Counting elliptic surfaces over finite fields

by A.J. de Jong, Massachusetts Institute of Technology

Respectfully dedicated to Y. Manin

CONTENTS

1. Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Heuristics and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Three kinds of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6. A geometric inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 7. Degree three surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 8. Appendix A: Genus 1 curves and divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9. Appendix B: Elliptic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 10. Appendix C: Estimate for Selmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1 Introduction and main result

1.1 In this paper we consider elliptic curves (E, 0) over k(t) = k(ℙ^{1}) where k is a finite field with q elements. The height h(E) of such an elliptic curve is defined as the minimal integer d ≥ 0 such that E can be defined by a Weierstrass equation

x^{3} + A_{2}x^{2} + A_{4}x + A_{6} = y^{2} + A_{1}xy + A_{3}y,

where A_{i} is a homogenous polynomial of degree id in the homogenenous coordinates on ℙ^{1}. For such an elliptic curve let r(E) denote the rank and let s(E; ℓ) (ℓ prime) denote the rank of the ℓ-Selmer group of E:

s(E; ℓ) = dim_{𝔽_{ℓ}} Sel_{ℓ}(E).

Note that one always has r(E) ≤ s(E; ℓ). In this paper we esthablish by a simple counting procedure the following inequality.

1.2 Theorem.

lim sup_{d→∞} q^{−10d−1} ( ∑_{(E,0), h(E)=d} \frac{3^{s(E;3)} − 1}{# Aut(E, 0)}) ≤ \frac{3q^{9}}{(q^{3} − 1)(q^{2} − 1)^{2}(q − 1)^{2}} .

The sum is over isomorphism classes of elliptic curves of height d. The proof of this inequality is a trivial combination of Propositions 5.14, 7.5, and 7.9. A heuristic explanation of the “3” in the theorem is in Section 2. We expect the inequality of the theorem to tend to an equality as q tends to ∞ (see Remark 7.6).

The main motivation for the author to write up this paper was to show that even though it is usually difficult to esthablish results in the limit for large degree but with fixed q, in this case it is not so. Of course, the theorem above has a corollary concerning the average Selmer rank of elliptic curves. To this end let us define the average ℓ-Selmer rank for elliptic curves of height d as follows

1.2.1  AS_{d}(ℓ) := \frac{∑_{(E,0), h(E)=d} s(E; ℓ)}{∑_{(E,0), h(E)=d} 1}.

A similar formula defines the average rank AR_{d}. Note that AR_{d} ≤ AS_{d}(ℓ) so that the following corollary also gives an upper bound for the average rank. The author believes the (asymptotic) inequality for the average rank of elliptic curves over k(t) which is given in the Corollary below to be the best known bound at least for q ≥ 7. See [2] for the previously known best bound.

1.3 Corollary.

lim sup_{d→∞} AS_{d} ≤ \frac{3}{2}q^{27} ( \begin{matrix}1 − q − 2 q^{2} + q^{3} + 2 q^{4} + q^{5} − 2 q^{6} − q^{7} + q^{8} − q^{9} \\ +3 q^{11} + q^{12} − 3 q^{13} − 3 q^{14} + q^{15} + 3 q^{16} − q^{18} + q^{19} \\ −q^{20} − 2 q^{21} + q^{22} + 2 q^{23} + q^{24} − 2 q^{25} − q^{26} + q^{27} \end{matrix} )^{−1}

In particular, for q ≥ 7 we have AS_{d}(3) < 2. In particular we have

lim sup_{q→∞} lim sup_{d→∞} AR_{d} ≤ lim sup_{q→∞} lim sup_{d→∞} AS_{d}(3) ≤ \frac{3}{2}.

Proof. The contributions to the numerator of 1.2.1 coming from elliptic curves with constant j-invariant is at most (cst) · d · q^{9d}, see Proposition 4.14 and Lemma 10.2. On the other hand the denominator is at least 2q^{10d−1}(q^{2} − 1), see Proposition 4.16 for example. Hence we may ignore these contributions. Then, by the simple inequality

\frac{3^{s} − 1}{2} ≥ s

and since # Aut(E, 0) = 2 for j(E) ∉ k we see that

q^{−10d−1} ( ∑_{(E,0), j(E)∉k, h(E)=d} s(E; 3)) ≤ \frac{3q^{9}}{(q^{3} − 1)(q^{2} − 1)^{2}(q − 1)^{2}}

by our main theorem. On the other hand, since # Aut(E, 0) ≥ 2 always we have

\begin{aligned}q^{−10d−1} ( ∑_{(E,0), h(E)=d} 1) & ≥ 2q^{−10d−1} ( ∑_{(E,0), h(E)=d} \frac{1}{# Aut(E, 0)}) \\ & = 2(1 + \frac{1 − q^{−8} − q^{−9} + q^{−18}}{q − 1} − q^{−8d−1} + q^{−8d−3}). \end{aligned}

The equality is Proposition 4.12. A simple algebraic calculation finishes the proof.

A note on the methods and structure of this paper. The author decided to use the theory of minimal models of elliptic surfaces as much as possible so as to avoid as many computations as possible. The reader should be familiar with the theory of surfaces and know about Weierstrass equations; see e.g. [4] and [9].

In Section 4 we count elliptic curves of height d. We use Weierstrass equations and induction. We also give a precise count of the number of elliptic curves with constant jinvariant and of the number of semi-stable elliptic curves. In Section 5 we relate elements in the n-Selmer group of an elliptic curve of height d to elements of a set A_{n,d} of (precisely described) surfaces fibred over ℙ^{1}. Readers who are only interested in bounding ranks of elliptic curves may, after reading up to Proposition 5.10, jump forward to Section 7. In Section 6 it is shown that A_{n,d} is always finite. In Section 7 we do the actual count. The remaining sections contain various lemmas used in the text. The author thinks that Section 8 deserves a look for those who like applications of the theory of duality.

Acknowledgements. The author would like to thank Nick Katz for stimulating conversations and for raising questions like the ones considered above. The work of the author was partially supported by a grant from the NSF. Furthermore, the author was a prize fellow of the Clay Mathematics Institute during the period in which this work was done.

2 Heuristics and questions

In this section we do not claim any originality. We quickly explain the heuristics that predict the value of the average

\frac{∑(E,0), h(E)=d \frac{3^{s(E;3)}−1}{# Aut(E,0)}}{∑_{(E,0), h(E)=d} 1} .

Namely, imagine that we have a finite group G acting on a finite dimensional vector space V over 𝔽_{ℓ}. Assume that G leaves invariant a nondegenerate quadratic form b on V, and that the orbits of G on V \{0} correspond bijectively to the level sets of b. Then the reader verifies easily that

\frac{1}{#G} ∑_{g∈G}(#V^{g} − 1) = ℓ.

To make this heuristics “work” one should replace G with the fundamental group of the stack of elliptic surfaces X (with section) of height d, and with V replaced with a certain subquotient of the cohomology group H^{2}(X ⊗ k̄, μ_{ℓ}).

2.1 Question. Does the number

lim sup_{d→∞} \frac{∑(E,0), h(E)=d \frac{3^{s(E;3)}−1}{# Aut(E,0)}}{∑_{(E,0), h(E)=d} 1}

depend on q?

It would be fascinating if such numbers could depend on q (especially if this were true for the average rank). See Remark 4.22 for a related example. There have been a number of heuristic and actual investigations into such questions (for example by Ekedahl, Katz and Katz-Sarnak), but little is know for certain. Since we do not know the answer to this question, we replace the limit by lim sup_{q→∞} lim sup_{d→∞}. It seems to be generally expected that this limit should model what happens over number fields.

2.2 Question. Can one obtain a result similar to the main theorem for other primes ℓ? (With 3 replaced by ℓ.)

The author thinks he knows how to approach this for ℓ = 5 and for ℓ = 4 (of course this is not a prime number), but has not worked it out. From a positive answer to this question, and the inequality (ℓ^{s} − 1)/2 ≥ s(ℓ − 1)/2, one would get the bound ℓ/(ℓ − 1) for the average rank of the ℓ-selmer groups:

lim sup_{q→∞} lim sup_{d→∞} AS_{d}(ℓ) ≤ ℓ/(ℓ − 1).

This would still not be good enough to get what we perceive as the main goal of more investigations into the behaviour of average ranks of elliptic curves. Namely, an estimate for the average rank of elliptic curves.

2.3 Question. What is lim sup_{q→∞} lim sup_{d→∞} AR_{d}? Is it < 1? Is it = 0.5?

Clearly, the methods of this article cannot prove too much about this. Perhaps any bound which comes about by bounding the selmer rank should be termed “soft” , and “hard” any estimate which goes beyond such an estimate.

3 Conventions

3.1 In this article k denotes a field of any characteristic. Often k is a finite field with q elements. We will write ℙ^{1} in stead of ℙ_{k}^{1}, etc.

3.2 A curve (resp. surface) is a geometrically integral variety over k having dimension 1 (resp. 2).

3.3 Let X be a nonsingular surface. A (−1) curve in X is a closed sub scheme C ⊂ X such that C ≅ ℙ_{k′}^{1} for some finite field extension k ⊂ k′ and such that N_{C}X ≅ O_{C}(−1).

3.4 An elliptic surface is a morphism f : X → C of a nonsingular projective surface X to a nonsingular projective curve C whose generic fibre X_{η} is a smooth curve of genus 1 over κ(η). If the curve C (and the morphism f) is clear from context then we simply say that X is an elliptic surface. An elliptic surface is relatively minimal if there are no (−1)-curves contained in the fibres of the morphism X → C.

4 Weierstrass Equations

4.1 In this section we count isomorphism classes of elliptic curves (E, O) of given height (see below) over k(t) = k(ℙ^{1}). As usual we will often count a pair (E, 0) with weight 1/# Aut(E, 0). It should always be clear from the context whether we are counting just isomorphism classes or weighted isomorphism classes.

In addition to counting all elliptic curves, there are two subsets we we will count. The first is the number of elliptic curves which are twists of constant elliptic curves over ℙ^{1} (i.e., which have constant j-invariant). The second is the number of semi-stable elliptic curves. We will find a closed formula for all three numbers.

4.2 For any elliptic curve (E, O) over k(ℙ^{1}) there exists an elliptic surface X → ℙ^{1} with a section σ : ℙ^{1} → X such that (X_{η}, σ(η)) ≅ (E, 0). We may choose X relatively minimal and then the pair (X, σ) is well determined up to isomorphism. This pair (X, σ) is called the minimal model for the elliptic curve (E, 0) (even though the surface X need not be minimal). Also, in this case we have Aut(X, σ) = Aut(E, 0).

4.3 Given an arbitrary elliptic surface X → ℙ^{1} with a section σ we define

ω = σ^{∗}ω_{X/ℙ^{1}} = σ^{∗}O_{X}(−σ) = f_{∗}(ω_{X/ℙ^{1}}).

The implied isomorphisms are canonical. The integer d ≥ 0 such that ω ≅ O_{ℙ^{1}}(d) is (in some sense) the only discrete invariant of the pair (X, σ); we will see later that there is an irreducible variety whose points correspond to relatively minimal pairs (X, σ) with given d (namely the variety of Weierstrass polynomials 4.5). That d is nonnegative will be seen below as well.

4.4 Definition. The height of the pair (X, σ) is the integer d such that ω ≅ O_{ℙ^{1}}(d). The height h(E) of the elliptic curve (E, 0)/k(t) is the height of the relatively minimal pair (X, σ) corresponding to (E, 0).

4.5 We recall the standard argument to get the Weierstrass equation of a pair (X, σ) as in 4.3. We leave out the verifications. First, we consider the canonical exact sequences

4.5.1  0 → O → f_{∗}O_{X}(2σ) → ω^{⊗−2} → 0,

and

4.5.2  0 → f_{∗}O_{X}(2σ) → f_{∗}O_{X}(3σ) → ω^{⊗−3} → 0.

For degree reasons these exact sequences split; upon choosing splittings we obtain an isomorphism

f_{∗}O_{X}(3σ) ≅ O ⊕ ω^{⊗−2} ⊕ ω^{⊗−3}.

Second, we remark that the natural map

Sym^{3} f_{∗}O_{X}(3σ) → f_{∗}O_{X}(9σ)

is surjective. Furthermore, by filtering f_{∗}O_{X}(9σ) by the pole order along σ we deduce that det (f_{∗}O_{X}(9σ)) = ω^{⊗−44}. Similarly det ( Sym^{3} f_{∗}O_{X}(3σ)) = ω^{⊗−50}. Thus the kernel of the multiplication map above is canonically isomorphic to ω^{⊗−6}. In other words, the Weierstrass equation F is a canonical nowhere vanishing section

F ∈ Γ(ℙ^{1}, ω^{⊗6} ⊗ Sym^{3} f_{∗}O_{X}(3σ))

which maps to the zero under the multiplication map.

Using the splittings chosen above we may write

\begin{matrix}ω^{⊗6} ⊗ Sym^{3} f_{∗}O_{X}(3σ) = \\ ω^{⊗−3} ⊕ ω^{⊗−2} ⊕ ω^{⊗−1} ⊕ O ⊕ O ⊕ ω ⊕ ω^{⊗2} ⊕ ω^{⊗3} ⊕ ω^{⊗4} ⊕ ω^{⊗6}. \end{matrix}

At this point we conclude that d ≥ 0; if not then Γ(ℙ^{1}, ω^{i}) = 0 for i > 0 and the polynomial F defines a surface with singular generic fibre. Thus we can write F symbolically as

A_{0}x^{3} + A_{2}x^{2} + A_{4}x + A_{6} − A′_{0}y^{2} − A_{1}xy − A_{3}y

with A_{i} ∈ Γ(ℙ^{1}, ω^{i}). Indeed, traditionally, in the direct sum decomposition f_{∗}O_{X}(3σ) = O ⊕ ω^{⊗−2} ⊕ ω^{⊗−3}, the first summand has the generator 1, the second has the generator x and the third has the generator y. The coefficients A_{0}, A′_{0} are not zero since X_{η} is a smooth curve. In [6, (2.2.5)] it is shown that A_{0} = A′_{0} = 1. Thus F has the following familiar shape

F = x^{3} + A_{2}x^{2} + A_{4}x + A_{6} − y^{2} − A_{1}xy − A_{3}y.

Finally, after choosing an isomorphism O(d) ≅ ω we may think of A_{i} as a section of O(id) on ℙ^{1}.

4.6 Definition. A Weierstrass polynomial is any polynomial

F = x^{3} + A_{2}x^{2} + A_{4}x + A_{6} − y^{2} − A_{1}xy − A_{3}y.

with A_{i} ∈ Γ(ℙ^{1}, O(id)). We say that F is minimal if it is obtained as above starting from a relatively minimal pair (X, σ).

4.7 The relation between the pair (X, σ) and any Weierstrass polynomial associated to it is as follows. Let X̄ = V (F) be the closed subscheme of

Proj (Sym^{∗} (O ⊕ O(−2d) ⊕ O(−3d)))

defined by F. The point “(0 : 1 : 0)” defines a section σ̄ : ℙ^{1} → X̄. The singular fibres of X̄ → ℙ^{1} are nodal or cuspidal cubics and σ̄ lies in the smooth locus of X̄ → ℙ^{1}. There is a canonical morphism X → X̄ which maps σ̄ to σ and which is birational. (The existence of the morphism follows from 8.4 and 8.5; birationality follows by looking at the generic fibres for example.) So X is a resolution of singularities of X̄. Note that X → X̄ is a minimal resolution if and only if any (−1)-curve in a fibre of X → ℙ^{1} meets σ.

4.8 How does the Weierstrass polynomial change when we change the choices made above? Changing the isomorphism ω → O(d) by a nonzero scalar λ correpsonds to the coordinate change x ↦ λ^{2}x, y ↦ λ^{3}y and changes A_{i} into λ^{i}A_{i}. A change of choice of splittings of 4.5.1 and 4.5.2 will lead to a coordinate change

\{ \begin{matrix}x & ↦ & x + B_{2} \\ y & ↦ & y + B_{1}x + B_{3} \end{matrix}

where B_{i} is a section of O(id). The formulae for the corresponding change of the coefficients A_{i} may be found in the book [9]. In any case the number of choices for a given (X, σ) is (q − 1)q^{6d+3}; some of these choices may give the same Weierstrass polynomial.

4.9 Lemma. Let (X, σ) as in 4.3 have height d. Assume that any (−1)-curve contained in a fibre of X → ℙ^{1} meets σ(ℙ^{1}). Then the number of Weierstrass polynomials corresponding to (X, σ) is

\frac{(q − 1)q^{6d+3}}{# Aut(X, σ)}.

Proof. Suppose that two triples of choices (of ω ≅ O(d) and splittings 4.5.1 and 4.5.2) give rise to the same polynomial F. These two triples correspond to distinct morphisms α, β : X → X̄ = V (F). Since both α and β are minimal resolutions of X̄ there is a unique automorphism φ : X → X such that α = β ∘ φ. The result follows.

4.10 Let N_{d} be the number of minimal Weierstrass polynomials for given d ≥ 0. Let I_{d} be the weigthed number of isomorphism classes of elliptic curves of height d; in a formula

I_{d} = ∑_{(E,O),h(E)=d} \frac{1}{#Aut(E, O)}.

The lemma above shows that

I_{d} = \frac{N_{d}}{(q − 1)q^{6d+3}} .

4.11 Given a nonnegative integer d and sections A_{i} of O(id) for i = 1, 2, 3, 4, 6 we obtain a Weierstrass polynomial F. This defines a closed subscheme X̄ of the threefold

Proj ( Sym^{∗} (O ⊕ O(−2d) ⊕ O(−3d))).

The surjection O ⊕ O(−2d) ⊕ O(−3d) → O(−3d) defines a section σ̄ of X̄ → ℙ^{1} with image in the smooth locus of X̄ over ℙ^{1}. There are two cases: (a) the generic fibre of X̄ is smooth, or (b) the generic fibre of X̄ is not smooth.

In case (b) there is a unique section τ : ℙ^{1} → X̄ whose image lies in the singular locus of the morphism X̄ → ℙ^{1}. Since τ is disjoint from σ, it never meets the line at infinity (in terms of the coordinates x, y). This means τ corresponds to a quotient invertible sheaf O ⊕ O(−2d) ⊕ O(−3d) → L such that the composition O → L generates L. Thus L ≅ O and τ is given by the row vector (1, C_{2}, C_{3}) seen as a map of locally free sheaves O ⊕ O(−2d) ⊕ O(−3d) → O, where C_{i} is a section of O(id). It follows that the following relations hold:

\{ \begin{matrix}2C_{3} + A_{1}C_{2} + A_{3} & = & 0 \\ A_{1}C_{3} & = & 3C_{2}^{2} + 2A_{2}C_{2} + A_{4} \\ C_{3}^{2} + A_{1}C_{2}C_{3} + A_{3}C_{3} & = & C_{2}^{3} + A_{2}C_{2}^{2} + A_{4}C_{2} + A_{6} \end{matrix}

Thus it is quite easy to count the number of Weierstrass polynomials of type (b). Namely, there are q^{8d+4} of them.

In case (a) the minimal resolution X of the surface X̄ is an elliptic surface, but it may not be a relatively minimal elliptic surface. In this case we say that the Weierstrass polynomial (of type (a)) is not minimal. Since X̄ is smooth along σ we see that σ lifts (uniquely) to a section σ of X. As mentionned in 4.7, X is relatively minimal over ℙ^{1} except for possibly (−1) curves in fibres meeting σ.

The preceding shows that every non-minimal Weierstrass polynomial of type (a) comes about as follows. Let (X, σ) be a relatively minimal elliptic surface with a section over ℙ^{1} of height d′ < d. Choose a closed point p_{1} ∈ ℙ^{1} and blow up the surface X in σ(p_{1}). We obtain an elliptic surface X_{1} which has a section σ_{1}. Repeat using a closed point p_{2} of ℙ^{1}. Say we stop after n steps. The isomorphism class of the final pair (X_{n}, σ_{n}) depends only on the divisor D_{n} = (p_{1}) + … + (p_{n}) on ℙ^{1}. It is quite easy to compute how the height changes:

h(X_{n}) = deg σ^{∗}O_{X_{n}}(−σ_{n}) = d′ + deg p_{1} + … + deg p_{n} = d + deg D_{n}.

For every pair (X_{n}, σ_{n}) of height d we produce (with choices as above) non-minimal Weierstrass polynomials F of type (a) with A_{i} ∈ Γ(ℙ^{1}, O(id)). Clearly we have Aut(X_{n}, σ_{n}) = Aut(X, σ). Thus the number of Weierstrass polynomials corresponding to (X_{n}, σ_{n}) is (q − 1)q^{6d+3}/# Aut(X, σ).

4.12 Proposition. The numbers N_{d} and I_{d} satisfy the following recursion relation:

N_{d} = q^{16d+5} − q^{8d+4} − ∑_{d′=0}^{d−1} I_{d′} · \frac{q^{d−d′+1} − 1}{q − 1} · (q − 1)q^{6d+3}.

The solutions for these recurrences are I_{0} = q, I_{1} = (q^{12} − q^{3})/(q − 1) − q^{2} − q, and for d ≥ 2 we have

I_{d} = q^{10d+1} + q^{10d} + … + q^{10d−7}
− (q^{10(d−1)+1} + q^{10(d−1)} + … + q^{10(d−1)−7}) − q^{2d} + q^{2d−2}
= q^{10d+1} [1 + \frac{1 − q^{−8} − q^{−9} + q^{−18}}{q − 1} − q^{−8d−1} + q^{−8d−3}]

Proof. Clearly the number N_{d} is equal to the number of all Weierstrass polynomials minus the number of polynomials of type (b) minus the number of non-minimal Weierstrass polynomials of type (a). The number of all Weierstrass polynomials is q^{16d+5}. The number of polynomials of type (b) is q^{8d+4}. The number of non-minimal polynomials of type (a) can be determined inductively using the above. (Recall that N_{d} = (q − 1)q^{6d+3}I_{d}, see 4.10.)

Let us explain this starting with the case d = 0. In this case there are q^{5} polynomials in all and q^{4} polynomials of type (b). There are no non-minimal polynomials. Hence N_{0} = q^{5} − q^{4} and I_{0} = q. This result corresponds to the well known fact that the number of points on the stack of elliptic curves over k is q. (In other words, the relatively minimal elliptic surfaces with height d = 0 correspond to constant curves.)

Let d = 1. The number of non-minimal Weierstrass polynomials in this case is

I_{0} · (#Div1_{ℙ^{1}}) · (q − 1)q^{6·1+3} = (q^{2} − 1)q^{10}.

Indeed, for every isomorphism class of a relatively minimal elliptic surface (X, σ) having height 0, we obtain q + 1 non-minimal surfaces (X_{1}, σ_{1}) having height d = 1. For each of these we get (q − 1)q^{6·1+3}/#Aut(X, σ) Weierstrass polynomials. Conclusion:

N_{1} = q^{21} − q^{12} − q(q + 1)(q − 1)q^{9} = q^{21} − 2q^{12} + q^{10}, I_{1} = \frac{q^{12} − q^{3}}{q − 1} − q^{2} − q.

The case d ≥ 2 is similar. Note that the factor (q^{d−d′+1} − 1)/(q − 1) represents the number of divisors of degree d − d′ on ℙ^{1}. (For another formula describing a similar recursion see 4.13.1.)

4.13 In this subsection we count elliptic curves with constant j-invariant. Let F = x^{3} + A_{2}x^{2}+A_{4}x+A_{6}−y^{2}−A_{1}xy−A_{3}y be a Weierstrass polynomial with A_{i} ∈ Γ(ℙ^{1}, O(id)) which defines an elliptic curve over k(ℙ^{1}) whose j-invariant is constant (i.e., F is a Weierstrass polynomial of type (a) as in subsection 4.11). Then there exists a coordinate change of the form

\{ \begin{matrix}x & ↦ & x + B_{2} \\ y & ↦ & y + B_{1}x + B_{3} \end{matrix}

with B_{i} ∈ Γ(ℙ^{1}, O(id)) such that F transforms into one of the normal forms described below. (a1) If the characteristic of k is not 2 or 3 and j ≠ 0, 1728 then there is a unique choice of

(B_{1}, B_{2}, B_{3}) so that we get

F = x^{3} + α_{4}(A′_{2})^{2}x + α_{6}(A′_{2})^{3} − y^{2}.

Here the pair (α_{4}, α_{6}) ∈ (k^{∗})^{2} is such that y^{2} = x^{3} +α_{4}x+α_{6} defines an elliptic curve, and A′_{2} is a nonzero section of O(2d). Note that there are (q − 2)(q − 1) such pairs (α_{2}, α_{3}) and that the triple (λ^{−2}α_{4}, λ^{−3}α_{6}, λA′_{2}) defines the same polynomial as the triple (α_{4}, α_{6}, A′_{2}).

(a2) If the characteristic of k is not 2 or 3 and j = 0 then there is a unique choice of

(B_{1}, B_{2}, B_{3}) so that we get

F = x^{3} + A′_{6} − y^{2}.

Here A′_{6} is a nonzero section of O(6d). (a3) If the characteristic of k is not 2 or 3 and j = 1728 then there is a unique choice of

(B_{1}, B_{2}, B_{3}) so that we get

F = x^{3} + A′_{4}x − y^{2}.

Here A′_{4} is a nonzero section of O(4d). (b1) If the characteristic of k is 3 and j ≠ 0 then there is a unique choice of (B_{1}, B_{2}, B_{3})

so that we get

F = x^{3} + A′_{2}x^{2} + α_{6}(A′_{2})^{3} − y^{2}.

Here α_{6} ∈ k^{∗} and A′_{2} is a nonzero section of O(2d). (b2) If the characteristic of k is 3 and j = 0 then there is a unique choice of the pair

(B_{1}, B_{3}) so that we get

F = x^{3} + A′_{4}x + A′_{6} − y^{2}.

Here A′_{4} is a nonzero section of O(4d) and A′_{6} is any section of O(6d). (c1) If the characteristic of k is 2 and j ≠ 0 then there is a unique choice of the pair

(B_{2}, B_{3}) so that we get

F = x^{3} + A′_{2}x^{2} + α_{6}(A′_{1})^{6} − y^{2} − A′_{1}xy.

Here α_{6} ∈ k^{∗}, A′_{1} is a nonzero section of O(d), A′_{2} is a section of O(2d). (c2) If the characteristic of k is 2 and j = 0 then there is a unique choice of B_{2} so that we

get

F = x^{3} + A′_{4}x + A′_{6} − y^{2} − A′_{3}y.

Here A′_{3} is a nonzero section of O(3d), A′_{4} is a section of O(4d), and A′_{6} is a section of O(6d).

We leave the verifications to the reader. We can use the above the count the number of Weierstrass polynomials with constant (finite) j-invariant. Namely, if the characteristic of k is not 2 or 3, then we see that the number of such polynomials is

M_{d} = (q − 2)(q^{2d+1} − 1)q^{6d+3} + (q^{6d+1} − 1)q^{6d+3} + (q^{4d+1} − 1)q^{6d+3}.

In characteristic 3 we obtain

M_{d} = (q − 1)(q^{2d+1} − 1)q^{6d+3} + (q^{4d+1} − 1)q^{6d+1}q^{4d+2}.

Finally, in characteristic 2 we get

M_{d} = (q − 1)(q^{d+1} − 1)q^{2d+1}q^{5d+2} + (q^{3d+1} − 1)q^{4d+1}q^{6d+1}q^{2d+1}.

Note that in each case for d = 0 we obtain q^{5} − q^{4} which is what we saw above also.

We want to know the number of minimal Weierstrass polynomials among these. Namely, let J_{d} denote the weighted number of isomorphism classes of elliptic curves (E, 0)/k(t) of height d whose j-invariant is constant. As before we obtain the relation

4.13.1  M_{d} = q^{6d+3}(q − 1) [J_{d} + (q + 1)J_{d−1} + … + (q^{d} + … + q + 1)J_{0}] .

This allows us to compute J_{d}.

4.14 Proposition. The weighted number of isomorphism classes of elliptic curves over k(t) with constant j-invariant and height d is given by the following formula.

(a) In case the charateristic of k is not 2 or 3 we have J_{0} = q, J_{1} = (q^{3} + q^{2} + 2 q + 3) q^{3}, and for d ≥ 2 we have

J_{d} = (q^{4 d+2} + q^{4 d+1} + q^{4 d} + q^{4 d−1} + q^{4 d−2}
− q^{4 d−4} − q^{4 d−5} − q^{4 d−6} − q^{4 d−7} − q^{4 d−8}
+ q^{2 d+2} + q^{2 d+1} + q^{2 d} − q^{2 d−2} − q^{2 d−3} − q^{2 d−4} + q^{3} − 2 q^{2} − q + 2)q^{2 d−2}

(b) In case the characteristic of k is 3 we have J_{0} = q, J(1) = q^{8} + q^{7} + q^{6} + q^{5} + q^{4} + q^{3} − q^{2} − q − 1 and for d ≥ 2 we have

J_{d} = (q^{6 d+2} + q^{6 d+1} + q^{6 d} + q^{6 d−1} + q^{6 d−2} + q^{6 d−3} + q^{6 d−4}
− q^{6 d−6} − q^{6 d−7} − q^{6 d−8} − q^{6 d−9} − q^{6 d−10} − q^{6 d−11} − q^{6 d−12}
− q^{2 d+1} − q^{2 d} − q^{2 d−1} + q^{2 d−3} + q^{2 d−4} + q^{2 d−5} + q^{3} − q^{2} − q + 1)q^{2 d−2}

(c) In case the characteristic of k is 2 we have J_{0} = q, J(1) = q^{9} +q^{8} +q^{7} +q^{6} +q^{3} −q^{2} −2q and for d ≥ 2 we have

J_{d} = (q^{7 d+2} + q^{7 d+1} + q^{7 d} + q^{7 d−1} + q^{7 d−2} + q^{7 d−3} + q^{7 d−4} + q^{7 d−5}
− q^{7 d−7} − q^{7 d−8} − q^{7 d−9} − q^{7 d−10} − q^{7 d−11} − q^{7 d−12} − q^{7 d−13} − q^{7 d−14}

− q^{4 d+1} − q^{4 d} − q^{4 d−1} − q^{4 d−2} − q^{4 d−3} + q^{4 d−5} + q^{4 d−6} + q^{4 d−7} + q^{4 d−8} + q^{4 d−9}

+ q^{3} − q^{2} − q + 1)q^{2 d−2}

4.15 In the rest of this section we count the number of semi-stable pairs (X, σ) of height d up to isomorphism. We say that an elliptic surface X over ℙ^{1} is semi-stable if every geometric fibre is either a smooth genus 1 curve or an n-gon of rational curves. Clearly, a semi-stable elliptic surface is relatively minimal. We say that an elliptic curve (E, 0) over k(t) is semi-stable if the corresponding relatively minimal elliptic surface is semi-stable. We let S_{d} be the weighted number of semi-stable elliptic curves over k(t) of height d.

The Weierstrass polynomial F of a semi-stable pair (X, σ) is characterized by the property that the singular fibres of X̄ = V (F) have at worst nodes as singularities (i.e., we have multiplicative reduction at the singular fibres). A Weierstrass polynomial of type (a) (see 4.11) will be called semi-stable if this is the case. Note that a semi-stable Weierstrass polynomial F is minimal: If X → X̄ = V (F) is a minimal resolution of singularities, and if C is the strict transform of a singular fibre X̄_{p}, i.e., of a nodal cubic curve, then C meets the other components of X_{p} in two points. Hence C is not a (−1)-curve. The other irreducible components of fibres X_{b} are produced by the resolution process, so they are not (−1)-curves either. Hence X is relatively minimal over ℙ^{1} as desired.

Let L_{d} be the number of semi-stable Weierstrass polynomials for given d ≥ 0. By the above we have the relation S_{d} = L_{d}/(q − 1)q^{6d+3}.

Since every elliptic curve of height 0 is constant (and hence semi-stable) we see that S_{0} = I_{0} = q. Every semi-stable elliptic surface (X, σ) of height d ≥ 1 has at least one singular fibre since the discriminant Δ has degree 12d. (See eg. the formula for Δ in Silverman’s book.) This implies that the j-function corresponding to this elliptic surface attains the value ∞ (since X̄ = V (F) has a nodal fibre, see formulae in Silverman’s book) as well as finite values. In particular it is not constant. We deduce that Aut(X, σ) = {id, −id}.

By the above the number of (not weighted) isomorphism classes of semi-stable elliptic curves of height d is 2S_{d} for d ≥ 1. In the subsections 4.17, 4.18 and 4.19 we will compute L_{d} for d ≥ 1; combined with the results of this subsection we obtain the following.

4.16 Proposition. Let k be any finite field. Let d ≥ 1. The number of isomorphism classes of semi-stable elliptic curves over k(t) of height d is equal to

2 · q^{10d+1} · \frac{(q − 1)(q + 1)}{q^{2}}.

4.17 In this subsection we calculate L_{d} assuming that the characteristic of k is not 2 or 3. For any Weierstrass polynomial there is a unique triple (B_{1}, B_{2}, B_{3}) such that F transforms into an polynomial of the form

F_{A,B} = x^{3} + Ax + B − y^{2}

with A a section of O(4d) and B a section of O(6d). Therefore L_{d} is equal to the number of semi-stable polynomials F_{A,B} times q^{6d+3}. On the other hand, it is quite easy to see that F_{A,B} is semi-stable if and only if the pair (A, B) is relatively prime. By Lemma 4.21 below we conclude that

L_{d} = q^{6d+3} · q^{10d+2} · \frac{(q − 1)^{2}(q + 1)}{q^{3}} = q^{16d+5} · \frac{(q − 1)^{2}(q + 1)}{q^{3}}

as desired.

4.18 In this subsection we calculate L_{d} assuming that the characteristic of k is 3. Here we can change coordinates using a unique pair (B_{1}, B_{3}) so that the resulting polynomial looks like

x^{3} + Ax^{2} + Bx + C − y^{2}.

This is semi-stable if and only if A and B are coprime. Thus we conclude that L_{d} = q^{16d+5}(q − 1)^{2}(q + 1)q^{−3} in the same way as above.

4.19 In this subsection we calculate L_{d} assuming that the characteristic of k is 2. We can change coordinates using a unique choice of B_{2} such that the resulting polynomial looks like

y^{2} + Axy + By + x^{3} + Cx + D.

This is semi-stable if and only if A and B are coprime. As above this leads to the same value of L_{d}.

4.20 Intermezzo. For any pair of integers n, m ≥ 1 we set

φ(n, m) := #{(A, B) ∈ Γ(ℙ^{1}, O(n)) × Γ(ℙ^{1}, O(m)) | A, B coprime}.

Further for any n ≥ 1 we set

φ(n) := ∑_{Z⊂ℙ^{1}, deg(Z)=n}#Γ(Z, O_{Z}^{∗}),

where the sum is over all closed sub schemes of ℙ^{1} of length n.

4.21 Lemma. With notations as above:

(a) If n ≥ m then

φ(n, m) = q^{n−m+1} · φ(m) · (q − 1),

(b) we have φ(n) = q^{2n−2}(q − 1)(q + 1), and

(c) φ(n, m) = q^{n+m−1}(q − 1)^{2}(q + 1).

Proof. It is a trivial calculation to see that (a) + (b) ⇒ (c).

Proof of (a). For every closed subscheme Z ⊂ ℙ^{1} of degree m there are q − 1 sections B of O(m) whose zero scheme is Z. This accounts for the factor (q − 1). The other two factors are evident upon considering the exact sequence

0 → Γ(ℙ^{1}, O(n − m)) → Γ(ℙ^{1}, O(n)) → O_{Z} → 0.

Proof of (b). The function φ satisfies a recursion relation. Namely

φ(n) = ∑_{deg(Z)=n} #O_{Z}^{∗}

= ∑_{deg(Z)=n}[#O_{Z} − #{s ∈ O_{Z}, s ∉ O_{Z}^{∗}}]

= \frac{q^{n+1} − 1}{q − 1} · q^{n} − ∑_{deg(Z)=n} ∑_{T ⊂Z,T ≠∅} #\{s ∈ O_{Z} | V (s) = T \}.

This follows as #O_{Z} = q^{n} for all Z and the number of divisors of degree n on ℙ^{1} is (q^{n+1} − 1)/(q − 1). Since O_{Z} is a principal ideal domain we can write the second term as

∑_{T,S⊂ℙ^{1},deg(T)+deg(S)=n,deg(T)≥1}#O_{S}^{∗} = ∑_{j=1}^{n} \frac{q^{j+1} − 1}{q − 1}φ(n − j).

Here we use the convention φ(0) = 1. All in all we obtain the recursion

φ(n) + ∑_{j=1}^{n} \frac{q^{j+1} − 1}{q − 1}φ(n − j) = \frac{q^{n+1} − 1}{q − 1} · q^{n}.

For example this gives φ(1) = (q + 1)(q − 1) = q^{2} − 1 as it should. The next step gives φ(2) + (q + 1)(q^{2} − 1) + (q^{2} + q + 1) = q^{4} + q^{3} + q^{2}, and so φ(2) = q^{4} − q^{2}. It is easy to prove that in general

φ(n) = q^{2n−2}(q^{2} − 1).

4.22 Remark. Let R_{d} be the weighted number of isomorphism classes of elliptic surfaces (X, σ) with reduced discriminant. (This means that all singular fibres of X → ℙ^{1} are 1-gons.) One can show that

γ_{q} := lim inf_{d→∞} \frac{R_{d}}{S_{d}} > 0.

We sketch the argument in the case of characteristic not 2 or 3. This amounts to counting those relatively prime pairs (A, B) such that Δ = −16(4A^{3} + 27B^{2}) is a square free polynomial. If Δ is not square free, then there is a closed point p ∈ ℙ^{1} such that Δ has a double zero at p. We bound the number of coprime pairs (A, B) such that Δ has a double zero at p. If (A, B) such a pair then neither A nor B will vanish at p. When 1 ≤ deg(p) ≤ 2d we have the exact sequences

0 → Γ(ℙ^{1}, O(4d − 2 deg(p))) → Γ(ℙ^{1}, O(4d)) → O_{2p} → 0,
0 → Γ(ℙ^{1}, O(6d − 2 deg(p))) → Γ(ℙ^{1}, O(6d)) → O_{2p} → 0.

Now O_{2p} ≅ κ(p)[ϵ]/(ϵ^{2}), and there are exactly (q^{i} − 1)q^{i} pairs α, β ∈ O_{2p}^{∗} such that 4α^{3} +27β^{2} = 0. Therefore there are at most q^{4d−2i+1}q^{6d−2i+1}(q^{i} −1)q^{i} = q^{10d−3i+2}(q^{i} −1) coprime pairs (A, B) such that Δ_{A,B} has a double zero at p. At this point we use the fact that there are at most (q^{i} + 1)/i closed points of degree i on ℙ^{1}. All in all we obtain at most

∑_{i=1}^{2d} \frac{q^{i} + 1}{i}q^{10d−3i+2}(q^{i} − 1) ≤ q^{10d+2} ∑_{i=1}^{∞} \frac{1}{iq^{i}}

comprime pairs (A, B) such that Δ has a double zero at a closed point p ∈ ℙ^{1} of degree ≤ 2d.

We leave it to the reader to estimate similarly those coprime pairs (A, B) whose discriminant has a double zero at a point p ∈ ℙ^{1} of degree > 2d. I claim that, upon dividing by the total number of comprime pairs, and taking the limit as d → ∞ this term vanishes. At this point the diligent reader will have obtained the inequality

γ_{q} ≥ 1 + \frac{q^{3} log(1 − 1/q)}{(q − 1)^{2}(q + 1)}.

For example when q = 5 (lowest possible under the current assumptions) then γ ≥ .7094485010.

In particular we see that lim_{q→∞} γ_{q} = 1. However, it is not true that γ_{q} = 1. One can precisely determine γ_{q} in this case, see e.g. [8].

5 Three kinds of surfaces

5.1 In this section we introduce three sets of elliptic surfaces and compare them. Note that the surfaces occuring in 5.2 may be singular and hence are not strictly speaking elliptic surfaces.

We fix two integers: d ≥ 0 and n ≥ 3. The integer d will indicate the height of the surface. The integer n will be a bound for the order of the surface seen as an element of the Weil-Chatelet group of the Jacobian of the generic fibre. For the purposes of this paper the reader may take n = 3, but since there is no additional complication for larger n we decided to work out the general case.

5.2 In this subsection we consider pairs (g : Y → ℙ^{1}, D) with the following properties:

(a) The morphism g : Y → ℙ^{1} is flat, proper and its generic fibre is a smooth curve,

(b) the equality g_{∗}O_{Y} = O holds universally (i.e., after any base change),

(c) the fibres of g are Gorenstein and ω_{Y/ℙ^{1}} ≅ g^{∗}O(d),

(d) D ⊂ Y is an effective Cartier divisor, flat over ℙ^{1} such that D · F = n, where F is a (k-rational) fibre of g,

(e) the sheaf O(D) is relatively very ample for g, i.e., we obtain a canonical closed im-

mersion

Y ↪ Proj(g_{∗}O_{Y} (D)).

Two pairs (Y, D) and (Y′, D′) are called equivalent if and only if there exists an isomorphism ψ : Y → Y′ over ℙ^{1} such that ψ^{∗}(D′) ∼ D + aF (rational equivalence) for some integer a ∈ ℤ. Note that this is an equivalence relation as we can compose morphisms like this. For a pair (Y, D) ∈ A_{n,d} we define the automorphism group Aut(Y, D) as the set of automorphisms ψ : Y → Y over ℙ^{1} such that ψ^{∗}(D) ∼ D + aF for some a ∈ ℤ (actually a = 0 always in this case).

The pair (Y, D) is called degenerate if there exists an integer n′ < n, n′|n and a divisor D′ of degree n′ on the generic fibre Y_{η} such that D_{η} ∼ (n/n′)D′ (rational equivalence). The idea behind this definition is that the degenerate pairs come up when we count pairs (Y′, D′) with smaller n.

Let A_{n,d} be the set of equivalence classes of nondegenerate (!) pairs (Y, D) satisfying (a)–(e) above. This is a finite set; see Section 6. Conform the usual counting procedure we set

A_{n,d} = ∑_{(Y,D)∈A_{n,d}} \frac{1}{# Aut(Y, D)}.

5.3 In this subsection we consider pairs (f : X → ℙ^{1}, D) with the following properties:

(a) X is a relatively minimal elliptic surface over ℙ^{1},

(b) the equality f_{∗}O_{X} = O holds universally,

(c) we have ω_{X/ℙ^{1}} ≅ f^{∗}O(d),

(d) D ⊂ X is an effective Cartier divisor, flat over ℙ^{1} such that D · F = n, where F is a fibre of f,

The concept of equivalence, of automorphism group and of degeneracy is defined in the same way as in 5.2. We denote B_{n,d} the set of equivalence classes of nondegenerate pairs (X, D) satisfying (a)–(d) above. The weighted number of elements of B_{n,d} is denoted B_{n,d}.

5.4 In this subsection we consider triples (X, σ, τ) where f : X → ℙ^{1} is a relatively minimal elliptic surface and (σ, τ) is a pair of sections of f. Let us denote (E, 0) = (X_{η}, σ(η)) the associated elliptic curve over k(t). Of course we assume that the height of (X, σ) is equal to d. The integer n does not appear in the definitions of the objects, but it appears in the definition of the concept of equivalence.

We say that (X, σ, τ) is equivalent to (X′, σ′, τ′) if there exists an isomorphism φ : X → X′ over ℙ^{1} and a section ρ : ℙ^{1} → X′ of f′ such that

(a) φ ∘ σ = σ′,

(b) on the generic fibre (E′, O′) = (X′_{η}, σ′(η)) we have nρ = τ′ − φ ∘ τ (addition in the group law of E′ with obvious notational abuse), and

(c) for every closed point t ∈ ℙ^{1} the sections ρ and σ′ meet the same irreducible component of X′_{t}.

Correspondingly, the automorphism group Aut(X, σ, τ) is the set of pairs (φ, ρ), with φ : X → X an isomorphism, and ρ : ℙ^{1} → X a section satisfying (a), (b), and (c) above.

We say that (X, σ, τ) is degenerate if τ is divisible by n/n′ in the Mordell-Weil group of E for some integer n′ < n, n′|n.

We denote C_{n,d} the set of equivalence classes of nondegerate triples (X, σ, τ) as above. As before we denote C_{n,d} the weighted number of equivalence classes.

5.5 In this subsection we construct a map B_{n,d} → A_{n,d}. Let (X⟶^{f}ℙ^{1}, D) corresponding to an element of B_{n,d}. By Lemma 8.4 (a) the sheaf f_{∗}O_{X}(D) is locally free (of rank n), its formation commutes with change of base, and f^{∗}f_{∗}O_{X}(D) → O_{X}(D) is surjective. Furthermore, as n ≥ 3, the sheaf of O-algebras ⨁_{m≥0} f_{∗}O_{X}(mD) is generated in degree 1, see Lemma 8.4 (a). Set

Y = Proj_{ℙ^{1}} (⨁_{m≥0} f_{∗}O_{X}(mD)) ⟶^{g}ℙ^{1}.

There is a morphism π : X → Y by the above. We define the divisor D_{Y} ⊂ Y to be the divisor corresponding to the canonical section 1 ∈ Γ(X, O_{X}(D)) = Γ(Y, O_{Y} (1)). By Lemma 8.5, the morphism is birational, the fibres of Y → ℙ^{1} are Gorenstein, g_{∗}(O_{Y}) = O universally and g^{∗}g_{∗}ω_{Y/ℙ^{1}} = ω_{Y/ℙ^{1}}. By Remark 8.6 we see that there is a canonical isomorphism g_{∗}ω_{Y/ℙ^{1}} = f_{∗}ω_{X/ℙ^{1}}. From these results the reader easily deduces that properties (a)-(e) of 5.2 hold. Note that X_{η} ≅ Y_{η} so that (X, D) is nondegenerate if and only if (Y, D) is nondegenerate.

The morphism π is a minimal resolution of singularities of Y (as X is relatively minimal over ℙ^{1}). On the other hand, the construction of Y depends only on the divisor class of D up to rational equivalence and up to adding multiples of F. These two remarks imply that, given two pairs (X, D) and (X′, D′) corresponding to elements of B_{n,d}, with associated pairs (Y, D_{Y}) and (Y′, D_{Y′}) in A_{n,d} we have

Mor_{B_{n,d}}((X, D), (X′, D′)) = Mor_{A_{n,d}}((Y, D_{Y}), (Y′, D_{Y′})).

(Obvious notations.) In other words, we may think of B_{n,d} as a “weighted” subset of A_{n,d}. In particular we obtain the inequality

5.5.1  B_{n,d} ≤ A_{n,d}.

5.6 Remark. Which pairs (Y, D) ∈ A_{n,d} are not in the image of the map B_{n,d} → A_{n,d}? Clearly, if Y is singular in codimension 1 (i.e., Y is singular along a component of a fibre, i.e., Y is not normal) then this is the case. In addition, there are the “non-minimal” normal cases: a pair (Y, D) with Y normal is not in the image exactly when Y has a singularity which is nonrational. To see this let π : X → Y be the minimal resolution of singularities. There is a canonical short exact sequence

0 → π_{∗}(ω_{X/ℙ^{1}}) → ω_{Y/ℙ^{1}} → Ext_{O_{Y}}^{2} (R^{1}π_{∗}O_{X}, ω_{Y/ℙ^{1}}) → 0.

From this we see that ω_{X/ℙ^{1}} ≅ f̸^{∗}O(d) if R^{1}π_{∗}O_{X} ≠ (0). In other words, B_{n,d} is identified with those pairs of objects (Y, D) ∈ A_{n,d} having at worst rational singularities.

5.7 In this subsection we define a map C_{n,d} to B_{n,d}. Namely, to a triple (X, σ, τ) we associate the pair (X, D), where D = (n−1)[σ]+[τ]). Here [σ] is short hand for the divisor σ(ℙ^{1}) of X. Since X has a section, it is easy to verify conditions (a)–(d) of Subsection 5.3 for the pair (X, D) (see also Section 8).

Suppose we are given two triples (X, σ, τ) and (X′, σ′, τ′) as above. Let us show that there is a bijection

Mor_{B_{n,d}}((X, D), (X′, D′)) = Mor_{C_{n,d}}((X, σ, τ), (X′, σ′, τ′)).

We construct the map from the LHS to the RHS. Given an equivalence ψ : X → X′ we set ρ = ψ ∘ σ and φ = T_{−ρ} ∘ ψ. Here T_{−ρ} denotes the translation by −ρ on (E, O) = (X_{η}, σ_{η}) which extends to an automorphism T_{−ρ} of X by the uniqueness of relatively minimal models. By definition of B_{n,d} there exists an a ∈ ℤ such that

5.7.1  (n − 1)[ρ] + [φ ∘ τ +_{E′} ρ] ∼ (n − 1)[σ′] + [τ′] + aF

(rational equivalence of divisors on X′). Since n ≥ 3 this implies that ρ and σ′ meet the same irreducible component of every (geometric) fibre of X′. This proves that (φ, ρ) is in the RHS. Next let us construct the map from the RHS to the LHS. Suppose we are given an equivalence (φ, ρ) from (X, σ, τ) to (X′, σ′, τ′). We set ψ = T_{ρ} ∘ φ and we claim that it is an equivalence between (X, D) and (X′, D′). To see this we have to show that there exists some a ∈ ℤ such that ψ^{∗}(D′) ∼ D + aF. In other words we have to prove the following rational equivalence holds

5.7.2  (n − 1)[σ′] + [τ′] ∼ (n − 1)[σ′ +_{E′} ρ] + [φ ∘ τ +_{E′} ρ] + aF.

If we restrict this to the generic fibre E′ of X′ then this amounts to the equality τ′ = (n − 1)ρ + φ ∘ τ + ρ in the group law of E′, which is a consequence of condition (b) of Subsection 5.4. To show that 5.7.2 holds it now suffices to show that for any irreducible component C of a fibre X′_{t} the intersection number of C with either side of 5.7.2 is the same. Here we use that by condition (b) of 5.4, ρ always meets the same irreduble component of X′_{t} as σ′. Thus translation by ρ fixes (not pointwise) the irreducible components of the reducible fibres of X′ → ℙ^{1}. We conclude that σ′ +_{E′} ρ meets the same irreducible components as σ′ and that τ′ = φ ∘ τ +_{E′} nρ = φ ∘ τ +_{E′} ρ +_{E′} (n − 1)ρ meets the same irreducible components as φ ∘ τ +_{E′} ρ. From this the desired equalities readily follow.

We have proved that C_{n,d} can be considered a subgroupoid of B_{n,d} and in particular that

5.7.3  B_{n,d} ≥ C_{n,d}.

5.8 Remark. Which pairs (X, D) ∈ B_{n,d} are not in the image of the map C_{n,d} → B_{n,d}? Clearly, any pair (X, D) such that X → ℙ^{1} does not have a section is not in the image. Now, suppose that X → ℙ^{1} does have a section, call it σ. Since the generic fibre is an elliptic curve (with σ as its zero), we get a unique section τ such that D ∼ (n − 1)[σ] + [τ] + G, where G is a divisor which is a linear combination of fiber components. If all fibres of X → ℙ^{1} are irreducible (e.g. if X is semi-stable with reduced discriminant) then (X, D) is in the image. On the other hand, if for example there is exactly one fibre which is not irreducible and which is a 2-gon, then one can show there are approximately n/2 choices of D (up to equivalence) for a given pair (σ, τ) as above.

5.9 Suppose that (E, 0) is an elliptic curve of height d over k(t) and that (X, σ) is its minimal model. We will denote r(E) the rank of E over k(t), i.e., the ℤ-rank of the Mordell-Weil group

E(k(t)) ≅ ℤ^{r(E)} ⊕ (torsion).

Let P_{1}, …, P_{r(E)} ∈ E(k(t)) be generators of the free part. Let ā = (a_{1}, …, a_{r(E)}) ∈ {0, 1, …, n − 1}^{r(E)} be a vector such that gcd(n, a_{1}, a_{2}, …, a_{r(E)}) = 1. Note that the number of such ā is n^{r(E)} − 1 if n is prime. Let τā be the section of X corresponding to the point Pā = ∑ a_{i}P_{i} (addition in the group law of E). For each ā we get an object Xā = (X, σ, τā) ∈ C_{n,d}. (Nondegeneracy is the reason for requiring gcd(n, a_{1}, …, a_{r(E)}) = 1.) By our definitions in 5.4 we see that if Xā is equivalent to Xb̄ then there exists some φ ∈ Aut(E, 0) such that Pā − φ(Pb̄) ∈ nE(k(t)). Furthermore, there is a map Aut(Xā) → {φ ∈ Aut(E, 0) | φ(Pā) − Pā ∈ nE(k(t))}. By Lemma 5.15 below and condition (c) of Subsection 5.4 this map is injective, at least if d ≥ 1. We conclude that the number of equivalence classes of Xā in C_{n,d} is at least the number of orbits of Aut(E, 0) on the set of Pā, and that # Aut(Xā) is at most the number of elements of the stabilizer of Pā in Aut(E, 0). Clearly, if E′ is another elliptic curve, E ≅ E̸ and if X′_{ā′} is one of the objects associated to E′, then Xā is not equivalent to Xā. All in all we have proved that

∑_{(E,0), h(E)=d} \frac{n^{r(E)} − 1}{# Aut(E, 0)} ≤ C_{n,d}

if n is prime. By inequalities 5.7.3 and 5.5.1 we obtain the following.

5.10 Proposition. For n ≥ 3 prime, and d ≥ 1 we have

∑_{(E,0), h(E)=d} \frac{n^{r(E)} − 1}{# Aut(E, 0)} ≤ A_{n,d}

5.11 Remarks. (a) Of course it is a simple matter to work out the inequality for any given integer n. For example, in case n = 4 one should take the quantity 4^{r(E)} − 2^{r(E)} in the denominator.

(b) In stead of the exponent r(E) the exponent r(E; n) := dim_{ℤ/nℤ} E(k(t))/nE(k(t)) may be used in the proposition. Note that r(E; n) = r(E) + dim_{ℤ/nℤ} E[n](k(t)). For an improvement involving the Selmer group, see Proposition 5.12.

5.12 We can construct elements in B_{n,d} starting with elements of the n-Selmer groups of elliptic curves over k(t) of height d. We explain this construction here, but we advise the reader to skip this Subsection on a first reading.

Let (E, 0) be an elliptic curve of height d over k(t). Let

0 → E(k(t))/nE(k(t)) → Sel_{n}(E) → \underline{|||}(E)[n] → 0

be the usual exact sequence. Recall that Sel_{n}(E) ⊂ H^{1}(k(ℙ^{1}), E[n]) is the set of cohomology classes which map to zero in H^{1}(k(t)_{v}, E) for every completion k(t)_{v} of k(t). Observe that we use flat cohomology in case n is not prime to the characteristic of k. We think of elements α ∈ H^{1}(k(t), E[n]) as represented by pair (C, θ), where C is a principal homogenous space for E and θ is a k(t)-rational point of the Picard scheme \underline{Pic}_{C}^{n}. If P is a rational point of E then the pair (C, T_{P}^{∗}(θ)) represents the same element α in H^{1}(k(t), E[n]). Here T_{P} : C → C is translation by P on C. There may not be a divisor on C representing θ, but if α ∈ Sel_{n}(E) then there always is one (see [7]). So assume α ∈ Sel_{n}(E) and let Θ ⊂ C be an effective divisor of degree n representing θ. If (C′, Θ′) is another pair representing α then there is an isomorphism ψ : C → C′ (of homogenous spaces) such that ψ^{∗}(Θ′) ∼ Θ (rational equivalence).

Next, we assume in addition that α has order n in Sel_{n}(E). For each pair (C, Θ) representing α as above we construct a pair (X, D) ∈ B_{n,d} as follows. Let f : X → ℙ^{1} be a relatively minimal elliptic surface with a given identification X_{η} = C. Since C has a point over every completion of k(t), we see that every fibre X_{t} has at least one component of multiplicity 1. By Lemma 8.4 we have properties (a), (b) of 5.3 and f_{∗}ω_{X/ℙ^{1}} ≅ O(d′) for some d′ ∈ ℤ. We claim that d′ = d = h(E).

There are two ways to prove the claim. The first, and perhaps more elegant, is to note that the scheme \underline{Pic}_{X/ℙ^{1}}^{0} is the connected component of the Néron model E of E (see e.g. Theorem 4, page 267 of [1]). Furthermore, E is the smooth locus of a minimal model of E over ℙ^{1} (see Proposition 1, page 21 of [1]). We conclude that the relative tanget bundle T_{σ}E to E along the zero section σ is canonically isomorphic to R^{1}f_{∗}O_{X} (since T_{0}(\underline{P ic}_{X_{t}}^{0}) = H^{1}(X_{t}, O)). As T_{σ}E is dual to σ^{∗}(ω_{E/ℙ^{1}}) and as R^{1}f_{∗}(O_{X}) is dual to f_{∗}ω_{X/ℙ^{1}} we get d = deg(σ^{∗}(ω_{E/ℙ^{1}})) = deg(f_{∗}(ω_{X/ℙ^{1}})) = d′.

The second method consists in finding a finite (ramified) covering S → ℙ^{1} such that the base change X_{S} → S has a section. We leave it to the reader to show that one can choose S such that the morphism S → ℙ^{1} is unramified over the discriminant locus of X → ℙ^{1} and over the discriminant locus of a relatively minimal model for E (actually these loci are the same). In this case X_{S} is a minimal model for both C ⊗ k(S) and E ⊗ k(S). The equality then follows since, by our choice of S → ℙ^{1}, the height of C (resp. E) over k(S) is [k(S) : k(t)] times d′ (resp. d). Details left to the reader.

5.13 Lemma. In the situation above. There exists an effective divisor D ⊂ X such that (a) D_{η} ∼ Θ (rational equivalence) and (b) for every closed point t ∈ ℙ^{1} there are sections σ and τ of X over O = Ô_{ℙ^{1},t} with the property that D ∼ (n − 1)[σ] + [τ] on X_{O}.

Proof. Our first approach is to set D′ ⊂ X equal to the scheme theoretic closure of Θ ⊂ X_{η} in X. We then modify D′ as follows. For each closed point t ∈ ℙ^{1} we choose a section σ of X over O; this is possible since C corresponds to an element of \underline{|||}(E). If K_{t} denotes the field of fractions of O, then we clearly have D ≅ (n − 1)[σ] + [τ] on X ⊗ K_{t} for a unique point τ ∈ X(K_{t}) = X(O). Thus D ∼ (n − 1)[σ] + [τ] + ∑ a_{t,i}C_{t,i} where the C_{t,i} are the irreducible components of X_{t}. If X_{t} is a smooth genus 1 curve then there is only one i and we may take a_{t,i} = 0.

Look at the divisor G = D−∑ a_{t,i}C_{t,i} on X and consider F = f_{∗}O_{X}(G). The sheaf F is locally free of rank n on ℙ^{1} and f^{∗}F → O_{X}(G) is surjective: namely, after base change to O we are computing f_{∗}O_{X}((n − 1)[σ] + [τ]) and Lemma 8.4 applies. Take a large twist F(a) and a section s ∈ Γ(ℙ^{1}, F(a)) = Γ(X, O_{X}(G+aF)) which does not vanish identically on any irreducible component of any singular fibre of f (again this is possible by Lemma 8.4). Then the zero scheme V (s) = D + ∑ m_{i}F_{i} is the union of a horizontal divisor D and multiples m_{i} of nonsingular fibres F_{i} of f. We leave it to the reader to show that the divisor D =⊂ X satisfies all the required properties.

All in all we conclude that (X, D) is an object of B_{n,d}. In order to count these objects we have to understand the equivalences between them. Suppose that we have another pair (E′, α′) giving rise to a pair (C′, θ′), a pair (C′, Θ′), and hence a pair (X′, D′) as above. Let ψ : X → X′ be an equivalence as in 5.2. Since C is a principal homogenous space for E and C′ is a principal homogenous space for E′, we obtain a unique isomorphism of elliptic curves φ : E → E′ (compatible with the actions and with ψ). The condition that ψ^{∗}(D′) = D + aF for some a ∈ ℤ implies that φ_{∗} : H^{1}(k(t), E[n]) → H^{1}(k(t), E′[n]) maps α to α′. In other words, the set of equivalence classes of pairs (X, D) obtained by the construction above is at least as large as the disjoint union of the orbit spaces Sel_{n}(E)/ Aut(E, 0) taken over all isomorphism classes of elliptic curves of height d.

Finally, we consider the automorphism group of the pair (X, D) in the category B_{n,d}. As we saw above there is a map

Aut(X, D) → Aut(E, O).

We claim this is injective. An automorphism ψ : X → X which induces the identity on E = J(C) is given by translation with a section ρ of the Néron model E of E. The condition that ψ^{∗}D = D + aF implies that ρ is a section of order dividing n. (This can be seen by looking at the generic fibre.) Since we chose D as in the Lemma 5.13, we can argue as in Subsection 5.7 (see the discussion involving equation 5.7.1) and we see that ρ is a section in the connected component of the Néron model of E. By Lemma 5.15 below we’re done.

The above combined with inequality 5.5.1 gives the following.

5.14 Proposition. Assume that n ≥ 3 is a prime and that d ≥ 1. Then

∑_{(E,0), h(E)=d} \frac{n^{s(E;n)} − 1}{#Aut(E, 0)} ≤ A_{n,d}.

Here we put s(E; n) = dim_{ℤ/nℤ} Sel_{n}(E).

5.15 Lemma. Let (E, 0) be an elliptic curve over ℙ^{1}. Let P be a rational torsion point of E which extends to a section of the connected component of the Néron model E of E. Then either P = 0, or the height of E is d = 0.

Proof. Let X be the minimal model of E and let σ resp. τ be the section corresponding to 0 resp. P. Consider the divisor [σ] − [τ] on X. This divisor is orthogonal to the group generated by the irreducible components of fibres. Furthermore, if P has order m on E, then m([σ] − [τ]) is trivial on the generic fibre also, hence m([σ] − [τ]) is numerically equivalent to zero. We conclude that the intersection product ([σ] − [τ])^{2} = 0. On the other hand it is equal to −2d − 2[σ] · [τ] (since both −[σ]^{2} and −[τ]^{2} compute the height of E). This is negative unless d = 0 or [σ] = [τ].

6 A geometric inequality

6.1 Propositions 5.10 and 5.14 show there is some interest in bounding the number A_{n,d}. It turns out that A_{n,d} is in a natural way a sum, and each summand is the weighted number of k-rational points in a certain Hilbert scheme. To see this, let (Y ⟶^{g}ℙ^{1}, D) be an object of A_{n,d}. By Lemma 8.4 the higher direct image R^{1}g_{∗}O_{Y} (D) = (0) and g_{∗}O_{Y} (D) is a locally free sheaf of rank n on ℙ^{1}. By Grothendieck’s Theorem on vector bundles on ℙ^{1} we may write

g_{∗}O_{Y} (D) = O(a_{1}) ⊕ O(a_{2}) ⊕ … ⊕ O(a_{n}).

We will assume that a_{1} ≤ a_{2} ≤ … ≤ a_{n}. The equivalence class of the pair (Y, D) does not determine the sequence a_{1}, …, a_{n} uniquely. Namely, the rational equivalence class of D is only well defined up to adding multiples of F, hence (a_{1}, …, a_{n}) is only well defined up to adding integer multiples of the vector (1, 1, …, 1). A complete system of invariants is (∑ a_{i} mod n, a_{2} − a_{1}, a_{3} − a_{2}, …, a_{n} − a_{n−1}) ∈ ℤ/nℤ × ℤ^{n−1}. (Or we can normalize so that a_{1} = 0 for example.) Note that for such a vector ā the scheme

Pā = Proj_{ℙ^{1}} (Sym^{∗} (O(a_{1}) ⊕ O(a_{2}) ⊕ … ⊕ O(a_{n})))

is up to isomorphism independent of the choice of ā in its equivalence class. Property (e) of Subsection 5.2 implies that Y maps to a closed subscheme of Pā. The computation of A_{n,d} therefore amounts to evaluating a sum ∑_{[ā]} (weighted # of pairs (Y, D) occuring in Pā). In this section we show that only a finite number of equivalence classes [ā] of vectors ā contribute to this sum.

The vanishing of R^{1}g_{∗}O_{Y} (D) and the splitting of g_{∗}O_{Y} (D) shows that χ(Y, O_{Y} (D)) = χ(ℙ^{1}, ⨁ O(a_{i})) = n + ∑ a_{i}. Since Y is Gorenstein, with K_{Y} = (d − 2)F we have by Riemann-Roch on Y:

χ(Y, O_{Y} (D)) = (1/2)(D^{2} − K_{Y} D) + χ(Y, O_{Y}) = (1/2)(D^{2} − (d − 2)n) + χ(Y, O_{Y})

Our assumptions and duality imply that R^{1}g_{∗}O_{Y} ≅ O(−d) and hence we can compute χ(Y, O_{Y}) = χ(ℙ^{1}, O)−χ(ℙ^{1}, O(−d)) = d. We obtain n+∑ a_{i} = (1/2)D^{2}−(1/2)n(d−2)+d. In other words D^{2} = (n − 2)d + 2(∑ a_{i}).

The formula for g_{∗}O_{Y} (D) shows that there is an effective representative G in the rational equivalence class D − a_{n}F. Note that G^{2} = (n − 2)d + 2 ∑(a_{i} − a_{n}). By the lemma below and the nondegeneracy of the pair (Y, D) we have G^{2} ≥ −[(n − 1)^{2} + 1^{2}]d, which means that

6.1.1  ∑(a_{n} − a_{i}) ≤ \binom{n}{2}d.

Clearly, this gives a bound for the number of equivalence classes of n-tuples (a_{1}, …, a_{n}).

Before we state and prove the lemma we would like to remark that in the case n = 3 we will give a direct proof of the finiteness of the set of possible triples (a_{1}, a_{2}, a_{3}) that can occur for objects of A_{3,d}. The inequalities we obtain in Section 7 are stronger than inequality 6.1.1 above. In fact they describe precisely which triples occur. The author would be interested to know which vectors ā occur for n = 4 and n = 5.

6.2 Lemma. Assume that g : Y → C is a proper flat morphism of a Gorenstein surface to a nonsingular projective curve C. Assume that ω_{Y/C} ≅ g^{∗}ω for some ω ∈ Pic(C), that g_{∗}O_{Y} = O_{C} holds universally and that the generic fibre Y_{η} of g is smooth. Let D ⊂ Y be an effective Cartier divisor such that DF = n and DC ≥ 0 for every irreducible component of every fibre of g. Then

D^{2} ≥ −n^{2} deg_{C}(ω).

If the divisor D_{η} is not n times a rational point on Y_{η} then

D^{2} ≥ −[(n − 1)^{2} + 1^{2}] deg_{C}(ω).

Proof. As a Weil divisor we can write D = D^{h} + D^{v} as a sum of a horizontal part and a vertical part. We may replace C by a finite (ramified) covering C′ of C, and hence we may assume that the horizontal part of D is a union of sections with multiplicities:

D^{h} = ∑ m_{i}σ_{i}.

(Equality of Weil divisors.) Since by assumption we have DD^{v} ≥ 0 we see that

D^{2} ≥ ∑ m_{i}D · σ_{i}.

We claim that Dσ_{i} ≥ −m_{i} deg_{C}(ω).

Fix i, set Z = σ_{i}(C) ⊂ Y and let J ⊂ O be the ideal sheaf of Z. Then from the canonical isomorphisms

O_{Z} ≅ ω_{Z/C} ≅ R Hom_{O_{Y}} (O_{Z}, ω_{Y/C}) ≅ Coker[ω_{Y/C} → Hom_{O_{Y}} (J , ω_{Y/C})] we obtain a map

J /J^{2} → ω_{Y/C} ⊗ O_{Z}

which is an isomorphism up to torsion (this and similar statements later on follow by looking at the generic fibre where g is smooth). Furthermore, since Z occurs with multiplicity m_{i} in D there is an injection O(−D) → J^{m_{i}}. Upon composing these maps we obtain a map

O(−D) ⊗ O_{Z} → J^{m_{i}}/J^{m_{i}+1} → ω_{Y/C}^{m_{i}} ⊗ O_{Z} ≅ ω^{m_{i}}

which is an isomorphism up to torsion. The claim follows.

From the claim and the inequality directly above it we get D^{2} ≥ −(∑ m_{i}^{2}) deg_{C}(ω). The result of the lemma follows from elementary number theory.

7 Degree three surfaces

7.1 In this section we fix integers d ≥ 0 and b ≥ a ≥ 0. As in Subsection 6.1 we consider the projective variety

P = P_{a,b} = Proj_{ℙ^{1}} (Sym^{∗} (O ⊕ O(a) ⊕ O(b))) ⟶^{π}ℙ^{1}.

As a matter of normalization we take O_{P} (1) to be the invertible sheaf on P_{a,b} such that π_{∗}O_{P} (1) = O ⊕ O(a) ⊕ O(b). Recall that there is a canonical isomorphism ω_{P/ℙ^{1}} = O_{P} (−3) ⊗ π^{∗}(O(a + b)). (Here O(a + b) is more canonically the highest exterior power of O ⊕ O(a) ⊕ O(b).)

As explained in Section 6 we want to count closed subschemes Y ⊂ P_{a,b}, so that (Y, D) occurs in A_{3,d}, where O_{Y} (D) ≅ O_{P} (1)|_{Y}. Since P_{a,b} is a smooth projective threefold, any such Y is a divisor on P_{a,b}.

7.2 Lemma. For any Y as above there is a nonzero section F of Γ(P_{a,b}, O_{P} (3)⊗π^{∗}O(d− a − b)) such that Y is the zero scheme of F: Y = V (F).

Proof. Clearly Y corresponds to a section of O_{P} (3) ⊗ π^{∗}O(ℓ) for some integer ℓ. Adjunction for the immersion Y → P_{a,b} shows that ω_{Y/ℙ^{1}} is the restriction of ω_{P/ℙ^{1}} ⊗ O_{P} (Y) ≅ π^{∗}(a + b + ℓ) to Y. On the other hand, by the axioms for objects of A_{3,d}, we have ω_{Y/ℙ^{1}} ≅ g^{∗}O(d). The result follows.

7.3 Next, we work out the shape of the polynomial F, much like in the case of Weierstrass polynomials. Namely, there are canonical isomorphisms

π_{∗} [O_{P} (3) ⊗ π^{∗}O(d − a − b)] = Sym^{3} (O ⊕ O(a) ⊕ O(b)) ⊗ O(d − a − b)
= O(d − a − b) ⊕ O(d − b) ⊕ O(d − a) ⊕ O(d + a − b) ⊕ O(d)
⊕O(d − a + b) ⊕ O(d + 2a − b) ⊕ O(d + a) ⊕ O(d + b) ⊕ O(d − a + 2b).

What this really means is that we can write symbolically

F = Ay^{3} + By^{2}x + Cy^{2}z + Dyx^{2} + Exyz + F yz^{2} + Gx^{3} + Hx^{2}z + Ixz^{2} + Jz^{3},

where A is homogenous of degree d−a−b, B is homogenous of degree d−b, C is homogenous of degree d − a, D is homogenous of degree d + a − b, etc. (In other words, we think of the summands of O ⊕ O(a) ⊕ O(b) as having the generators y, x and z.) Observe that

• If a + b > d then A = 0, hence the point “(0 : 1 : 0)” defines a section σ of Y over ℙ^{1}.

• If a > d, then A, B and C are all zero (recall that b ≥ a). Thus Y is singular along σ which is a contradiction.

• If d + a − b < 0, then A = 0, B = 0 and D = 0. This means that σ is a flex point of the generic fibre of Y → ℙ^{1}. (The flex line is z = 0.) This contradicts the nondegeneracy assumption for elements of A_{3,d}.

By the above we see that the pairs (a, b) that will occur are subject to the following system of inequalities

7.3.1  \{ \begin{matrix}0 ≤ a ≤ b, \\ a ≤ d, b ≤ d + a. \end{matrix}

7.4 Let G_{a,b} = Aut(O ⊕O(a)⊕O(b)) be the automorphism group of the sheaf O ⊕O(a)⊕ O(b) over ℙ^{1}. Clearly, k^{∗} ⊂ G_{a,b} is a central subgroup and we set P G_{a,b} = G_{a,b}/k^{∗}. The group G_{a,b} acts on the pair (P_{a,b}, O_{P} (1)), and hence it acts on the vector space

V_{a,b} = Γ(P, O_{P} (3) ⊗ π^{∗}O(d − a − b)).

The action of the subgroup k^{∗} ⊂ G_{a,b} on V_{a,b} is by scalars and hence the group P G_{a,b} acts on the projective space ℙ(V_{a,b}) of lines in V_{a,b}.

Suppose that Y ⊂ P_{a,b} and Y′ ⊂ P_{a,b} are two closed subvarieties with divisors D and D′ having the property (as in 7.1) that O_{Y} (D) ≅ O_{P} (1)|_{Y} and O_{Y′}(D) ≅ O_{P} (1)|_{Y′}. Suppose moreover that these correspond to objects of A_{3,d}. Hence we obtain corresponding elements F and F′ of V_{a,b} and corresponding lines [F] and [F′] in ℙ(V_{a,b}). We claim that

7.4.1  \begin{matrix}Mor_{A_{n,d}}((Y, D), (Y′, D′)) = & \{g ∈ P G_{a,b} | g(Y) ⊂ Y′\} \\ = & \{g ∈ P G_{a,b} | g([F]) = [F′]\}. \end{matrix}

We leave this to the reader as an easy consequence of (all) the definitions.

7.5 Proposition. The number A_{3,d} can be bounded above by the following expression

∑_{(a,b)∈7.3.1} \frac{#ℙ(V_{a,b})}{#P G_{a,b}}.

Proof. This is a combination of the results obtained so far. In 6.1 we explained that A_{3,d} can be computed as a sum over equivalence classes of vectors (a_{1}, a_{2}, a_{3}). In this section we have normalized these such that a_{1} = 0 (possible). In 7.3 we showed that the inequalities 7.3.1 hold. Finally, the formula 7.4.1 shows that 1/#P G_{a,b} is the correct weight to associate to points of ℙ(V_{a,b}).

7.6 Remark. We expect that, for fixed q and large d, the bound of the proposition is asyptotically γ_{q}A_{3,d} for some constant γ_{q}. Furthermore, we expect that the constant γ_{q} will tend to 1 as q → ∞. The only evidence we are able to give the reader is the similar phenomenon that is explained in Remark 4.22. Furthermore, we expect similar factors γ′′_{q} and γ′′_{q} such that A_{3,d} ∼ γ′_{q}B_{3,q} and such that B_{3,d} = γ′′_{q} ∑_{h(E)=d} [(3^{s(E;3)} − 1)/ Aut(E, 0)]. This would imply that the inequality in the main theorem of the introduction is an asymptotic equality up to a factor γ_{q}γ′_{q}γ′′_{q} which tends to 1 as q → ∞. However, as the major goal of the author was to prove an upper bound, we have not worked this out.

7.7 In this last part we compute the numbers #ℙ(V_{a,b}) and #P G_{a,b} and sum up the quotients to get an explicit upper bound for A_{3,d} using the inequality above.

Trivial calculations show that for pairs satisfying 7.3.1 we have

dim V_{a,b} = \{ \begin{matrix}10d + 10 & if & d ≥ a + b \\ 10d + 10 + (a + b − d − 1) & if & d < a + b and d ≥ b \\ 10d + 10 + (a + b − d − 1) + (b − d − 1) & if & d < b \end{matrix}

Further, the number of elements of P G_{a,b} is given as follows:

#P G_{a,b} = \{ \begin{matrix}(q − 1)^{2}q^{2b+3} & if & 0 < a < b, \\ (q − 1)^{2}(q + 1)q^{2b+3} & if & 0 = a < b, \\ (q − 1)^{2}(q + 1)q^{2b+3} & if & 0 < a = b, \\ (q^{3} − 1)(q^{3} − q)q^{2} & if & 0 = a = b. \end{matrix}

7.8 Remark. It is reasonable to associate to a pair (a, b) a dimension, namely the highest exponent of q in the expression #ℙ(V_{a,b})/#P G_{a,b} (when written out as a Laurent series in q). It turns out, as the reader can verify by hand, that (for a fixed d) the maximum dimension is 10d + 1, and this is attained by the three triples (0, 0), (1, 1) and (0, 1). All other triples (satisfying 7.3.1) have smaller dimension (expect in the case d = 1 where there is the exception (a, b) = (1, 2)). Morally speaking this means that the moduli space of objects as in Subsection 5.2 has three irreducible components of dimension 10d + 1. Clearly, for the large q limit version of the problem this is enough information to proceed and obtain a reasonable upper bound. It is the fact that we can estimate precisely all the error terms that gives us the upper bound for fixed q and large d.

7.9 Proposition.

lim_{d→∞} q^{−10d−1} ( ∑_{(a,b)∈7.3.1} \frac{#ℙ(V_{a,b})}{#P G_{a,b}} ) = \frac{3q^{9}}{(q^{3} − 1)(q^{2} − 1)^{2}(q − 1)^{2}}

Proof. We claim that the sum over the pairs (a, b) satisfying 7.3.1 and a + b > d and d ≥ b is vanishingly small. Namely the contribution of this term is (if 0 < a < b)

q^{−10d−1} \frac{q^{10d+10+(a+b−d−1)} − 1}{(q − 1)^{3}q^{2b+3}} ≤ \frac{q^{5−d}}{(q − 1)^{3}} .

The number of pairs like this is ≤ d^{2} which proves the result. (In the cases a = 0 or a = b the inequality is even a little better.) Similarly, the sum over the pairs (a, b) satisfying 7.3.1 and d < b is vanishingly small; again if 0 < a < b we have a contribution of

q^{−10d−1} \frac{q^{10d+10+(a+b−d−1)+(b−d−1)} − 1}{(q − 1)^{3}q^{2b+3}} ≤ \frac{q^{4−d}}{(q − 1)^{3}} .

The same argument applies.

Now we may assume that d ≥ a + b. The cases (a, b) = (0, 0), (1, 1), and (0, 1) contribute in the limit

\frac{q^{6}}{(q^{3} − 1)(q^{2} − 1)(q − 1)} + \frac{2q^{4}}{(q^{2} − 1)(q − 1)^{2}} = \frac{3q^{6} + 2q^{5} + 2q^{4}}{(q^{3} − 1)(q^{2} − 1)(q − 1)}.

For every fixed b ≥ 2 the pairs (a, b) with 0 ≤ a ≤ b contribute to the limit the quantity

\frac{(b − 1)q^{9}}{(q − 1)^{3}q^{2b+3}} + \frac{2q^{9}}{(q^{2} − 1)(q − 1)^{2}q^{2b+3}} = q^{−2b} \frac{b(q^{9} + 2 q^{8} + 2 q^{7} + q^{6}) − q^{9} + q^{6}}{(q^{3} − 1)(q^{2} − 1)(q − 1)} .

Since it is easy to sum the series ∑ bq^{−2b} and ∑ q^{−2b} it is a simple computation to reach the result as stated in the proposition.

8 Appendix A: Genus 1 curves and divisors

In this appendix we prove a few lemmas on linear systems on curves of genus 1. We find it amusing to deduce these from certain natural results on zero dimensional Gorenstein algebras (we prove these first). The reader can find her/his own proofs of the lemmas which avoid the use of duality theory if she/he so desires.

8.1 Let k be a field and let A be a finite dimensional k-algebra. Set ω_{A} = Hom_{k}(A, k) and let Tr : ω_{A} → k be the evaluation at 1 map. As usual we say that A is Gorenstein if ω_{A} is isomorphic to A as an A-module.

8.2 Lemma. If dim_{k}(A) ≥ 2 then V = Ker(Tr) generates ω_{A} as an A-module.

Proof. If V does not generate ω_{A} then it is an A-submodule. Since dim_{k}(A) ≥ 2 we see that V ≠ (0). However, the pairing A × ω_{A} → k, (a, η) ↦ Tr(aη) is nondegenerate. Hence, ω_{A} has no nontrivial submodules contained in Ker(Tr).

8.3 Lemma. If A is Gorenstein, and dim_{k}(A) ≥ 3 then the multiplication map

Ker(Tr) ⊗_{k} Ker(Tr) → ω_{A} ⊗_{A} ω_{A} ≅ A,

is surjective.

Proof. The previous lemma shows that V = Ker(Tr) contains a generator of ω_{A}, at least if k is infinite. Thus we can choose an isomorphism φ : A → ω_{A} which maps 1 into V. The map in the lemma is not surjective, if and only if B = φ^{−1}(V) is a subalgebra. In this case V is a B-submodule of ω_{A}. Also, the trace map factors as

ω_{A} → ω_{B} → k

where the first map is a B-module map. Since dim B ≥ 2 we see that Ker(Tr) is not a B-submodule, a contradiction.

If k is finite, then the result for A ⊗_{k} k̄/k̄ implies the result for A/k.

8.4 Lemma. Let X be a projective Gorenstein curve over a field k such that ω_{X/k} ≅ O_{X} and H^{0}(X, O_{X}) = k. Let D ⊂ X be an effective Cartier divisor.

(a) If ℓ(D) ≥ 2, then O_{X}(D) is globally generated and H^{1}(X, O(D)) = (0).

(b) If ℓ(D) ≥ 3, then the graded k-algebra R := ⨁_{n≥0} Γ(X, O(nD)) is generated in degree 1.

Proof. We may work with the invertible O_{X}-module ω_{X}(D) in stead of O_{X}(D). We consider the canonical exact sequence

0 → ω_{X} → ω_{X}(D) → ω_{D} → 0.

The boundary map composed with the trace map on H^{1}(ω) is the trace map Tr_{D} on ω_{D}:

ω_{D}⟶^{δ}H^{1}(X, ω_{X})⟶^{Tr}k

See [5]. The map H^{1}(X, ω_{X}) → k is an isomorphism as it is dual to the map k → H^{0}(X, O_{X}). Hence (a) follows from Lemma 8.2. Similarly the surjectivity of the map Γ(X, ω(D)) ⊗ Γ(X, ω(D)) → Γ(X, ω^{⊗2}(2D)) follows from Lemma 8.3 upon considering the following commutative diagram

Γ(X, ω_{X}(D)) ⊗ Γ(X, ω(D)) → ω_{D} ⊗_{k} ω_{D}
↓  ↓
Γ(X, ω_{X}(D)) →  Γ(X, ω_{X}^{⊗2}(2D))  →  ω_{D}^{⊗2}.

Indeed, the upper horizontal arrow has image Ker(Tr_{D}) ⊗ Ker(Tr_{D}). The surjectivity of the maps Γ(X, ω^{⊗n}(nD)) ⊗ Γ(X, ω(D)) → Γ(X, ω^{⊗(n+1)}((n + 1)D)) is easier and left to the reader.

8.5 Lemma. In the situation described in (b) of the lemma above, the scheme Y = Proj(R) is a Gorenstein curve with ω_{Y} ≅ O_{Y} and H^{0}(Y, O_{Y}) = k. The morphism X → Y is dominant and induces an isomorphism of an open subscheme of X with a dense open subscheme of Y.

Proof. We may assume that k is algebraically closed. Let T = 1 ∈ Γ(X, O(D)) be the canonical section. We think of T as an element of degree 1 in R. The proof of the lemma above shows that T is a not a zero divisor and that

R/T R = k ⊕ V ⊕ ω_{D}^{2} ⊕ ω_{D}^{3} ⊕ …

where V = Ker(Tr_{D}) ⊂ ω_{D}. Recall also that V contains a generator, call it S̄, of ω_{D}. We will use this generator to identify ω_{D} with A = O_{D}. (See the proof of Lemma 8.3.) Clearly, S̄ is a not a zero divisor on R/T R. Let S ∈ R be an element of degree 1 mapping to S̄. Then {T, S} is a regular sequence and

R/(T, S) ≅ k ⊕ V/k ⊕ A/V.

It is easy to see that this is a graded Gorenstein ring with dualizing module ω_{R/(T,S)} ≅ R/(T, S)(2). It follows from [3, 3.1.19(b) & 3.6.14] that R is Gorenstein with ω_{R} ≅ R (no shift). This implies that ω_{Y} ≅ O_{Y}.

The argument above implies that Y has dimension 1. It is a general fact that φ : X → Y is dominant; in fact O_{Y} → φ_{∗}O_{X} is an isomorphism. (This follows as O_{Y} (1) is ample and Γ(Y, O_{Y} (n)) = Γ(X, O_{X}(nD)).) This implies that H^{0}(Y, O_{Y}) = k. Let Z ⊂ Y be a finite closed subset which contains the images of irreducible components of X which are collapsed under the morphism φ : X → Y. It is easy to show that φ : X \ φ^{−1}Z → Y \ Z is an isomorphism.

8.6 Remark. By Riemann-Roch we have dim_{k} H^{1}(Y, O_{Y}) = 1. The fact that O_{Y} = φ_{∗}O_{X} implies, via the Leray spectral sequence, that R^{1}φ_{∗}O_{X} = (0) and that the canonical map H^{1}(Y, O_{Y}) → H^{1}(X, O_{X}) is an isomorphism. The dual of this map is a canonical isomorphism Γ(X, ω_{X}) = Γ(Y, ω_{Y}).

8.7 Remark. In case ℓ(D) ≥ 4 the equations for Y in the imbedding given by O_{Y} (1) are quadratic. Presumably, when ℓ(D) ≥ 5 the relations between the quadratic equations will all be linear, etc. This remark is not used in the text.

9 Appendix B: Elliptic surfaces

The following condition on elliptic surfaces f : X → C turns out to be useful in the text

(†) For every closed point t ∈ ℙ^{1} we have H^{0}(X_{t}, O) = κ(t) and ω_{X_{t}} ≅ O. Another way to say this is that f is cohomologically flat in dimension 0, and that ω_{X/C} ≅ f^{∗}ω for some invertible O_{C}-module ω. Thus the height of X is defined, namely h(X) = deg_{C}(ω). An elliptic surface satisfying (†) is automatically relatively minimal over ℙ^{1}, and, in characteristic 0 at least, this condition excludes multiple elliptic fibres. The following lemma shows in particular that (†) holds whenever X → C has (étale locally) a section.

9.1 Lemma. Write X_{t} = ∑ r_{i}C_{i}, and assume that κ(t) is perfect. If X is relatively minimal and gcd(r_{i}) = 1, then (†) holds for X_{t}.

Proof. We will use that the intersection form on the group of divisors ⨁ ℤC_{i} is seminegative definite with kernel the (rational) multiples of X_{t}.

Since X_{t} is geometrically connected and κ(t) is perfect, we see that Γ(X_{t}, O) is an Artin local ring with residue field κ(t). If f ∈ Γ(X_{t}, O) is nilpotent, then let Z ⊂ X_{t} be the largest divisor such that f|_{Z} = 0. Set Y = X_{t}−Z and note that f is a section of O_{X}(−Z)|_{Y} with a finite number of zeros. Hence we deduce that deg(O_{X}(−Z)|_{Y}) = Y^{2} ≥ 0. This is possible only if Y is a (fractional) multiple of X_{t}. Since gcd(r_{i}) = 1 we conclude that Y = ∅ or Y = X_{t}. In the first case f = 0 (and this is what we want) and in the second case f is not nilpotent (a contradiction).

The fact that ω_{X_{t}} ≅ O_{X_{t}} follows in the same way. Since χ(X_{t}, O) = χ(ω_{X_{t}}) = 0 we see that Γ(X_{t}, ω_{X_{t}}) ≠ (0). Take any nonzero section s of ω_{X_{t}} and consider the largest divisor Z ⊂ X_{t} on which s vanishes. Set Y = X_{t} − Z and note that deg(ω(−Z)|_{Y}) ≥ 0. Since X is minimal we have deg(ω|_{C_{i}}) = 0 for all i. Thus we conclude that Y^{2} = 0 and Y is a (fractional) multiple of X_{t}. We finish as above.

10 Appendix C: Estimate for Selmer

In the proof of Corollary 1.3 we need an estimate for the size of the Selmer group of an elliptic curve (E, 0) over k(t) with constant j-invariant. This is well known, but since it is amusing to deduce it from the description of these curves given in 4.13 we do this here.

In the following lemma, the variety C may not be geometrically irreducible over k (hence our conventions do not allow us to call C a curve).

10.1 Lemma. There exists an absolute constant κ such that every elliptc curve (E, 0) over k(t) of height d and constant j-invariant there is a finite morphism of nonsingular projective schemes C → ℙ^{1} such that (a) the extension k(t) ⊂ k(C) is separable and [k(C) : k(t)] ≤ 24, (b) dim_{k} H^{1}(C, O_{C}) ≤ κd, and (c) E ⊗_{k(t)} k(C) ≅ E_{0} ⊗_{k} k(C) for some elliptic curve (E_{0}, 0) over k.

Proof. This follows directly from the shape of the Weierstrass polynomials in Subsection 4.13. Let X_{0}, X_{1} be homogenous coordinates on ℙ^{1}. Then, for example in the case (a1) it is clear that if we take C to be the curve in the weighted projective space ℙ(1, 1, d) (with coordinates X_{0}, X_{1}, X_{2}) given by the equation X_{2}^{2} = A′_{2}. The curve E_{0} will have the Weierstrass equation y^{2} = x^{3} + α_{4}x + α_{6}. Similarly for the cases (a2), (a3), (b1).

The cases (b2), (c1), and (c2) are a little harder. For example, in the hardest case (c2) one takes C to be the normalization of an irreducible component of closed subscheme Z of the weigthed projective space ℙ(1, 1, d, d, d) with coordinates X_{0}, X_{1}, B_{1}, B_{3}, X_{2} given by the following system of equations

\{ \begin{matrix}A′_{4} + A′_{3}B_{1} + B_{1}^{4} & = 0, \\ A′_{6} + B_{3}^{2} + A′_{3}B_{3} + A′_{4}B_{1}^{2} + A′_{3}B_{1}^{3} & = 0, \\ X_{2}^{3} & = A′_{3}. \end{matrix}

The nonvanishing of A′_{3} implies that π : Z → ℙ^{1} is generically étale; it is clearly finite. The degree of π is 4 · 2 · 3 = 24. Furthermore, it is easy to see that π_{∗}O_{Z} = ⨁_{0≤ℓ≤3,0≤m≤1,0≤n≤2} O_{ℙ^{1}}(−(ℓ + m + n)d) (the summand with index (ℓ, m, n) corresponds somehow to the monomial B_{1}^{ℓ}B_{3}^{m}X_{2}^{n}). This proves that dim_{k} H^{1}(Z, O_{Z}) is bounded by 72d. This implies the bound for the genus of C. (Presumably the constant κ is actually a little better than 72.)

10.2 Lemma. Same hypothesis as in Lemma 10.1. There exists an absolute constant κ′ such that for any prime number ℓ the rank of the ℓ-Selmer group of E is bounded as follows: s(E; n) ≤ κ′d.

Proof. Choose C → ℙ^{1} as in the previous lemma. Let L ⊃ k(t) be a Galois extension with group G which is a normal closure of k(C) over k(t). Clearly #G ≤ 24. Elements of the kernel of H^{1}(k(t), E[n]) → H^{1}(k(C), E[n] ⊗ k(C)) correspond to elements of the group H^{1}(G, E[n](L)) which is universally bounded. Since the map H^{1}(k(t), E[n]) → H^{1}(k(C), E[n] ⊗ k(C)) maps Sel_{n}(E/k(t)) into Sel_{n}(E ⊗ k(C)/k(C)) it now suffices to bound Sel_{n}(E_{0} ⊗ k(C)/k(C)).

Let k′ ⊂ k(C) be the algebraic closure of k in k(C). Then C is a nonsingular curve over k′ (see 3.2). Clearly the genus of C (as a curve over k′) is ≤ dim_{k} H^{1}(C, O_{C}) and hence we are reduced to estimating the Selmer group of a constant elliptic curve over the function field of a curve C in terms of the genus of C.

For a constant elliptic curve E_{0} over a function field k(C) of a nonsingular projective curve C, there is a canonical equality Sel_{n}(E_{0}/k(C)) = H^{1}(C, E_{0}[n]). Here we use flat cohomology in case n is the characteristic of k. In the case n is prime to the characteristic of k a bound can easily be obtained by using that dim_{ℤ/nℤ} H^{1}(C ⊗ k̄, E_{0}[n]) = 4g(C). In the case n = p is the characteristic of k we have, in case E_{0} is ordinary, H^{1}(C ⊗k̄, E_{0}[p]) = H^{1}(C ⊗ k̄, μ_{p}) ⊕ H^{1}(C ⊗ k̄, ℤ/pℤ). Again we obtain the desired bound since each of the summands has rank at most g. In case E_{0} is supersingular, we have an exact sequence 0 → α_{p} → E_{0}[p] → α_{p} → 0 and H^{1}(C, α_{p}) has at most rank g + 1 (use the Artin-Schreier sequence over C, not over C ⊗ k̄).

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