finite fields have index 1 (this follows from Lemma 2.1(ii); see [26, Cor. V.1.11] for a more elementary proof). Over some infinite fields, higher indices are possible; for instance, a conic over ℚ with no rational points has index 2.

Proposition 2.2. Let C be a curve of genus g over a field k and let I be its index. If B → C is a degree-2 cover of C of genus h, then h ≡ 1 (mod I). Conversely, if h is an integer satisfying h ≥ 4g and h ≡ 1 (mod I), then there exists a degree-2 cover B → C of genus h.

Proof. The index divides 2g − 2, because 2g − 2 is the degree of the canonical divisor. If the characteristic of k is not 2, then k(B) ≃ k(C)(\sqrt{f}) for some f ∈ k(C)^{∗}, and the total degree of the set of points where f has odd valuation must be a multiple of 2I, since the divisor of f has degree zero; now the Hurwitz formula implies h ≡ 1 (mod I). If k has characteristic 2, we have k(B) ≃ k(C)(z), where z satisfies z^{2} + z = f for some f ∈ k(C). A local calculation shows that each point of B occurs with even multiplicity in the different, so the degree of the different is divisible by 2I, and Hurwitz again implies h ≡ 1 (mod I).

Now we prove the converse assertion. Proposition 1.4 handles the case where k is finite, so we assume k is infinite. Let d = h − 2g + 1. Then d ≡ 2 − 2g ≡ 0 (mod I), so we can choose a divisor D on C of degree d. Since d ≥ 2g + 1, the divisor D is very ample, and therefore determines an embedding C → ℙ^{d−g} whose image has degree d. The composition of this embedding with a generically chosen linear projection to ℙ^{1} is a degree-d map f: C → ℙ^{1}. The linear projection may be chosen so as not to kill a chosen tangent vector at some point of C, so we may assume the map f induces a separable extension of function fields. By composing f with an automorphism of ℙ^{1}, we may further assume that 0 and ∞ are not branch points of f. Now let B be the cover obtained by adjoining to k(C) a root y of y^{2} + y = f or y^{2} = f, according as the characteristic of k equals 2 or not. The different of B → C has degree 2d in each case, so by Hurwitz we have g(B) = 2g − 1 + d = h. □

3. Asymptotic lower bounds

In this section we prove Theorem 1.1 and a weakened version of Theorem 1.2. (The full version of Theorem 1.2 will be proved in Section 5.2.) As we noted in the introduction, Serre used class field towers to exhibit a positive lower bound on A^{+}(q) for every q, and Ihara used Shimura curves to produce a better lower bound for square q. We will briefly recall these constructions, and include some necessary complements. Our theorems will follow from these constructions and Proposition 1.4.

3.1. Preliminaries. We begin by relating Proposition 1.4 to rational points.

Proposition 3.1. Let C be a curve over 𝔽_{q} of genus g. For every integer h ≥ 4g, we have N_{q}(h) ≥ #C(𝔽_{q}).

Proof. By Proposition 1.4, there is a genus-h curve B over 𝔽_{q} that admits a degree-2 covering map B → C over 𝔽_{q}. The result follows, since either B or its quadratic twist over C has at least as many 𝔽_{q}-rational points as does C. □

Corollary 3.2. If h ≥ 4g then N_{q}(h) ≥ N_{q}(g). □

Now we give a general technique for producing lower bounds for A^{−}(q). We will call a sequence C = {C_{0}, C_{1}, … } of curves over 𝔽_{q} ascensive if lim_{i→∞} g(C_{i}) = ∞. If C is an ascensive sequence of curves, define

γ(C) := lim inf_{i→∞} \frac{#C_{i}(𝔽_{q})}{g(C_{i+1})}.

Note that we divide the number of points of each curve in the sequence by the genus of the next curve. It is not hard to show that 0 ≤ γ(C) ≤ A^{+}(q).

Proposition 3.3. If C = {C_{0}, C_{1}, … } is an ascensive sequence of curves over 𝔽_{q}, then A^{−}(q) ≥ γ(C)/4.

Proof. For any h ≥ 4g(C_{0}), let i be the largest integer for which h ≥ 4g(C_{i}). Proposition 3.1 implies that N_{q}(h) ≥ #C_{i}(𝔽_{q}); thus,

\frac{N_{q}(h)}{h} ≥ \frac{#C_{i}(𝔽_{q})}{4g(C_{i+1})},

and the desired result follows. □

3.2. Class field towers. In this section we prove Theorem 1.1 using Proposition 3.3. To do this, we must produce sequences C with γ(C) > 0. We accomplish this using class field towers.

Let F be the function field of a curve over 𝔽_{q}. Let S be a nonempty set of rational places of F, and let ℓ be a prime number. Let \overline{F} be an algebraic closure of F. Then the (S, ℓ)- class field tower of F is the sequence F = {F_{0}, F_{1}, … } of function fields over 𝔽_{q} defined inductively as follows. We start with F_{0} = F. Then for every integer i ≥ 0 we let F_{i+1} be the maximal unramified abelian extension of F_{i} in \overline{F}, of degree a power of ℓ, in which every place in S splits completely into rational places. The tower is called infinite if each extension F_{i+1}/F_{i} is nontrivial.

The following result demonstrates the utility of curves whose function fields have infinite (S, ℓ)-class field towers.

Lemma 3.4. Let C be a curve over 𝔽_{q} of genus g(C) > 1, let S be a nonempty set of rational places of 𝔽_{q}(C), and let ℓ be a prime. Suppose the (S, ℓ)-class field tower of 𝔽_{q}(C) is infinite. Then there exists an ascensive sequence C of curves over 𝔽_{q} such that

γ(C) ≥ \frac{#S}{g(C) − 1} · \frac{1}{ℓ}.

Proof. Let F = {F_{0}, F_{1}, … } be the (S, ℓ)-class field tower of 𝔽_{q}(C). Since each extension F_{i+1}/F_{i} is Galois of ℓ-power degree, there is a refinement of the tower of fields

F_{0} ⊂ F_{1} ⊂ F_{2} ⊂ ⋯

to a tower of degree-ℓ field extensions

F_{0} = F̂_{0} ⊂ F̂_{1} ⊂ F̂_{2} ⊂ ⋯

such that each F_{i} occurs as some F̂_{j}. Clearly F̂_{n} is an unramified extension of F_{0} in which every place in S splits completely. Thus, the genus of F̂_{n} is 1+ℓ^{n}(g(C)−1), and the number of rational places of F̂_{n} is at least ℓ^{n} · #S. Let C_{n} be a curve over 𝔽_{q} such that 𝔽_{q}(C_{n}) ≅ F̂_{n} and let C = {C_{0}, C_{1}, …}. Then we have

\frac{#C_{n}(𝔽_{q})}{g(C_{n+1})} ≥ \frac{ℓ^{n} · #S}{1 + ℓ^{n+1}(g(C) − 1)},

so that C is an ascensive sequence satisfying the conclusion of the lemma. □

Theorem 1.1 follows from Lemma 3.4 and Proposition 3.3. Indeed, for every fixed q, Serre has constructed a curve C over 𝔽_{q} of genus g > 1 and a nonempty set S of rational places of 𝔽_{q}(C), such that the (S, 2)-class field tower of C is infinite. (The appendix contains a version of Serre’s construction.)

Remark. In fact, Serre’s construction yields C and S with #S > c_{1}(log q)^{2} and g(C) < c_{2} log q, for some positive constants c_{1} and c_{2}. Let d = min(c_{1}/(8c_{2}), 1/(2c_{2})). By considering hyperelliptic curves when g ≤ 4c_{2} log q, and degree-2 covers of the curves from Serre’s tower when g > 4c_{2} log q, we conclude that N_{q}(g) > (d log q)g for all q and g. This justifies the claim we made in the introduction, just before the statement of Theorem 1.2.

For some nonprime q, Serre’s proof yields better bounds (as does the method of Zink [31] based on degenerate Shimura surfaces). If q is a square, one gets even better results by using Shimura curves (among other methods — see [5, 6, 7]), as we explain in the following section.

3.3. Classical modular curves and Shimura curves. In this section we will prove the following weakened version of Theorem 1.2:

Theorem 3.5. For every square q, we have A^{−}(q) ≥ (\sqrt{q} − 1)/4.

Proof. We first sketch a proof in the special case q = p^{2} using the classical modular curves X_{0}(ℓ). If q = p^{2}, let C = {X_{0}(ℓ_{1}), X_{0}(ℓ_{2}), X_{0}(ℓ_{3}), … }, where ℓ_{1} < ℓ_{2} < … are the primes distinct from p. Then, writing g_{i} = g(X_{0}(ℓ_{i})) and N_{i} = #X_{0}(ℓ_{i})(𝔽_{q}), we have

N_{i} ≥ (p − 1)(ℓ_{i} + 1)/12 and g_{i} = \begin{cases}(ℓ_{i} − 13)/12 & if ℓ_{i} ≡ 1 (mod 12) \\ ⌊(ℓ_{i} + 1)/12⌋ & otherwise, \end{cases}

where the formula for g_{i} is classical, and the bound on N_{i} holds because all supersingular points on X_{0}(ℓ_{i}) are defined over 𝔽_{p^{2}} and their number is at least (p − 1)(ℓ_{i} + 1)/12. We compute

γ(C) = lim inf_{i→∞} \frac{N_{i}}{g_{i+1}} ≥ lim inf_{i→∞} \frac{(p − 1)(ℓ_{i} + 1)}{ℓ_{i+1} + 1} = p − 1.

(In fact, using the Drinfel’d-Vlăduţ bound one can show that γ(C) = p − 1.) Now Proposition 3.3 implies that A^{−}(p^{2}) ≥ (p − 1)/4. This proves Theorem 3.5 for the case q = p^{2}.

A different proof for the case q = p^{2} can be given using the Igusa-Ihara field of modular functions of level n over 𝔽_{p^{2}}, where n ranges over the positive integers coprime to p. (This field is the function field of a twist of X(n) defined over 𝔽_{p^{2}} such that all its supersingular points are defined over 𝔽_{p^{2}}: an open subset of this twist parameterizes elliptic curves E with a pairing-respecting isomorphism from E[n] to the Galois module (ℤ/nℤ)^{2} on which the p^{2}-Frobenius automorphism acts as multiplication by −p.) The relevant facts about these fields were announced in [8] and proved in [10, pp. 166–170] (see also [11]).

For more general square fields, we use Shimura curves. Suppose q = p^{2m} where p is prime and m ≥ 1. Let F be a totally real number field that has a prime 𝔭 whose norm is p^{m}. Let F_{𝔭} be the completion of F at 𝔭 and let 𝔬 denote the ring of integers of F. Let B be a quaternion algebra over F whose ramification locus contains all but one of the infinite places of F and does not contain 𝔭; standard results of class field theory show that there are infinitely many such B. As is explained in Ihara’s survey article [14], associated to the data (F, 𝔭, B) there is a family of Shimura curves with many points over 𝔽_{q} — see [25], [12], and [18] for further details. We will show that this family contains enough curves for us to find a subfamily C, indexed by the positive integers, for which γ(C) = \sqrt{q} − 1.

Let N_{B/F} denote the reduced norm map of the quaternion algebra B and let 𝔒 be an arbitrary 𝔬-order in B. Let 𝔬[𝔭^{−1}] denote the ring obtained by adjoining to 𝔬 all of the elements of the fractional ideal 𝔭^{−1}, and let 𝔒[𝔭^{−1}] = 𝔒⊗_{𝔬}𝔬[𝔭^{−1}]. We define the quaternionic arithmetic group

Γ := { α ∈ 𝔒[𝔭^{−1}] : N_{B/F} (α) = 1}/{±1},

which can be embedded in PSL_{2}(ℝ)×PSL_{2}(F_{𝔭}) as a discrete subgroup. The group Γ contains a torsion-free subgroup Γ_{0} of finite index. Associated to Γ_{0} there is a curve X_{0} over 𝔽_{q} and a set S_{0} of 𝔽_{q}-places of X_{0} such that #S_{0} ≥ (\sqrt{q} − 1)(g_{0} − 1), where g_{0} is the genus of X_{0}. To every finite-index congruence subgroup Γ_{1} ⊂ Γ_{0} there is an associated unramified cover X_{1} → X_{0} of degree (Γ_{0} : Γ_{1}), defined over 𝔽_{q}, such that all the places of S_{0} split completely. Consequently, to obtain a family of 𝔽_{q}-curves C with γ(C) = \sqrt{q} − 1 it suffices to show that Γ_{0} has a sequence of finite-index congruence subgroups such that the index d_{i} of the ith subgroup tends to infinity and such that lim_{i→∞} d_{i+1}/d_{i} = 1. We will construct such a sequence.

Let p_{1} < p_{2} < ⋯ be the sequence of primes that lie below degree-1 primes 𝔭_{i} of F. Chebotarev’s density theorem shows that the sequence {p_{i}} is infinite and has positive density in the sequence of prime numbers; in particular, the sequence (p_{i+1} + 1)/(p_{i} + 1) approaches 1 as i approaches infinity. Throwing away finitely many primes, we may suppose, for each i, that p_{i} ≠ p and that 𝔒 ⊗_{𝔬} (𝔬/𝔭_{i}) is a central simple algebra. By Wedderburn’s theorem, we may choose an isomorphism 𝔒[𝔭^{−1}] ⊗_{𝔬} (𝔬/𝔭_{i}) ≅ M_{2}(𝔽_{p_{i}}). Let U(p_{i}) ⊂ PSL_{2}(𝔽_{p_{i}}) denote the subgroup of upper-triangular matrices and define Γ_{0}(𝔭_{i}) to be the intersection of Γ_{0} with the subgroup

\{ α ∈ 𝔒[𝔭^{−1}] : N_{B/F} (α) = 1 and \overline{α} ∈ U(p_{i}) \}/\{±1\}

of Γ, where \overline{α} denotes the image of α in M_{2}(𝔽_{p_{i}}) under the chosen isomorphism.

We claim that the index d_{i} := [Γ_{0} : Γ_{0}(𝔭_{i})] is equal to p_{i} + 1 when i is sufficiently large. To see this, note that for each i the map α ↦ \overline{α} is a homomorphism τ_{i}: Γ → PSL_{2}(𝔽_{p_{i}}). A strong approximation theorem [24, (5.12)] applies in this case, and as a consequence τ_{i} is surjective for every i. When p_{i} > 3 the group PSL_{2}(𝔽_{p_{i}}) is simple with order divisible by p_{i} and hence has no subgroups of index less than p_{i}. Thus, for i sufficiently large, the restriction of τ_{i} to Γ_{0} is surjective as well. It follows that the index d_{i} is equal to p_{i} + 1, as claimed.

The sequence d_{i+1}/d_{i} approaches 1. As we noted above, this is enough to prove Theorem 3.5 for the field 𝔽_{q}. □

4. Curves with many points and large n-rank

In our proof of Theorem 1.1 we started with an ascensive sequence of curves with many points and used double covers to produce curves of every sufficiently large genus with many points. In this section we will use the same basic method to prove Theorem 1.3: We will start with an ascensive sequence of curves with many points and large n-rank, and use double covers to produce curves with the same properties in every sufficiently large genus.

Let us begin by introducing some notation. Suppose C = {C_{0}, C_{1}, …} is an ascensive sequence of curves over 𝔽_{q}. We define ρ_{n}(C) (for every integer n > 1) and β(C) by

ρ_{n}(C) := lim inf_{i→∞} \frac{r_{n}(C_{i})}{g(C_{i+1})} and β(C) := lim inf_{i→∞} \frac{g(C_{i})}{g(C_{i+1})}.

Note that in each expression we are dividing an invariant of one curve in the sequence by the genus of the next.

We will prove two lemmas. The first shows how we can obtain ascensive sequences having positive values of ρ_{n}.

Lemma 4.1. Suppose C is an ascensive sequence over 𝔽_{q}. Then for every integer n > 1 there is an ascensive sequence C_{n} over 𝔽_{q} such that

γ(C_{n}) ≥ \frac{min(γ(C), β(C))}{7n}  and  ρ_{n}(C_{n}) ≥ \frac{min(γ(C), β(C))}{7n}.

The second lemma shows how we can get curves with many points and large n-rank from the sequences produced by Lemma 4.1.

Lemma 4.2. Suppose C is an ascensive sequence over 𝔽_{q}. Then for every ε > 0 the following statement holds: For every sufficiently large h there is a curve B of genus h such that

#B(𝔽_{q}) > (1 − ε)γ(C)h/4  and  r_{n}(B) > (1 − ε)ρ_{n}(C)h/4.

Our refined versions of Theorem 1.1 show that there is a constant c such that for every q there are ascensive sequences C over 𝔽_{q} with β(C) = 1 and γ(C) > c. To prove Theorem 1.3 we need merely apply Lemma 4.1 to such a C and then apply Lemma 4.2 to the resulting sequences C_{n}.

Proof of Lemma 4.1. We may assume that C = {C_{0}, C_{1}, …} is an ascensive sequence over 𝔽_{q} with γ(C) > 0 and β(C) > 0. By eliminating a finite number of terms from C, if necessary, we may assume that g(C_{i}) > 1 and #C_{i}(𝔽_{q}) > 0 for all i. Now suppose we are given a positive integer n. Our goal is to define a sequence C_{n} = {B_{0}, B_{1}, …} satisfying the conclusions of the lemma. For every i, we define B_{i} as follows.

Let g = g(C_{i}), let d = min(#C_{i}(𝔽_{q}), g), and let S be a subset of C_{i}(𝔽_{q}) with #S = d. Let Q be a place of C_{i} of degree 2g +1; such a place exists by Lemma 2.1(ii). The Riemann-Roch theorem, applied to the divisor Q, shows that there is a function f_{Q} ∈ 𝔽_{q}(C_{i}) whose polar divisor is Q. For every P ∈ S, Riemann-Roch applied to Q and P + Q shows that there is a function f_{P} ∈ 𝔽_{q}(C_{i}) with simple poles at P and Q and no other poles. Let

f := ∑_{P ∈S} f_{P} ,

and if f has no pole at Q then replace f with f +f_{Q}. Then f is a function of degree d+2g+1 with a simple pole at every P ∈ S.

Write n = p^{e}r where p is the characteristic of 𝔽_{q} and (r, q) = 1. Let B′ be the curve obtained by adjoining a root of y^{r} = f to 𝔽_{q}(C_{i}). If e = 0, let B_{i} = B′; otherwise let B_{i} be the curve obtained by adjoining a root of z^{p^{e}} − z = f to 𝔽_{q}(B′). We note several facts about the curve B_{i}: First, if we apply the Hurwitz formula to the covers B′ → C_{i} and B_{i} → B′ (the formulas from [26, Cor. III.7.4] and [26, Prop. III.7.10(e)] are helpful here) and apply some easy estimates that depend on the fact that d ≤ g, we find that g(B_{i}) < 7ng. Second, the cover B_{i} → C_{i} is totally ramified at every point in S, so #B_{i}(𝔽_{q}) ≥ d. Third, we have r_{n}(B_{i}) ≥ d − 2, as the following argument shows.

Let π be the natural map from B′ to C_{i}. Let G denote the group of degree-zero divisors of B′ supported on S′ := π^{−1}(S), and let G_{0} be the subgroup of G consisting of those degreezero divisors D with ord_{P} D ≡ ord_{P′} D (mod r) for every P, P′ ∈ S′. Note that G is a free abelian group of rank d − 1, that G_{0} ⊇ rG, and that G_{0}/rG is naturally isomorphic to a subgroup of ℤ/rℤ.

We claim that the kernel of the natural map

G → (Jac B′)(𝔽_{q})/π^{∗}((Jac C_{i})(𝔽_{q}))

contains rG and is contained in G_{0}. Clearly rG is in the kernel, since if D ∈ G we have rD = π^{∗}π_{∗}D. On the other hand, suppose that D ∈ G is in the kernel. Then D = (h) + π^{∗}E for some h ∈ 𝔽_{q}(B′) and divisor E on C_{i}. The function h is preserved up to scalar multiple by the action of Gal(B′/C_{i}) (by which we mean the Galois group of the covering of curves over \overline{𝔽}_{q}), and writing h in terms of the basis 1, y, …, y^{r−1} shows that h ∈ y^{j}𝔽_{q}(C_{i})^{∗} for some j. It follows that the multiplicities of the points of S′ in D are all congruent to each other modulo r, so D is contained in G_{0}. This proves the claim.

It follows from the previous paragraph that the image of G in the quotient group

(Jac B′)(𝔽_{q})/π^{∗}((Jac C_{i})(𝔽_{q}))

contains (ℤ/rℤ)^{d−2}. Group theory then shows that (Jac B′)(𝔽_{q}) also contains (ℤ/rℤ)^{d−2}. But since the natural map B_{i} → B′ has degree coprime to r, the kernel of the natural map Jac B′ → Jac B_{i} has order coprime to r, so (Jac B_{i})(𝔽_{q}) also contains a copy of (ℤ/rℤ)^{d−2}.

The same proof shows that (Jac B_{i})(𝔽_{q}) contains (ℤ/p^{e}ℤ)^{d−2}. (The only difference in the argument is that now when the action of Gal(B_{i}/B′) on the analogous h is by scalar multiples, the scalars must be p^{e}-th roots of 1, hence equal to 1, so h comes from 𝔽_{q}(B′).) Combining this with the previous paragraph and applying the Chinese Remainder Theorem, we find r_{n}(B_{i}) ≥ d − 2, as we had noted above.

Combining the several facts we have noted about the curve B_{i}, we find that for every i ≥ 0 we have

\frac{#B_{i}(𝔽_{q})}{g(B_{i+1})} > \frac{min(#C_{i}(𝔽_{q}), g(C_{i}))}{7ng(C_{i+1})}  and  \frac{r_{n}(B_{i})}{g(B_{i+1})} > \frac{−2 + min(#C_{i}(𝔽_{q}), g(C_{i}))}{7ng(C_{i+1})}.

It follows that

γ(C_{n}) ≥ \frac{min(γ(C), β(C))}{7n} and ρ_{n}(C_{n}) ≥ \frac{min(γ(C), β(C))}{7n}.

□

The proof of the second lemma is almost identical to the proof of Proposition 3.3.

Proof of Lemma 4.2. Suppose we are given an ascensive sequence C = {C_{0}, C_{1}, …} and an ε > 0 as in the statement of the lemma. We may certainly assume that ε < 1. Then there is an integer j such that for all i > j we have

#C_{i}(𝔽_{q}) ≥ (1 − ε)γ(C)g(C_{i+1}) and r_{n}(C_{i}) ≥ (1 − ε)ρ_{n}(C)g(C_{i+1}).

Now suppose we are given an integer h ≥ 4g(C_{j+1}). Let i be the largest integer such that h ≥ 4g(C_{i}). Proposition 1.4 shows that there is a curve B of genus h that is a double cover of C_{i}. By replacing B with its quadratic twist over C_{i}, if necessary, we may assume that #B(𝔽_{q}) ≥ #C_{i}(𝔽_{q}). Also, the Hurwitz formula shows that the double cover B → C_{i} must be totally ramified at some point, so class field theory for curves shows that the natural map (Jac B)(𝔽_{q}) → (Jac C_{i})(𝔽_{q}) is surjective. It follows that r_{n}(B) ≥ r_{n}(C_{i}). Since we also have h < 4g(C_{i+1}), we find that

#B(𝔽_{q}) ≥ #C_{i}(𝔽_{q}) ≥ (1 − ε)γ(C)g(C_{i+1}) > (1 − ε)γ(C)h/4.

Similarly, we have

r_{n}(B) ≥ r_{n}(C_{i}) ≥ (1 − ε)ρ_{n}(C)g(C_{i+1}) > (1 − ε)ρ_{n}(C)h/4,

as we were to show. □

Remark. The reader who objects to the intrusion of class field theory into the proof of Lemma 4.2 can remove it at the expense of slightly weakening the lemma. Consider the double cover π: B → C_{i} introduced at the end of the proof. We have π_{∗}π^{∗} = 2 on Jac C_{i}, so r_{n}(B) ≥ r_{2n}(C_{i}). Thus, without class field theory, we can show that for every sufficiently large h there is a curve B of genus h such that #B(𝔽_{q}) > (1 − ε)γ(C)h/4 and r_{n}(B) > (1 − ε)ρ_{2n}(C)h/4.

Theorem 1.3 has the following consequence for class field towers:

Proposition 4.3. Fix a prime power q and a prime ℓ. For g ≫ 0, there exists a genus-g curve C over 𝔽_{q} with P ∈ C(𝔽_{q}) such that the ({P }, ℓ)-class field tower of 𝔽_{q}(C) is infinite.

Proof. Suppose g is at least 5ℓ/c, where c is the constant that appears in Theorem 1.3. Then Theorem 1.3 shows that there is a genus-g curve C over 𝔽_{q} whose n-rank is greater than 5 and that has at least 5 rational points. Pick P ∈ C(𝔽_{q}) and embed C in its Jacobian J using P as the base point. Let F ∈ End J be the q-power Frobenius endomorphism and let D be the inverse image of C under (1 − F): J → J. Class field theory for curves shows that D is an irreducible curve and that D → C is a unramified Galois covering with Galois group J(𝔽_{q}), and clearly P splits completely in this covering. By Galois theory, there exists a subcovering E → C with Galois group (ℤ/ℓℤ)^{r}, where r = r_{n}(C) > 5. The function field analogue of the Golod-Shafarevich criterion (the lemma in our appendix, with S = {P }) implies that the ({P }, ℓ)-class field tower of 𝔽_{q}(C) is infinite. □

5. Quantitative refinements

Our arguments in Section 3 required asymptotic results about special sequences of curves, but the tool we applied to these results — Proposition 1.4 — makes no use of any special properties these curves might have. Indeed, the proposition makes reference only to the genera of the curves it mentions. In this section we will show that we can produce better bounds on A^{−}(q) by replacing Proposition 1.4 with sharper asymptotic results. At the end of this section we prove Theorem 1.2.

5.1. Bounding data. Let C = {C_{0}, C_{1}, …} be an ascensive sequence of curves over 𝔽_{q} and let H and M be real numbers. We will say that the pair (H, M) is bounding data for C (over 𝔽_{q}) if for every ε > 0 there is an L_{ε} such that following statement holds:

If i > L_{ε} and h is an integer with h > (H + ε)g(C_{i}), then there exists a genus-h curve B over 𝔽_{q} such that #B(𝔽_{q}) ≥ (M − ε)#C_{i}(𝔽_{q}).

Bounding data for a sequence C can be used to give a lower bound on A^{−}(C), as the following proposition shows.

Proposition 5.1. If (H, M) is bounding data for an ascensive sequence C over 𝔽_{q}, then A^{−}(q) ≥ γ(C) · M/H.

Proof. This is a generalization of Proposition 3.3, and the proof is essentially identical to the proof of that result. Indeed, Proposition 3.3 amounts to the observation that (4, 1) is bounding data for every ascensive C, which follows from Proposition 1.4. □

5.2. Improved bounds on A^{−}(q) for square q. In this section we will prove Theorem 1.2 by showing that every ascensive sequence over 𝔽_{q} has bounding data of the form (H, 1) for an appropriate value of H. To show that (H, 1) is bounding data for C, we must show that for every ε > 0, for every sufficiently large i, if h > (H + ε)g(C_{i}) then there is a genus-h curve B with #B(𝔽_{q}) ≥ (1 − ε)#C_{i}(𝔽_{q}). In this section we will restrict ourselves to the case where B is a degree-2 cover of C_{i}. This restriction entails that h ≥ 2g(C_{i}) − 1, so this line of argument cannot possibly work for values of H less than 2.

If C = {C_{0}, C_{1}, …} is an ascensive sequence over 𝔽_{q}, define

R_{2}(C) := lim sup_{i→∞} \frac{r_{2}(C_{i})}{g(C_{i})},

where, as before, r_{2}(C_{i}) denotes the 2-rank of (Jac C_{i})(𝔽_{q}). (Note that we are dividing the 2-rank of each curve in the sequence by its own genus, and not the genus of the next curve!) Also define

H_{C} := 2 + R_{2}(C)\frac{log 2}{log q}.

Proposition 5.2. If C is an ascensive sequence over 𝔽_{q} then (H_{C}, 1) is bounding data for C.

Proof. Our proof depends on the parity of q. First suppose that q is odd. Let C be a genus-g curve over 𝔽_{q}. Let J be the Jacobian of C. Let j = #(J(𝔽_{q})/2J(𝔽_{q})). Lemma 2.1(iv) shows that if g ≥ 900 and d ≥ log_{q}(j log j +1)+\sqrt{g}, then n_{d}(C) > j. Since log j ≤ log(2^{2g}) = O(g), the condition on d can be rewritten as

d ≥ log_{q} j + o(g) = r_{2}(C)\frac{log 2}{log q} + o(g)

as g → ∞. Proposition 5.2 (for odd q) follows by applying the argument in the final paragraph of the proof of Proposition 1.4.

Now suppose q is even. Fix a curve C/𝔽_{q} of genus g and let d be an arbitrary integer with d ≥ d(C), where

d(C) := max (r_{2}(C)\frac{log 2}{log q}, \frac{2 log(6g + 3)}{log q}) .

We see from Lemma 2.1(i) that C has a place of degree d, and it follows from Lemma 5.3 (below) that there exists a degree-2 cover B → C with g(B) = 2g + d − 1. As C ranges through the curves in C, we have

lim sup \frac{d(C)}{g(C)} = R_{2}(C)\frac{log 2}{log q}.
□

Lemma 5.3. Let q be a power of 2, and let C be a curve over 𝔽_{q}. Suppose C has a place of degree d, where q^{d} ≥ #J[2](𝔽_{q}). Then there exists a degree-2 cover B → C such that g(B) = 2g(C) + d − 1.

Proof. Let 𝔭 be a place of degree d on C and let P be a geometric point of C lying in 𝔭, so that 𝔭 consists of the d conjugates of P. Let u be a uniformizing parameter at P. For every integer m ≥ 0 let S_{m} be the additive group of functions in 𝔽_{q}(C) that have no poles outside 𝔭 and whose polar expansions at P are of the form

c_{m}u^{−2^{m}} + c_{m−1}u^{−2^{m−1}} + ⋯ + c_{1}u^{−2} + c_{0}u^{−1},

where the c_{i} are elements of \overline{𝔽}_{q} and where c_{m} is not necessarily nonzero. A standard Riemann-Roch argument shows that S_{m} contains nonconstant functions when m ≫ 0, and in fact it is not hard to see that #S_{m} = q^{d}#S_{m−1} for m ≫ 0. For m > 0 let T_{m} ⊆ S_{m} be the subgroup

T_{m} := {g^{2} + g + c : g ∈ S_{m−1}, c ∈ 𝔽_{q}}.

Then #T_{m} = #S_{m−1}, so for m ≫ 0 we have #(S_{m}/T_{m}) = q^{d}.

Artin-Schreier theory shows the following:

(1) If f ∈ S_{m} \T_{m} then the degree-2 cover of C obtained by adjoining a root of y^{2} −y −f to 𝔽_{q}(C) is geometrically irreducible and is ramified at most at 𝔭.

(2) If f, g ∈ S_{m} \ T_{m} then the degree-2 covers obtained as in (1) from f and g are geometrically isomorphic (as curves equipped with their maps to C) if and only if f − g ∈ T_{m}.

Thus, from the set S_{m} we obtain −1 + #(S_{m}/T_{m}) geometrically nonisomorphic irreducible degree-2 covers of C. By taking m large enough, we find that there are at least q^{d} − 1 such covers. But the number of geometrically distinct unramified degree-2 covers of C is equal to #J[2](𝔽_{q}) − 1, so there must be a ramified degree-2 cover B → C over 𝔽_{q} coming from S_{m}, and it must be ramified exactly at 𝔭. The Hurwitz formula then shows that g(B) = 2g(C) + d − 1. □

Proof of Theorem 1.2. We showed in Section 3.3 that for every square q there exists an ascensive sequence C of curves over 𝔽_{q} with γ(C) = \sqrt{q} − 1. In light of Propositions 5.1 and 5.2, to prove Theorem 1.2 it will be enough for us to show that

R_{2}(C) ≤ \begin{cases}1 & when q is even; \\ 2 & when q is odd. \end{cases}

But these inequalities follow from the fact that for every curve C over 𝔽_{q} we have

r_{2}(C) ≤ \begin{cases}g(C) & when q is even; \\ 2g(C) & when q is odd. \end{cases}

□

6. Variants of our argument

Section 3 gave a lower bound on A^{−}(q) in terms of the value of γ(C) of an ascensive sequence. Section 5.1 gave an improved bound taking into account also the size of q, but the full power of Proposition 5.2 was not used, because we used only the trivial upper bound on R_{2}(C). It would be extremely interesting to obtain better bounds on R_{2}(C) when C is one of the sequences of modular curves or Shimura curves used in Section 3.3. Ideally, one would like to show that R_{2}(C) = 0 for such an ascensive sequence, so that (2, 1) would be bounding data for C. Theorem 1.3 shows, however, that R_{2}(C) can be positive, even for sequences of curves with many points. Thus, proving that R_{2}(C) = 0 for a sequence of Shimura curves would require using special properties of the curves.

It is possible to improve the 4g in Proposition 1.4 if we require only that the proposition apply to curves of sufficiently large genus. Indeed, the proof of Proposition 5.2 shows that we can replace the 4g with (2 + (log 2/ log q))g if q is even and with (2 + (log 4/ log q))g if q is odd. But the following argument limits the possible further improvements we might hope to obtain: Given m and q, there exists a curve C over 𝔽_{q} such that every place has degree at least m. (For instance, nonsingular plane curves with this property exist [19].) It then follows from Proposition 1.4 that such C exist in every sufficiently large genus. An argument similar to that proving Proposition 2.2 shows that such C do not admit degree-2 covers of any genus h < 2g + 2m − 1 except possibly for unramified degree-2 covers of genus h = 2g − 1. In particular, we cannot improve the 4g in Proposition 1.4 to 2g + s for any fixed constant s, even if we ask only that the proposition hold for g ≫ 0. We do not know whether there always exists a cover of genus 2g + o(g).

It is conceivable that A^{−}(q) = \sqrt{q} −1. In order to prove this using our methods, we would need an ascensive sequence C with γ(C) = \sqrt{q} − 1 for which we could provide bounding data (H, M) with H/M = 1. But the smallest ratio H/M that we can possibly attain using degree-2 covers B → C is 2, unless we can make M > 1 by forcing more than half of the points in C(𝔽_{q}) to split. For q < 207, it is in fact possible to show in this way that every ascensive sequence over 𝔽_{q} has bounding data (H, M) with H/M < 2; plugging this into Proposition 5.1 improves the lower bounds on A^{−}(q). We give the argument below, beginning with the following variant of Proposition 1.4.

Proposition 6.1. Let q be an odd prime power and let m = (log 2/ log q)(\sqrt{q} + 1). Fix a positive real ε. Then for all g ≫ 0, if C/𝔽_{q} is a curve of genus g, and if h is an integer with h ≥ (2 + m + ε)g, then there exists a genus-h curve B/𝔽_{q} that admits a degree-2 covering map B → C in which every rational place of C splits.

Proof. Let d = h − 2g + 1 > (m + ε)g. Thus d > 1 when g ≫ 0. As in the proof of Proposition 1.4, let T_{d} denote the set of places on C of degree d. By Lemma 2.1(i), we have #T_{d} ≥ q^{d}/(2d) when g ≫ 0. Thus g ≫ 0 implies #T_{d} ≥ q^{(m+ε/2)g}. Consider the map T_{d} → J(𝔽_{q})/2J(𝔽_{q}) defined by P ↦ [P − P_{0}], where P_{0} is any fixed degree-d divisor on C. Since #J(𝔽_{q})/2J(𝔽_{q}) ≤ 2^{2g}, there exists x ∈ J(𝔽_{q})/2J(𝔽_{q}) having at least q^{(m+ε/2)g}/2^{2g} preimages in T_{d}. Fix one of these preimages P_{1}. Then for each preimage P of x, fix f_{P} ∈ 𝔽_{q}(C) having divisor P − P_{1} + 2D for some D.

Let K = 𝔽_{q}(C). For each Q ∈ C(𝔽_{q}), let K_{Q} be the completion of K at Q, and let O_{Q} ⊂ K_{Q} be the valuation ring. Since d > 1, f_{P} has even valuation at each Q ∈ C(𝔽_{q}). Hence the f_{P} map into the group

S := ∏_{Q∈C(𝔽_{q})} \frac{O_{Q}^{∗}K_{Q}^{∗2}}{K_{Q}^{∗2}}

of order 2^{#C(𝔽_{q})}. The Drinfel’d-Vlăduţ bound gives #C(𝔽_{q}) < (\sqrt{q} − 1 + ε/2)g for g ≫ 0, but then the definition of m implies

\frac{q^{(m+ε/2)g}}{2^{2g}} > 2^{(\sqrt{q}−1+ε/2)g} > #S,

so there exist distinct P, P′ ∈ T_{d} in the preimage of x such that f_{P} and f_{P′} have the same image in S. Let e = f_{P} /f_{P′}.

Let B be the curve with 𝔽_{q}(B) ≃ 𝔽_{q}(C)(\sqrt{e}). The divisor of e has the form P − P′ + 2D with P, P′ ∈ T_{d}, so Hurwitz shows that B has genus h. By construction, e ∈ K_{Q}^{∗2} for all Q ∈ C(𝔽_{q}), so every rational point of C splits in the degree-2 cover B → C. □

Corollary 6.2. Let q be an odd prime power and let m = (log 2/ log q)(\sqrt{q} + 1). Let C be an ascensive sequence over 𝔽_{q}. Then A^{−}(q) ≥ γ(C)/(1 + m/2).

Proof. Proposition 6.1 shows that C has bounding data (2+m, 2). Now apply Proposition 5.1. □

We have 1+m/2 < 2 for all odd prime powers q < 207, and 1+m/2 < 2+log 4/ log q (the trivial upper bound on H_{C}) for all odd prime powers q < 417. We showed in Section 3.3 that, for every square q, there are ascensive sequences C over 𝔽_{q} with γ(C) = \sqrt{q}−1. Thus, for odd square q we have A^{−}(q) ≥ (\sqrt{q} − 1)/(1 + m/2), which improves the bound in Theorem 1.2 if q < 417. For example, if q = 9 then 1 + m/2 < 1.631, and the ascensive sequence {X_{0}(ℓ)} over 𝔽_{9} gives A^{−}(9) > A^{+}(9)/1.631 = (\sqrt{9} − 1)/1.631 > 1.226.

Appendix: class field towers

In this appendix we present a version of Serre’s proof that A^{+}(q) > 0 for all q. Previously Serre’s proof has appeared only in the unpublished lecture notes [23]. The version below uses an idea from [16] at one step to simplify the argument.

As in Section 3.2, it suffices to show that for every q there is a curve C over 𝔽_{q} of genus g > 1 such that for some nonempty set S of rational places on 𝔽_{q}(C), the (S, 2)-class field tower of 𝔽_{q}(C) is infinite. To do this, we use a function field analogue of the Golod-Shafarevich criterion:

Lemma. Let C be a curve over 𝔽_{q}, let ℓ be a prime number, and let S be a nonempty set of rational places of 𝔽_{q}(C). Suppose there is an unramified Galois extension F/𝔽_{q}(C), with Galois group (ℤ/ℓℤ)^{r}, such that every place in S splits completely into 𝔽_{q}-rational places of F. If r ≥ 2 and

\frac{(r − 2)^{2}}{4} ≥ \begin{cases}1 + #S & if ℓ | (q − 1); \\ #S & otherwise, \end{cases}

then the (S, ℓ)-class field tower of 𝔽_{q}(C) is infinite.

Proof. See [20]. □

Theorem (Serre). We have A^{+}(q) > 0 for all prime powers q. In fact, there exists c > 0 such that A^{+}(q) > c log q for all q.

Proof. First suppose q is odd. Let f = f_{1}f_{2} … f_{6}, where f_{1}, f_{2}, …, f_{6} are distinct monic irreducible polynomials in 𝔽_{q}[x] having even degrees. Then 𝔽_{q}(x, \sqrt{f_{1}}, …, \sqrt{f_{6}}) is an unramified Galois extension of 𝔽_{q}(x, \sqrt{f}), with Galois group (ℤ/2ℤ)^{5}, in which both places of 𝔽_{q}(x, \sqrt{f}) which contain 1/x are completely split. It follows that A^{+}(q) > 0.

For odd q it remains only to prove the second statement for q ≫ 0. Let α_{1}, …, α_{r+1} be distinct elements of 𝔽_{q}, and let f(x) = (x − α_{1}) … (x − α_{r+1}). Suppose r is even. Then F := 𝔽_{q}(\sqrt{x − α_{1}}, …, \sqrt{x − α_{r+1}}) is an unramified Galois extension of 𝔽_{q}(x, \sqrt{f}), with Galois group (ℤ/2ℤ)^{r}. By Hurwitz, 𝔽_{q}(x, \sqrt{f}) has genus g := r/2 and F has genus 1 + 2^{r}(r/2 − 1). By the Weil lower bound, the number of rational places N on F satisfies

N ≥ q + 1 − 2\sqrt{q} · 2^{r}(r/2 − 1).

Exactly N/2^{r} rational places of 𝔽_{q}(x, \sqrt{f}) split completely in F. Choose r as the even integer nearest (log_{2} q)/3. For q ≫ 0, we have r ≥ 4 and N/2^{r} ≥ (r − 2)^{2}/4 − 1 ≥ 0, so it is possible to choose a subset S of these rational places with #S = ⌊(r − 2)^{2}/4 − 1⌋. By the lemma, the (S, 2)-class field tower is infinite, so a proof similar to that of Lemma 3.4 shows that

A^{+}(q) ≥ \frac{#S}{g − 1} = \frac{r}{2} + O(1) = c′ log q + O(1)

as q → ∞, for some c′ > 0.

The case of even q can be treated in a completely analogous manner, using Artin-Schreier covers. □

The proof of the theorem shows that there exists c > 0 such that for every q, there exists a curve C over 𝔽_{q} with g(C) > 1 and a set S of rational places on 𝔽_{q}(C) such that #S > (c log q)·(g(C)−1) and such that the (S, 2)-class field tower of 𝔽_{q}(C) is infinite. As in Section 3.2, it follows that for every q we have A^{−}(q) ≥ (c/4) log q. We do not know whether this bound can be improved significantly when q is prime.

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Department of Mathematics, Harvard University, Cambridge, MA 02138–2901

E-mail address: elkies@math.harvard.edu

Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121–1967

E-mail address: however@alumni.caltech.edu

Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104–6395

E-mail address: kresch@math.upenn.edu

Department of Mathematics, University of California, Berkeley, CA 94720–3840

E-mail address: poonen@math.berkeley.edu

Department of Mathematics, University of Southern California, Los Angeles, CA 90089– 1113

Current address: Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121–1967

E-mail address: jlwether@alum.mit.edu

Center for Communications Research, 805 Bunn Drive, Princeton, NJ 08540–1966

E-mail address: zieve@idaccr.org