Counting the number of curves over a finite field

by A.J. de Jong, Massachusetts Institute of Technology and N.M. Katz, Princeton University

1 Introduction.

Let 𝔽_{q} be a finite field having q elements and let M_{g} be the algebraic stack classifying smooth proper curves of genus g ≥ 2. We would like to know the number of points of M_{g} over 𝔽_{q}. Recall that for a stack this number is defined as

#M_{g}(𝔽_{q}) = ∑ \frac{1}{#Aut(C)},

where the sum is over the isomorphism classes of smooth projective genus g curves C over 𝔽_{q} and Aut(C) denotes the automorphism group of C over 𝔽_{q}.

To give a precise formula for this number seems out of reach at the moment. On the other hand it is known that for g fixed and q → ∞ one has #M_{g}(𝔽_{q}) ∼ q^{3g−3}. However, Nick Katz asked [1] what the behaviour is of #M_{g}(𝔽_{q}) when q is fixed and g grows.

In this note we show that there exist positive constants c_{1}, c_{2} such that

#M_{g}(𝔽_{q}) ≤ q^{c_{1}g}e^{c_{2}g log(g)}.

The constants c_{1}, c_{2}, … that appear in this paper are positive absolute constants. We give two proofs of this result. The first method bounds the number of isomorphism classes of degree d morpisms to ℙ^{1} with a fixed discriminant. The main term in the resulting bound comes from a Hurewitz number. The second method uses the Lefschetz fixed point formula for Frobenius and proceeds by estimating the betti numbers of M_{g} ⊗ ℂ.

It is perhaps interesting to note that neither method can be used to obtain better bounds (see 3.1 and 5.2). Thus the question arises whether a better bound exists. For example, does there exist a function φ(q) and a constant c_{3} > 0 such that

#M_{g}(𝔽_{q}) ≤ φ(q)q^{c_{3}g}?

2 Easy bounds.

In this section we give a quick sketch of the first thing one tries.

Let X be a smooth projective curve of genus g over 𝔽_{q}. There exists an invertible O_{X}-module L on X of degree g + 1. By Riemann-Roch we have dim Γ(X, L) ≥ 2. Taking two linearly independent sections gives a morphism

f : X ⟶ ℙ1_{𝔽_{q}}

of degree ≤ g + 1 (the degree may drop if the sections have common zeros). If f is not generically étale, then we may factor f as X⟶^{f′}ℙ^{1} → ℙ^{1}, where the second map is the geometric Frobenius morphism of ℙ^{1} (of degree p). We replace f by f′ and we observe that the degree has decreased. Continue untill f is generically étale. By the Riemann Hypothesis for X, there exists a closed point x ∈ X of degree at most

deg(x/𝔽_{q}) ≤ 2 log_{q}(g) + 5

such that f is étale at x. (Use that the number of ramification points of f is at most 4g.) By Riemann-Roch applied to Γ(X, O_{X}(nx)) for suitable n, we obtain a morphism

h : X ⟶ ℙ1_{𝔽_{q}}

with h(x) = 0 and h^{−1}({0}) = {x} (i.e. totally ramified at x) of degree

deg(h) ≤ g + 2 log_{q}(g) + 5.

We claim that the morphism (f, h) : X → ℙ1_{𝔽_{q}} × ℙ1_{𝔽_{q}} is birational onto its image. Indeed, let y ∈ ℙ1_{𝔽_{q}} × ℙ1_{𝔽_{q}} be the image of x under (f, h). By construction, we have (f, h)^{−1}({y}) = {x} and κ(y) ≅ κ(x). This implies the claim.

We conclude that the number of isomorphism classes of curves of genus g is at most the number of curves C ⊂ ℙ1_{𝔽_{q}} × ℙ1_{𝔽_{q}} of bidegree ≤ (g + 1, g + 2 log_{q}(g) + 5). The number of curves of bidegree (a, b) in ℙ1_{𝔽_{q}} × ℙ1_{𝔽_{q}} is q^{(a+1)(b+1)} − 1/q − 1. We conclude that there exists a constant c_{4} > 0 such that

#M_{g}(𝔽_{q}) ≤ g^{2g+2}q^{g^{2}}e^{c_{4}g}.

The main term is q^{g^{2}} which is too large as we shall see later.

3 Tame ramification.

Let X be a smooth projective curve of genus g over 𝔽_{q}. Given a nonconstant morphism f : X → ℙ1_{𝔽_{q}} of degree d we get a discriminant divisor

D = discr(f) ⊂ ℙ1_{𝔽_{q}}

of degree deg(D) = 2g − 2 + 2d. Since it is easy to count the number of possible D we try to count the number of isomorphism classes of f’s with given D.

Over ℂ this is given by a Hurewitz number. We define

H(g, d) = #{f : X → ℙ1_{ℂ} | discr(f) = D}/isomorphism.

Here D ⊂ ℙ1_{ℂ} is any fixed divisor consisting of 2g − 2 + 2d distinct points. We remark that

H(g, d) ≤ \frac{1}{d!}\binom{d}{2}^{2g−2+2d}.

We use here that H(g, d) is equal to the number of ways the free group F ⟨γ_{1}, …, γ_{2g−2+2d}⟩ can act transitively on an (unordered) set of d elements such that each γ_{i} acts as a transposition τ_{i} and such that τ_{1} ⋯ τ_{2g−2+2d} = 1. This clearly gives the estimate above. On the other hand, by choosing

τ_{1} = τ_{2} = (12), τ_{3} = τ_{4} = (13), τ_{5} = τ_{6} = (14), …, τ_{2d−3} = τ_{2d−2} = (1d),

and τ_{2i−1} = τ_{2i} freely for g − 1 + d ≥ i ≥ d we see that

(3.1)  H(g, d) ≥ \binom{d}{2}^{g}.

It follows from the deformation theory of curves that for any divisor D ⊂ ℙ1_{ℂ} of degree 2g − 2 + 2d we have

#{f : X → ℙ1_{ℂ} | discr(f) = D}/isomorphism ≤ H(g, d).

If we consider only tame coverings, then we can extend this result to characteristic p.

3.2 Lemma. Let k be an algebraically closed field of characteristic p. Let D ⊂ ℙ_{k}^{1} be a divisor of degree 2g − 2 + 2d. Then the number of isomorphism classes of nonconstant morphisms f : X → ℙ_{k}^{1} of smooth projective curves over k of degree d which are tamely ramified and have discriminant D is at most H(g, d).

The proof is by lifting to characteristic zero. We omit it here since we prove a similar result later (see Proposition 4.2).

3.3 Lemma. Let X → ℙ_{k}^{1} be a morphism of degree d, where k is a finite field. The number of isomorphism classes of Y → ℙ_{k}^{1} such that Y ⊗_{k} k̄ ≅ X ⊗_{k} k̄ over ℙ1 ; k̄ is at most d.

Proof. The set in question is bijective to H^{1}(Gal(k̄/k), Aut(X ⊗_{k} k̄/ℙ1 ; k̄)). Since the group Aut(X ⊗_{k} k̄/ℙ1 ; k̄) has at most d elements we get the result from the fact that Gal(k̄/k) is pro-cyclic.

Let us assume for the moment that there exists a constant c_{5} > 0 such that every curve X of genus g over 𝔽_{q} has a nonconstant map X → ℙ1_{𝔽_{q}} of degree at most c_{5}g. If so, then we can estimate

#M_{g}(𝔽_{q}) ≤ c_{5}gH(g, c_{5}g)#{divisors D of degree ≤ 2g − 2 + 2c_{5}g}.

The factor c_{5}g comes from Lemma 3.3, the factor H(g, c_{5}g) comes from Lemma 3.2, and the last factor counts the number of possible discriminants. Still hypothetically speaking, the arguments above would give constants c_{6}, c_{7} > 0 such that

#M_{g}(𝔽_{q}) ≤ q^{c_{6}g}e^{c_{7}g log(g)}.

The assumption of the preceding paragraph seems a little strong. For example, in characteristic 2 the general morphism of a curve to ℙ^{1} is wildly ramified. In the next section we will estimate the number of wildly ramified coverings with given discriminant D. This estimate will suffice to obtain the above bound.

4 Wild ramification.

We fix integers g ≥ 2 and d ≥ 2g − 1. We will consider finite morphisms

f : X ⟶ ℙ_{k}^{1}

which are generically étale, have degree d, and with g(X) = g. Here k is a field of characteristic p and X is a smooth projective geometrically irreducible curve over k.

To such a map of curves we can associate a number of invariants. The first is the discriminant which we mentionned before. For every t ∈ discr(f) we can look at the set of closed points x ∈ X such that f(x) = t and such that f is wildly ramified at x (i.e. not tamely ramified). Let us write W (f, t) or W (X, t) for this set. Another invariant is the isomorphism class of the Ô_{ℙ^{1},t}-algebra

Ô_{X,W (f,t)} = ∏_{x∈W (f,t)} Ô_{X,x}.

We want to distinguish between curves which have different invariants.

4.1 Definition. We say that two such maps f : X → ℙ_{k}^{1} and h : Y → ℙ_{k}^{1} having the same discriminant have formally isomorphic wild ramification if for all points t ∈ discr(f) there exists an isomorphism

Ô_{X,W (f,t)} ≅ Ô_{Y,W (h,t)}

as Ô_{ℙ^{1},t}-algebras.

If f and h are tame, and if they have the same discriminant, then they have formally isomorphic wild ramification.

4.2 Proposition. Let k be an algebraically closed field of characteristic p, and recall that g ≥ 2, d ≥ 2g − 1. Let f : X → ℙ_{k}^{1} be given. The number of isomorphism classes of h : Y → ℙ_{k}^{1} which have the same discriminant and have formally isomorphic wild ramification is at most H(g, d).

Proof. Let W = W (k) be the Witt-ring of k. Let K be the fraction field of W, and let us choose an algebraic closure K̄ of K. Finally, we denote by W̄ the ring of integers of K̄.

Choose a lift X̃/W of X/k. This is possible since the deformation theory of X is unobstructed. The morphism f is given by an invertible O_{X}-module L of degree d on X, and a pair of sections s_{0}, s_{1} which generate it. Let L̃ be a lift of L to X̃ and let s̃_{0}, s̃_{1} be two general (how general we will see later) lifts of s_{0}, s_{1}. Lifting L is possible as H^{2}(X, O_{X}) = (0). Lifting s_{0}, s_{1} is possible as H^{1}(X, L) = (0) by our assumption that d = deg(L) > 2g − 2. These choices define a lift

f̃: X̃ ⟶ ℙ_{W}^{1}

of f. Let D̃ ⊂ ℙ_{W}^{1} be the discriminant of f̃. Since s̃_{0} and s̃_{1} are general the discriminant D̃_{η} of f̃_{η} : X̃_{η} → ℙ_{η}^{1} is a reduced divisor (i.e. all multiplicities are 1). Here η is the generic point of Spec(W).

Let h : Y → ℙ_{k}^{1} be any morphism which has the same discriminant as f and which has wild ramification formally isomorphic to f. We will show that h has a lift h̃ : Ỹ → ℙ1 ; W̄ over W̄ with discriminant discr(f̃) = D̃ ⊗ W̄ ⊂ ℙ1 ; W̄. Let us assume this for the moment and show that this implies the result.

Suppose that h_{i} : Y_{i} → ℙ_{k}^{1} are two such morphisms with lifts h̃_{i} : Ỹ_{i} → ℙ1 ; W̄ having discriminant D̃ ⊗ W̄. We have to show that if Ỹ_{1,η̄} ≅ Ỹ_{2,η̄} (over ℙ1 ; K̄), then Y_{1} ≅ Y_{2} (over ℙ_{k}^{1}). This is well known. The morphisms h̃_{i} and the isomorphism Ỹ_{1,η̄} ≅ Ỹ_{2,η̄} are defined over a finite extension K ⊂ K′ ⊂ K̄ of the fraction field K of W and extends to the regular minimal models Ỹ_{1,O_{K′}} ≅ Ỹ_{2,O_{K′}}. Taking the special fibre of this isomorphism gives the isomorphism Y_{1} ≅ Y_{2}. Here we use the fact that k is algebraically closed, as this implies that the residue field of O_{K′} is k as well.

To finish the proof of the proposition, we have to prove the existence of the lifting Ỹ of Y. We first do some preliminary study on the lifting f̃. The divisor D is f_{∗}(D′) where D′ ⊂ X is the ramification divisor on X. We can write D′ = D′_{t} + D′_{w} as the sum of the tame part D′_{t} and the wild part D′_{w}. Correspondingly the ramification divisor of f̃ is D̃′ ⊂ X̃ is split up into a tame and wild part (note that D̃′ is finite flat over W). In this way we obtain a decomposition D̃ = D̃_{t} + D̃_{w}.

Pick a point t ∈ D. We first want to solve the local problem of lifting the Ô_{ℙ_{k}^{1},t}-algebra

(∗)  Ô_{Y,W (Y,t)} × Ô_{Y,T (Y,t)} = ∏_{y∈h^{−1}(t)} Ô_{Y,y}.

Here T (Y, t) denotes the set of y ∈ Y lying over t at which h is tamely ramified or étale. In the following R will denote a discrete valuation ring which is a finite extension of W. We would like to lift the algebra (∗) to a finite flat Ô_{R,t} = Ô_{ℙ_{R}^{1},t}-algebra whose discriminant is the trace of D̃ on Ô_{R,t}. Since we are assuming that Ô_{Y,W (Y,t)} ≅ Ô_{X,W (X,t)}, we can use Ô_{X̃,W (X,t)} to obtain a finite flat Ô_{W,t}-algebra lifting the first factor of (*) whose discriminant is the trace of D̃_{w} on Ô_{W,t}. Furthermore, the trace of D̃_{t} on Ô_{W,t} is an effective Cartier divisor whose reduction modulo π_{R} is the discriminant of Ô_{Y,T (Y,t)} over Ô_{ℙ_{k}^{1},t}. This is true since the discriminant of f is the discriminant of h and since f and h have formally isomorphic wild ramification. By Lemma 4.5 below, we obtain an extension W ⊂ R and a lift of Ô_{Y,T (Y,t)} over Ô_{R,t} having as discriminant the trace of D̃ on Ô_{R,t}.

Since D has finite support, we may choose a single R which works for all t ∈ D simultaneously. We fix choices of lifts Ã_{t} ⊃ Ô_{R,t} of Ô_{Y,f^{−1}(t)} ⊃ O_{ℙ_{k}^{1},t} as above. Let π = π_{R} be a uniformizer of R. We will prove by induction on n the existence of flat R/π^{n}-schemes Y_{n} and morphisms h_{n} : Y_{n} → ℙ_{R/π^{n}}^{1} such that

(i) Y_{1} = Y, h_{1} = h, (ii) there is an isomorphism Y_{n} ⊗ R/π^{n−1} ≅ Y_{n−1} compatible with the morphisms h_{n} and

h_{n−1}, (iii) for t ∈ D, the product of the complete local rings of Y_{n} at the points of h_{n}^{−1}(t) is

isomorphic to Ã_{t}/π^{n}Ã_{t} as a Ô_{R,t}-algebra.

There are (at least) two ways to prove this. The first is to use formal patching (à la Harbater). Indeed, the lift of the formal completion of Y along h^{−1}(D) is given to us by (iii). Lifting (Y \ h^{−1}(D)) → (ℙ_{k}^{1} \ D) presents no difficulty since h is finite étale over ℙ_{k}^{1} \ D. Glueing is easy; details left to the reader. Compare [3, Subsection 3.9].

The second method is to use deformation theory. The deformations of Y → ℙ_{k}^{1} are captured by the cohomology of the complex h^{∗}T_{ℙ^{1}} → T_{Y} on Y. This complex is quasi-isomorphic to a scyscraper sheaf placed in degree 1, and supported on h^{−1}(D). Therefore, any prescribed local deformations (at all t ∈ D) can be followed along by a global deformation. Details left to the reader.

Once we have the sequence of schemes Y_{n} we set 𝔜 = lim Y_{n} (a formal scheme). Grothendieck’s algebraization of formal schemes [2, III 5.1.4] applies and we obtain the lift Ỹ over R as desired.

4.3 With k and W as in the proposition. Let φ : k[[t]] → A be a tame finite extension of discrete valuation rings of degree n. The discriminant D of φ is D = V (t^{n−1}). Let C denote the category of local Artin W -algebras with residue field k. Consider the functors F, H : C → \underline{Sets}, defined by F (R) = the set of isomorphism classes of pairs (φ_{R} : R[[t]] → A_{R}, α : A_{R} ⊗k → A), where

ϕ_{R} is finite flat and α is an isomorphism, and

H(R) = the set of effective Cartier divisors D_{R} ⊂ Spec(R[[t]]) such that D_{R} ⊗ k = D. There is a natural transformation of functors F → H which associates to φ_{R} the discriminant divisor discr(φ_{R}) ⊂ Spec(R[[t]]).

4.4 Lemma. With the notations above. The functors F and H are pro-representable by W [[a_{0}, …, a_{n−2}]] and W [[b_{0}, …, b_{n−2}]] respectively. The transformation F → H is represented by a finite flat ring map W [[a_{0}, …, a_{n−2}]] → W [[b_{0}, …, b_{n−2}]].

Proof. Since k is algebraically closed, we have A ≅ k[[s]] and φ(t) = s^{n} for a suitable choice of s. (And p |̸ n.) We leave it to the reader to check Schlessinger’s conditions [5] for F and H. Having checked these it is easy to see that the universal deformation of φ is given by

\begin{matrix}W [[a_{0}, …, a_{n−2}]][[t]] & ⟶ & W [[a_{0}, …, a_{n−2}]][[s]] \\ t & ⟼ & s^{n} + a_{n−2}s^{n−2} + … + a_{0}. \end{matrix}

The universal deformation of D is the effective Cartier divisor defined by the element

t^{n−1} + b_{n−2}t^{n−2} + … + b_{0}

in W [[b_{0}, …, b_{n−2}]][[t]]. Finiteness and flatness of the ring map W [[a_{0}, …, a_{n−2}]] → W [[b_{0}, …, b_{n−2}]] follows from the fact that W [[b_{0}, …, b_{n−2}]]/(a_{i}, p) has finite length. This in turn follows from the fact that if the map t ↦ s^{n}+a_{n−2}s^{n−2}+…+a_{0} is totally ramified, then a_{i} = 0.

4.5 Lemma. Suppose that k[[t]] → A is a finite flat ring extension such that A is a product of tame extensions of k[[t]] as above. Let D̃ ⊂ Spec(W [[t]]) be any effective Cartier divisor such that D̃ ⊗ k is the discriminant of A over k[[t]]. Then there is a finite extension of discrete valuation rings W ⊂ R and a finite flat extension R[[t]] ⊂ Ã with the following properties: (1) the discriminant of Ã over R[[t]] is D̃⊗R and (2) Ã⊗_{R} R/𝔪_{R} ≅ A as k[[t]]-algebras.

Proof. Suppose that A is a product of r extensions A_{1} × … A_{r} of degrees n_{1}, …, n_{r}. Then D̃ is the zero set of a monic polynomial P of degree d = ∑(n_{i} − 1) congruent to t^{d} modulo p. After replacing W by a finite extension R′ as in the lemma, we may assume that P factors as P_{1} … P_{r} where each P_{i} is monic and has degree n_{i} −1. For each i, we can by the previous lemma find a finite flat ring extension R → R_{i} and a lift of the extension A_{i} with discriminant given by P_{i}. Now take an extension R of R′ which dominates all the R_{i}.

4.6 Proposition. Fix integers f, e, c ≥ 1 such that p | e, p |̸ c and c > e. Let k be a finite field with q = p^{a} elements. Consider finite flat extensions k[[t]] ⊂ A with the following properties: (i) A is a discrete valuation ring with residue field k′, (ii) [k′ : k] = f, (iii) ord_{A}(t) = e, and (iv) length(Ω_{A}^{1}/Adt) = c − 1. The number of isomorphism classes of such extensions is at most

eq^{f(1+ \frac{c−e}{p−1})} ≤ eq^{\frac{f(c−1)}{p−1}} .

Any finite extension of discrete valuation rings k[[t]] ⊂ A which is wildly ramified has invariants f, e, c as described in the proposition.

Proof. By the Cohen structure theorem we can choose an isomorphism A ≅ k′[[s]]. The map k[[t]] → k′[[s]] = A is determined by the powerseries

(∗)  t = f(s) = a_{e}s^{e} + a_{e+1}s^{e+1} + …

where a_{e}, a_{e+1}, … ∈ k′. We are going to choose a uniformizer s ∈ A such that t = f(s) has a particularly ”nice” shape.

The power series f has the following properties: the leading coefficient a_{e} ≠ 0, for e < i < c, p |̸ i we have a_{i} = 0, and the coefficient a_{c} ≠ 0. Reason: in this case and in this case only do we have dt = ca_{c}s^{c−1}ds+ higher order terms, so Ω_{A}^{1}/Adt has k′-basis ds, sds, …, s^{c−2}ds.

We introduce the induction hypothesis (H_{n}): There exists a choice of s such that for all i ≤ n we have

\{ \begin{matrix}a_{i} = 0 & if & p |̸ i and i ≠ c \\ a_{i} = 0 & if & p | i and i > \frac{pc−e}{p−1} \end{matrix}

By the remarks above (H_{c}) is satisfied. We proceed by showing that (H_{n−1}) ⇒ (H_{n}), if n ≥ c + 1. \underline{Case 1: n > (pc − e)/(p − 1)} We set α = n − c + 1. Note α ≥ 2 since n > c. Consider the coordinate change s = s_{1} + δs_{1}^{α}. We consider the effect of this on the terms of f.

(a) The term a_{c}s^{c}.

a_{c}s^{c} = a_{c}(s_{1} + δs_{1}^{α})^{c} = a_{c} + ca_{c}δs_{1}^{α+c−1} + h.o.t. = a_{c}s_{1}^{c} + ca_{c}δs_{1}^{n} + h.o.t.

(b) The terms a_{i}s^{i} with i ≤ c − 1. For these i we have a_{i} = 0 if p |̸ i so we may assume that i = pj for some integer j. Then

a_{i}s^{i} = a_{i}(s_{1} + δs_{1}^{α})^{pj}
= a_{i}(s_{1}^{j} + jδs_{1}^{j−1+α} + h.o.t.)^{p}
= a_{i}s_{1}^{i} + ja_{i}δ^{p}s_{1}^{p(j−1+α)} + h.o.t
= a_{i}s_{1}^{i} + ja_{i}δ^{p}s_{1}^{i+p(α−1)} + h.o.t

Since α = n − c + 1 and i ≥ e we have (by our assumption on n)

i + p(α − 1) ≥ e + p(n − c) = pn − (pc − e) > n.

(c) The terms a_{i}s^{i} with i ≥ c + 1. For i ≥ c + 1 we have a_{i}s^{1} = a_{i}s_{1}^{i}+ terms of order ≥ α + i − 1 > n.

From (a), (b), (c) we get

f(s) = f(s_{1} + δs_{1}^{α}) = f(s_{1}) + ca_{c}δs_{1}^{n} + h.o.t.

Choosing δ = a_{n}/(−ca_{c}) gives (H_{n}) for the uniformizer s_{1} ∈ A. \underline{Case 2: c + 1 ≤ n ≤ (pc − e)/(p − 1)} There are two possibilities. If p|n there is nothing to prove. If p |̸ n, then we perform a coordinate change s = s_{1} +δs_{1}^{α}, δ ∈ k′ with α = n−c+1 as in Case 1. However, the computations above show that the terms (b) affect only terms λs^{j} where p|j. These do not play a role in (H_{n}) since n is smaller than (pc − e)/(p − 1). The computations in (a) and (c) are identical. Hence the conclusion follows as before.

By performing the coordinate changes described above ad infinitum we obtain a uniformizer s ∈ A such that

t = a_{e}s^{e} + a_{e+p}s^{e+p} + … + a_{e+ℓp}s^{e+ℓp} + a_{c}s^{c},

where ℓ is the largest integer such that

e + ℓp ≤ \frac{pc − e}{p − 1} = c + \frac{c − e}{p − 1}.

This implies that

ℓ ≤ \frac{c − e}{p − 1}.

Finally, we remark that changing s into λs with λ ∈ (k′)^{∗} changes a_{i} into λ^{i}a_{i}. Therefore we may assume a_{e} runs through a fixed set of representatives of (k′)^{∗}/((k′)^{∗})^{e}. Thus we conclude that the number we are looking for is bounded by the number of elements of

(k′)^{∗}/((k′)^{∗})^{e} × (k′)^{ℓ+1}.

This gives the first estimate of the proposition. To prove the inequality sign use that e ≥ p.

4.7 Theorem. Let k be a finite field with q elements. Let D ⊂ ℙ_{k}^{1} be a divisor of degree 2g − 2 + 2d where g ≥ 2 and d ≥ 2g − 1. Then the number of isomorphism classes of morphisms f : X → ℙ_{k}^{1} as above with discriminant D is at most

d · H(g, d) · e^{c_{8}(2g−2+2d)} · q^{(2g−2+2d)/(p−1)}.

Proof. To prove this we simply combine the results above. There is only a slight combinatorial question which we have to resolve before we are able to conclude the result.

Write D = ∑ n_{t}(t), the sum is over the closed points of ℙ_{k}^{1}. The degree of D is ∑ n_{t} deg(t) = 2g − 2 + 2d. Choose a morphism f : X → ℙ_{k}^{1} of the type we are considering. For each t ∈ D, and each x ∈ W (X, t) we have the invariants e_{x}, f_{x}, c_{x} (of the preceding proposition) associated to the ring extension

Ô_{ℙ_{k}^{1},t} ⊂ Ô_{X,x}.

We have the following relations (note that we are only considering the wildly ramified points lying over t)

c_{x} > e_{x} > 1, p|e_{x}, ∑_{x∈W (X,t)} f_{x}e_{x} ≤ d, ∑_{x∈W (X,t)}(c_{x} − 1)f_{x} ≤ n_{t}.

Let us say that the numerical data associated to this X at t consists of the integer r = #W (X, t) and the list of triples e_{i}, f_{i}, c_{i}, 1 ≤ i ≤ r satisfying the equations above. In particular ∑ f_{i}(c_{i} −1) ≤ n_{t}. By the proposition above, for each such set of numerical data there are at most

∏_{i=1}^{i=r} e_{i} (q^{deg(t)})^{\frac{f_{i}(c_{i}−1)}{p−1}} ≤ q^{n_{t} deg(t)/(p−1)} ∏ e_{i}

possible isomorphism classes of Ô_{X,W (X,t)} over Ô_{ℙ^{1},t} with these given numerical data.

We claim that ∏ e_{i} ≤ e^{n_{t}/e}. This is true because ∑ e_{i} ≤ ∑(c_{i} − 1) ≤ n_{t} and because of the following elementary fact: if λ_{1}, …, λ_{s} ∈ ℝ_{≥0} with ∑ λ_{i} ≤ m, then ∏ λ_{i} ≤ e^{m/e}.

Next, we want to count the number of possible numerical data above t. For every solution r, e_{i}, f_{i}, c_{i} of the inequalities above we consider the string from the alphabet {0, 1, |} composed as follows: binary expansion of r | binary expansion of e_{1} | binary expansion of f_{1} | binary expansion of c_{1} | … | binary expansion of c_{r}. Note that r + ∑(e_{i} + f_{i} + c_{i}) ≤ 5n_{t}. This implies that the string above has length at most 10n_{t}. Therefore the number of possible numerical data above t is at most 3^{10n_{t}}.

Putting the above together we see that for each t ∈ D there are at most

N_{t} := 3^{10n_{t}}e^{n_{t}/e}q^{n_{t} deg(t)/(p−1)}

possible isomorphism classes of the Ô_{ℙ^{1},t}-algebra Ô_{X,W (X,t)}. Of course this means there are at most ∏_{t∈D} N_{t} possible isomorphism classes of morphisms f : X → ℙ_{k}^{1} which have distinct isomorphism classes of wild ramification. By Proposition 4.2, for each of these there are at most H(g, d) distinct isomorphism classes over k̄ of X having the same wild ramification. The factor d of the theorem comes from descending from k̄ to k using Lemma 3.3. Since D has degree 2g − 2 + 2d (see above) we obtain

∏_{t∈D} N_{t} ≤ e^{c_{8}(2g−2+2d)}q^{(2g−2+2d)/(p−1)}

as desired.

4.8 Theorem. We have

#M_{g}(𝔽_{q}) ≤ \{ \begin{matrix}H(g, 2g)e^{c_{9}g}q^{c_{10}g} & p |̸ 2g \\ H(g, 2g + 1)e^{c_{9}g}q^{c_{10}g} & p | 2g \end{matrix} .

By the bound for Hurewitz numbers given in Section 3 we obtain the bound mentionned in the introduction

#M_{g}(𝔽_{q}) ≤ e^{c_{11}g log(g)} · q^{c_{12}g}.

Proof. The number of divisors D in ℙ^{1} over 𝔽_{q} of degree 2g−2+2d is (q^{2g−1+2d}−1)/(q−1). Thus the theorem follows from the above as soon as we prove that every curve of genus g has at least one nonconstant morphism of degree d = 2g (if p |̸ 2g) or d = 2g + 1 (if p | 2g) to ℙ^{1} over 𝔽_{q}. The morphism will be generically étale sincce p does not divide d.

We split the argument into two cases. If X has an 𝔽_{q}-rational point x, then we consider the invertible sheaf L = O_{X}(d(x)). Take any section s of L which doesn’t vanish at x. This is possible as H^{1}(X, O((d−1)(x))) = (0) by our choice of d. Then s seen as a rational function on X has polar divisor d(x) and hence defines a morphism of degree d to ℙ^{1}. If X has no 𝔽_{q}-rational points, then we take some (linearly normal) closed immersion X → ℙd−g_{𝔽_{q}} of degree d (using an invertible sheaf of degree d). Having this we start projecting X to lower and lower projective spaces from 𝔽_{q}-rational points. These points do not lie on X as X(𝔽_{q}) = ∅ and hence the degree doesn’t drop. In the end we obtain the morphism as desired.

5 Bounds using cohomology

In this section we explain briefly how to get bounds similar to the bounds in the previous sections using bounds for cohomology.

First we remark that the usual Lefschetz trace formula holds for the stack M_{g} over 𝔽_{q}. More precisely, we have

#M_{g}(𝔽_{q}) = ∑_{i=1}^{i=6g−6}(−1)^{i}Trace (π_{q}, H_{c}^{i}(M_{g} ⊗ 𝔽̄_{q}, ℚ_{ℓ}))

where π_{q} is the geometric Frobenius and ℓ is a prime which does not divide q. This can be deduced from the fact that M_{g} is a quotient of a scheme by a finite group and the usual Lefschetz trace formula for schemes.

Since M_{g} has a compactification M̄_{g} such that the boundary is a divisor with normal crossings relative to Spec(ℤ), we see that we can compute the betti numbers β_{i} := dim H_{c}^{i}(M_{g} ⊗ 𝔽̄_{q}, ℚ_{ℓ}) over the complex numbers. More precisely, we have

β_{i} = dim H_{c}^{i}(M_{g}(ℂ), ℚ).

(Usual cohomology with compact support.)

We are going to use the stratification of the moduli spaces of curves which can be obtained by using ribbon graphs, see [4]. To do this we have to consider the moduli space M_{g,1} of 1-pointed smooth curves of genus g. For our purposes, a ribbon graph Γ is a connected graph such that (1) every vertex has valency ≥ 3, and (2) for every vertex we are given a cyclic ordering of the germs of edges meeting this vertex. We can construct a topological surface with boundary out of Γ by replacing each edge by a ribbon, each vertex by a disc and glueing these together respecting the cyclical ordering given by Γ. The genus or the number of boundary components of Γ are the genus and number of boundary components of this surface. Finally, a metric on Γ determines (and is determined by) giving a positive real number ℓ (the length) for each edge of Γ.

The set of all metrized ribbon graphs of genus g having one boundary component is denoted M_{g,1}^{comb}. There is a natural structure of an orbifold on M_{g,1}^{comb}. Note that if Γ is a metrized ribbon graph, and if we let the length of the edge e connecting the vertices v_{1}, v_{2} decrease to 0, then we obtain in the limit the graph obtained from Γ by removing the edge e, identifying the vertices v_{1} and v_{2}, and cyclicly ordering the germs of edges meeting v_{1} and v_{2} in an obvious manner. If we let the length of e increase to ∞, then the limit doesn’t exist. This means that for every ribbon graph Γ there is an ”orbifold cell” (ℝ_{≥0})^{k}/G which is closed in M_{g,1}^{comb}. Here k is the number of edges of Γ and G is the automorphism group of Γ.

The fact we’re going to use is that there is a homeomorphism

M_{g,1}^{comb} ≅ M_{g,1}(ℂ) × ℝ_{+}.

See [4, page 5]. The pullback maps

H_{c}^{i}(M_{g} ⊗ ℂ, ℚ) ⟶ H_{c}^{i}(M_{g,1} ⊗ ℂ, ℚ)

are injective. Thus to bound the betti numbers β_{i} we bound the numbers dim H_{c}^{i}(M_{g,1}^{comb}). We claim that

∑_{i} β_{i} ≤ #{ribbon graphs}/isomorphism.

This follows from the stratification of M_{g,1}^{comb} by the ”orbifold cells” which are labeled by ribbon graphs. (Use the spectral sequence coming from this stratification.)

5.1 Lemma. The number of ribbon graphs of genus g with 1 boundary component is at most (12g)!.

Proof. Let Γ be such a ribbon graph. We first bound the number of vertices v in terms of the number of edges e: we have 2e ≥ 3v. Furthermore, the formula for the euler characteristic is 2 − 2g − 1 = v − e.

According to [4, page 6] a ribbon graph Γ can be described by a triple (X, s_{0}, s_{1}), where X is a set having 2e elements, s_{1} is a free involution of X, and S_{1} is any permutation of X. (These triples do not all give rise to connected ribbon graphs, or if connected do not necessarily have the right genus, etc.) We can number the elements of X such that s_{1} switches 2j and 2j − 1 for all j = 1…. , #X/2. Since we know that e ≤ 6g − 3 we can bound the number of possible graphs by counting the number of possible permutations s_{0}. This gives the desired bound.

Before we conclude, using the Weil conjectures proved by Deligne, let us mention that H_{c}^{6g−6}(M_{g}(ℂ), ℚ) is 1-dimensional and that the Frobenius eigenvalue (of π_{q}) on this space is q^{3g−3}. Furthermore, the other cohomology groups are mixed with Frobenius weights < 6g − 6. Hence from all of the above we obtain the estimate

#M_{g}(𝔽_{q}) ≤ q^{3g−3} + (12g)!q^{3g−3− \frac{1}{2}} ≤ q^{3g−3} + e^{c_{13}g log(g)}q^{3g−3− \frac{1}{2}} .

This is better than the estimate we obtained in the preceding section in terms of the behaviour in q.

5.2 Remark. Note that the euler characteristic of M_{g}, defined as e(M_{g}) = ∑_{i}(−1)^{i}β_{i}, is equal to B_{2g}/(4g^{2} − 4g), see [7]. The absolute value of the Bernouilli numbers B_{2g} grow rapidly with g. In fact, we have |B_{2g}| > exp(c_{14}g log(g)). This means that the method described above cannot give an essentially better bound (for q fixed and g → ∞).

However, this does not imply that the bound obtained above is the best possible. It is certainly conceivable that when computing #M_{g}(𝔽_{q}), for g much large than q, the eigenvalues of Frobenius cancel each other out to a large extend. We refer the reader to the article of Serre [6] for a similar phenomenon.

References

[1] N. M. Katz, What we don’t know about L-functions, lecture in Harvard-MIT algebraic geometry seminar, September 28, 1999.

[2] A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique I, II, III, IV, Publ. Math. I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967).

[3] A.J. de Jong and F. Oort, Purity of stratification by Newton Polygons, 13, no. 1, pp. 209–241 (2000).

[4] M. Kontsevich, Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function, Communications in Mathematical Physics 147, pp. 1 - 23 (1992).

[5] M. Schlessinger , Functors of Artin rings, Trans. Amer. Math. Soc. 130, pp. 208–222 (1968).

[6] J.-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke T_{p}, Journal of the American Mathematical Society 10, no. 1, pp. 75–102 (1997).

[7] J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Inventiones Mathematicae 85, no. 3, 457–485 (1986).