Characterizing projective spaces on deformations of Hilbert schemes of K3 surfaces

DAVID HARVEY

Courant Institute

BRENDAN HASSETT

Rice University

AND YURI TSCHINKEL

Courant Institute

Abstract

We seek to characterize homology classes of Lagrangian projective spaces embedded in irreducible holomorphic-symplectic manifolds, up to the action of the monodromy group. This paper addresses the case of manifolds deformationequivalent to the Hilbert scheme of length-three subschemes of a K3 surface. The class of the projective space in the cohomology ring has prescribed intersection properties, which translate into Diophantine equations. Possible homology classes correspond to integral points on an explicit elliptic curve; our proof entails showing that the only such point is two-torsion. \begin{matrix}c \\ ◯ \end{matrix} 2000 Wiley Periodicals, Inc.

1 Introduction

Let X be an irreducible holomorphic symplectic manifold, i.e., a compact Kähler simply-connected manifold admitting a unique nondegenerate holomorphic twoform. Let (,) denote the Beauville–Bogomolov form on the cohomology group H^{2}(X,ℤ), normalized so that it is integral and primitive. When X is a K3 surface this coincides with the intersection form. In higher dimensions, the form induces an inclusion

(1.1)  H^{2}(X,ℤ) ⊂ H_{2}(X,ℤ),

which allows us to extend (,) to a ℚ-valued quadratic form.

Lagrangian projective spaces play a fundamental rôle in the birational geometry of these classes of manifolds. If X contains a holomorphically embedded projective space ℙ^{dim(X)/2} we can consider the Mukai flop of X, obtained by blowing up the projective space and blowing down the exceptional divisor

E ≃ ℙ(Ω_{ℙ^{dim(X)/2}}^{1})

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0022 (2000) \begin{matrix}c \\ ◯ \end{matrix} 2000 Wiley Periodicals, Inc.

along the opposite ruling. Our goal is to characterize possible homology classes of such submanifolds, modulo the monodromy representation on the cohomology of X.

Assuming X contains a Lagrangian projective space ℙ^{dim(X)/2}, let ℓ ∈ H_{2}(X,ℤ) denote the class of a line in ℙ^{dim(X)/2}, and λ = Nℓ ∈ H^{2}(X,ℤ) a positive integer multiple. We can take N to be the index of H^{2}(X,ℤ) ⊂ H_{2}(X,ℤ). Hodge theory [19, 24] shows that the deformations of X containing a deformation of the Lagrangian space coincide with the deformations of X for which λ ∈ H^{2}(X,ℤ) remains of type (1,1). Infinitesimal Torelli implies this is a divisor in the deformation space, i.e.,

λ^{⊥} ⊂ H^{1}(X,Ω_{X}^{1}) ≃ H^{1}(X,T_{X}).

We seek to establish intersection theoretic properties of ℓ for various deformationequivalence classes of holomorphic symplectic manifolds. Previous results in this direction include

(1) If X is a K3 surface then (ℓ,ℓ)= −2.

(2) If X is deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface then (ℓ,ℓ)= −5/2. [11]

(3) If X is deformation equivalent to a generalized Kummer fourfold then

(ℓ,ℓ)= −3/2. [12] Here we prove

Theorem 1.1. Let X be a six-dimensional Kähler manifold, deformation equivalent to the Hilbert scheme of length-three subschemes of a K3 surface. Let ℙ^{3} ⊂ X be a smooth subvariety and ℓ ⊂ ℙ^{3} a line. Then (ℓ,ℓ)= −3 and ρ = 2ℓ ∈ H^{2}(X,ℤ). Furthermore, we have

[ℙ^{3}] = \frac{1}{48} (ρ^{3}+ ρ^{2}c_{2}(X)) .

This uniquely characterizes the class of the Lagrangian plane, modulo the monodromy action, which acts transitively on the ρ ∈ H^{2}(X,ℤ) with (ρ,ρ) = −12 and (ρ,H^{2}(X,ℤ))= 2ℤ [9, §3].

In general, we conjectured in [13] that if X is of dimension 2n then

(ℓ,ℓ)= −(n + 3)/2,

if X is deformation equivalent to a Hilbert scheme of a K3 surface. Our main motivation for making these conjectures is to achieve a classification of extremal rational curves on irreducible holomorphic symplectic varieties (i.e., generators of extremal rays of birational contractions) in terms of intersection properties under the Beauville-Bogomolov form.

The structure of this paper is as follows: Section 2 reviews the cohomology groups of Hilbert schemes of K3 surfaces; Section 3 focuses on the ring structure. We employ representation theory to get results on the Hodge classes in Section 4. The Hilbert scheme of length-three subschemes is studied in detail in Section 5. We extract the distinguished absolute Hodge class in the middle cohomology in Section 6. Here ‘absolute Hodge classes’ are classes that remain Hodge under arbitrary deformations of complex structure; their span is stable under the monodromy representation, which is why we use this terminology (cf. [4, p. 28]). The computation of the class of the Lagrangian three planes is worked out in Section 7, modulo a number theoretic result. This is proved in Section 8.

2 Cohomology of Hilbert schemes

Let X be deformation equivalent to the punctual Hilbert scheme S^{[n]}, where S is a K3 surface. For n > 1 the Beauville-Bogomolov form can be written [1, §8]

H^{2}(X,ℤ) ≃ H^{2}(S,ℤ)_{(,)}⊕_{⊥} ℤδ, (δ,δ) = −2(n − 1)

where 2δ is the class of the ‘diagonal’ divisor Δ^{[n]}⊂ S^{[n]}parameterizing nonreduced subschemes. For each cohomology class f ∈ H^{2}(S,ℤ), let f ∈ H^{2}(X,ℤ) denote the class of the locus parameterizing subschemes with some support along f. This is compatible with the lattice embedding above. Duality gives a ℚ-valued form on homology

H_{2}(X,ℤ) ≃ H_{2}(S,ℤ)_{(,)}⊕_{⊥} ℤδ^{∨}, (δ^{∨},δ^{∨}) = −\frac{1}{2(n − 1)} ,

where δ^{∨} is characterized as the homology class orthogonal to H^{2}(S,ℤ) and satisfying δ^{∨} · δ = 1.

Theorem 2.1. [8] Let S be a K3 surface and S^{[n]}its Hilbert scheme. Consider the Poincaré polynomial

p(S^{[n]},z) = ∑_{j=0}^{4n} β_{j}(S^{[n]})z^{j},

where β_{j}(S^{[n]}) denotes the jth Betti number of S^{[n]}. Then

∑_{n=0}^{∞} p(S^{[n]},z)t^{n} = ∏_{m=1}^{∞}(1 − z^{2m−2}t^{m})^{−1} (1 − z^{2m}t^{m})^{−22} (1 − z^{2m+2}t^{m})^{−1} .

To save space, we write

q(S^{[n]},z) = ∑_{j=0}^{n} β_{2j}z^{j},

which determines the Poincaré polynomial by Poincaré duality. We have

\begin{aligned}q(S,z) & = & 1 + 22z \\ q(S^{[2]},z) & = & 1 + 23z + 276z^{2} \\ q(S^{[3]},z) & = & 1 + 23z + 299z^{2}+ 2554z^{3}. \end{aligned}

A theorem of Verbitsky [23, Theorem 1.5] asserts that the homomorphism arising from the cup product

μ_{k,n}: Sym^{k}H^{2}(S^{[n]},ℚ) → H^{2k}(S^{[n]},ℚ)

is injective for k ≤ n. Thus its image has dimension

\binom{22 + k}{k} .

In light of the computations above, μ_{2,2} is an isomorphism, μ_{2,3} has cokernel of dimension 23, and μ_{3,3} has cokernel of dimension

2554 − \binom{25}{3} = 254 = \binom{23}{2} + 1.

The cup product also induces a homomorphism

coker(μ_{2,3}) ⊗ H^{2}(S^{[3]},ℚ) → coker(μ_{3,3}).

This homomorphism has been observed by Markman [16, p. 80]. More generally, he analyzes what classes are needed to generate the cohomology ring H^{∗}(S^{[n]},ℚ), beyond those coming from H^{2}(S^{[2]},ℚ). Markman uses Chern classes of universal sheaves over the product S^{[n]}×S; a detailed discussion of the n = 3 case is given in [16, Ex. 14].

3 The ring structure on cohomology

Lehn-Sorger [15] and Nakajima [18] described H^{∗}(S^{[n]},ℚ) in terms of H^{∗}(S,ℚ). We review the Lehn-Sorger formalism for the cup product on the cohomology ring.

Let S be a K3 surface and A = H^{∗}(S,ℚ)(1), the cohomology ring shifted so that it has weights −2,0, and 2; this is written as H^{∗}(S,ℚ)[2] in their paper. Shifting the weights changes the sign of the intersection form, which is denoted by ⟨,⟩; this has signature (20,4). Let T : A → ℚ denote the linear form

γ ↦ −∫_{S}γ

and ⟨,⟩the induced bilinear form

⟨γ_{1},γ_{2}⟩= T (γ_{1}γ_{2}) = −∫_{S}γ_{1}γ_{2}.

For each n ∈ ℕ, we endow A^{⊗n} with an analogous structure. We shall use the fact that A has only graded pieces of even degrees to simplify the description in [15]. In this situation, graded commutative multiplication rules are in fact commutative, given by the rule

(a_{1}⊗ ··· ⊗ a_{n}) · (b_{1}⊗ ··· ⊗ b_{n}) = (a_{1}b_{1}) ⊗ ··· ⊗ (a_{n}b_{n}).

The linear form

T : A^{⊗n} → ℚ

is defined by

T (a_{1}⊗ ··· ⊗ a_{n}) = T (a_{1})··· T (a_{n}).

Let ⟨,⟩denote the associated bilinear form

⟨a,b⟩= T (a · b).

The symmetric group 𝔖_{n} acts on A^{⊗n} by the rule

π(a_{1}⊗ ··· ⊗ a_{n}) = a_{π^{−1} (1)} ⊗ ··· ⊗ a_{π^{−1} (n)}.

Given a partition n = n_{1} + ...+ n_{k} with n_{1},...,n_{k} ∈ ℕ, we have a generalized multiplication map

\begin{matrix}A^{⊗n} & → & A^{⊗k} \\ a_{1}⊗ ··· ⊗ a_{n} & ↦ & (a_{1}··· a_{n_{1}}) ⊗ ··· ⊗ (a_{n_{1}+··· +n_{k−1} +1} ··· a_{n_{1}+··· +n_{k}} ). \end{matrix}

Given a finite set I ⊂ {1,...,n}, let A^{⊗I} denote the tensor power with factors indexed by elements of I. Given a surjection ϕ : I → J, there is an induced multiplication

ϕ^{∗}: A^{⊗I} → A^{⊗J}

defined as above. Let

ϕ_{∗} : A^{⊗J} → A^{⊗I}

denote the adjoint of ϕ^{∗}, i.e.,

⟨ϕ^{∗}a,b⟩= ⟨a,ϕ_{∗}b⟩

for a ∈ A^{⊗I} and b ∈ A^{⊗J}.

We have the composite

A →^{Δ_{∗}} A ⊗ A → A,

where the first map is adjoint comultiplication and the second is multiplication. Let e:= e(A) denote the image of 1 under the composed map.

Remark 3.1. We evaluate the signs of Δ_{∗}1 and e(A). Let Δ_{S} denote the fundamental class of the diagonal in H^{∗}(S × S,ℤ) = H^{∗}(S,ℤ) ⊗ H^{∗}(S,ℤ). Using the adjoint property, we have

⟨Δ_{∗}1,α ⊗ β⟩ = ⟨1,αβ⟩
= T (αβ)
= −∫_{S}αβ

whereas

⟨Δ_{S},α ⊗ β⟩ = ⟨∑_{j} e_{j} ⊗ e_{j}^{∨},α ⊗ β⟩
= ∑_{j} T (e_{j}α)T (e_{j}^{∨}β)
= ∫_{S}αβ,

where {e_{j}} is a homogeneous basis for H^{∗}(S,ℚ) with Poincaré-dual basis e_{j}^{∨}. Therefore, we find

(3.1)  Δ_{∗}1 = −[Δ_{S}].

Furthermore, we have

∫_{S}e(A) = −T (e(A)) = −⟨e(A),1⟩= −⟨Δ_{∗}1,Δ_{∗}1⟩ = −χ(S) = −24,

so e(A) is a negative multiple of the point class. Nevertheless, we still have (cf. [15, §2.2])

e(A) = χ(S)vol, where T (vol) = 1,

but vol differs from the standard volume form by sign.

Let ⟨π⟩\[n] denote the set of orbits of [n] = {1,2,...,n} under the action of π ∈ 𝔖_{n}. Set

A{𝔖_{n}} = ⊕_{π∈𝔖_{n}} A^{⊗⟨π⟩\[n]}· π

which admits an action of 𝔖_{n}. First, note that σ ∈ 𝔖_{n} induces a bijection

\begin{matrix}σ : ⟨π⟩\[n] & → & ⟨σπσ^{−1} ⟩ \[n] \\ x & ↦ & σx. \end{matrix}

Thus we obtain an isomorphism

\begin{matrix}σ̃ : A\{S_{n}\} & → & A\{S_{n}\} \\ aπ & ↦ & σ^{∗}(a)σπσ^{−1} . \end{matrix}

Example 3.2. [15, §2.9, 2.17] We have A{𝔖_{2}} = A^{⊗2} id⊕ A(12) and

A{𝔖_{3}} = A^{⊗3} id⊕ A^{⊗2} (12) ⊕ A^{⊗2} (13) ⊕ A^{⊗2} (23) ⊕ A(123) ⊕ A(132).

Let A^{[n]}⊂ A{𝔖_{n}} denote the invariants under this action. Then we have [15, §2]

A^{[n]}= ∑_{∥α∥=n} ⨂_{i} Sym^{α_{i}}A,

where α corresponds to a partition

\underbrace{1 + ··· + 1}_{α_{1} times}+\underbrace{2 + ··· + 2}_{α_{2} times}+···

and

n = ∥α∥ = α_{1}+ 2α_{2}+ ··· + nα_{n}.

Note that this is compatible with Hodge structures; in particular, A^{[n]}is a representation of the Hodge group of S and the special orthogonal group G_{S} associated with the intersection form on H^{2}(S,ℝ). We interpret this as acting on A, trivially on the summands H^{0}(S,ℝ) and H^{4}(S,ℝ).

Theorem 3.3. [15, Theorem 1.1] Let S be a K3 surface. Then there is a canonical isomorphism of graded rings

(H^{∗}(S,ℚ)[2])^{[n]}→^{∼} H^{∗}(S^{[n]},ℚ)[2n].

In the cohomology of the Hilbert scheme, the subring generated by H^{2}(S^{[n]}) plays a special role. We have an isomorphism

H^{2}(S^{[n]},ℤ) = H^{2}(S,ℤ) ⊕ ℤδ,

where 2δ parameterizes the non-reduced subschemes of S. We express this in terms of our presentation. Given D ∈ H^{2}(S,ℤ), the class

∑_{i=1}^{n} 1_{{1}}⊗ ··· ⊗ 1_{{i−1}} ⊗ D_{{i}}⊗ 1_{{i+1}} ⊗ ··· ⊗ 1_{{n}}(id)

is the corresponding class in H^{2}(S^{[n]},ℚ)[2n]. Using the explicit form of the isomorphism in [15, §2.10 (2.7)] and Nakajima’s correspondence (the isomorphism Ψ in [15, Thm. 3.5]), we find that

δ = ∑_{1≤i<j≤n} 1_{{1}}⊗...⊗1_{{i−1}} ⊗1_{{i,j}} ⊗1_{{i+1}} ⊗··· ⊗1_{{j−1}} ⊗1_{{j+1}} ⊗··· ⊗1_{{n}}(ij).

Here is the essence of the computation: the interpretation of the nonreduced subschemes via the correspondence

Z_{2} = {(ξ,x,ξ′) : |ξ′| − |ξ| = 2x} ⊂ S^{[n−2]}× S × S^{[n]}

allows us to express δ in terms of Nakajima’s creation and annihilation operators, and thus in

H^{∗}(S^{[n]},ℚ)[2n].

We describe the general rule for evaluating the fundamental class in A^{[n]}. Let

[pt] ∈ H^{4}(S,ℤ)[2] ⊂ A

be the point class, which is of degree −2. Let

[pt]_{{1}}⊗ ··· ⊗ [pt]_{{n}}(id) ∈ A^{[n]}

denote the unique class of degree −2n up to scalar. Then the class of a point in S^{[n]} is equal to [15, §2.10]

(3.2)  [pt_{S^{[n]}}] = n![pt]_{{1}}⊗ ··· ⊗ [pt]_{{n}}(id).

4 Decomposition of the cohomology representation

We summarize general results on representations of complex (or split) orthogonal groups. Consider the orthogonal group of odd dimension 2r + 1, i.e., one associated with a quadratic form of rank 2r + 1. Let V (λ) denote the representation with highest weight λ = (λ_{1},...,λ_{r}), where λ is a vector consisting entirely of integers (or half integers) in the fundamental chamber

{λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{r−1} ≥ λ_{r} ≥ 0}.

For orthogonal groups of even dimension 2r, we also use V (λ) to denote the representation of highest weight λ, taken from the fundamental chamber

{λ_{1} ≥ λ_{2} ≥ ... ≥ λ_{r−1} ≥ |λ_{r}| ≥ 0}.

We only consider representations where the λ_{j} are integers.

Recall that • V (1,0,...) is the standard representation V.

• We have

V (\underbrace{1,···,1}_{k times},0,··· ) = ⋀^{k} V,

provided k < r (in the even case) or k ≤ r (in the odd case); see, for instance, [7, Thms. 19.2 and 19.14].

• V (k,0,...) = Sym^{k}(V)/Sym^{k−2}(V), embedded via the dual to the quadratic form on V.

• For the odd orthogonal group, we have

dimV (λ) = ∏_{i<j} \frac{ℓ_{i}− ℓ_{j}}{j − i} ∏_{i≤j} \frac{ℓ_{i}+ ℓ_{j}}{2n + 1 − i − j}

where ℓ_{i} = λ_{i}+ n − i + \frac{1}{2} [7, p. 408].

• For the even orthogonal group, we have

dimV (λ) = ∏_{i<j} \frac{ℓ_{i}^{2}− ℓ_{j}^{2}}{(j − i)(2n − i − j)}

where ℓ_{i} = λ_{i}+ n − i [7, p. 410].

• Let V_{X}(λ) denote an irreducible representation of an orthogonal group G_{X} of dimension 2r+1, G_{S}⊂ G_{X} the orthogonal subgroup of dimension 2r fixing a non-isotropic vector, and V_{S}(\overline{λ}) the representation of G_{S} with highest weight \overline{λ}. Then we have the branching rule [7, p. 426]

Res_{G_{S}}^{G_{X}}V_{X}(λ) = ⊕\overline{λ}V_{S}(\overline{λ}),

where the sum ranges over all \overline{λ} with

λ_{1} ≥ \overline{λ}_{1} ≥ λ_{2} ≥ \overline{λ}_{2} ≥ ··· λ_{r} ≥ |\overline{λ}_{r}|,

with the λ_{i} and \overline{λ}_{i} simultaneously all integers or half-integers.

Let X be a generic deformation of S^{[n]}, where S is a K3 surface. Our goal is to decompose H^{∗}(X,ℚ) into irreducible representations for the action of the identity component G_{X} of the special orthogonal group associated with the Beauville-Bogomolov form on H^{2}(X,ℚ). Let G_{S} denote the identity component of the special orthogonal group associated with the intersection form on H^{2}(S,ℚ). The decomposition

H^{2}(S^{[2]},ℤ) = H^{2}(S,ℤ) ⊕_{⊥} ℤδ

induces an inclusion G_{S} ⊂ G_{X}.

Proposition 4.1. Let X be deformation equivalent to S^{[n]}for some n, where S is a K3 surface. Then G_{X} admits a representation on the cohomology ring of X.

Proof. Let Mon ⊂ Aut(H^{∗}(X,ℤ)) denote the monodromy group, i.e., the group generated by the monodromy representations of all connected families containing X. Let Mon^{2} ⊂ Aut(H^{2}(X,ℤ)) denote its image under projection to the second cohomology group, so we have an exact sequence

1 → K → Mon→ Mon^{2} → 1.

Markman has shown [17, §4.3] that K is finite.

Note that G_{X} is a connected component of the Zariski closure of Mon^{2} (see, for example [17, §1.8]). Since Mon and Mon^{2} differ only by finite subgroups, it follows that the universal cover G̃_{X}→ G_{X} acts on the cohomology ring of X. Since the cohomology of X is nonzero only in even degrees, this representation passes to G_{X}. □

In principle, we can decompose H^{∗}(X,ℝ) explicitly into isotypic components as follows:

(1) Fix an embedding G_{S} ⊂ G_{X}, e.g., using the isomorphism

H^{2}(X,ℤ) ≃ H^{2}(S,ℤ) ⊕_{⊥} ℤδ,

and compatible maximal tori (both of which have rank 11).

(2) Identify the highest-weight irreducible G_{S}-representation V_{S}(λ) ⊂ H^{∗}(S^{[n]},ℝ), which is a summand of the restriction of an irreducible V_{X}(λ) ⊂ H^{∗}(X,ℝ). Decompose V_{X}(λ) into irreducible G_{S}-representations.

(3) Repeat step two for H^{∗}(X,ℝ)/V_{X}(λ) and subsequent quotients.

First consider X = S^{[2]}. We have decompositions

H^{∗}(S^{[2]}) = A ⊕ Sym^{2}(A)

inducing

H^{2}(S^{[2]}) = H^{0}(S) ⊕ (H^{0}(S) ⊗ H^{2}(S)) = 1_{S}⊕ V_{S}(1,0,...)
H^{4}(S^{[2]}) = H^{2}(S) ⊕ (H^{0}(S) ⊗ H^{4}(S)) ⊕ Sym^{2}(H^{2}(S))
= V_{S}(1,0,...) ⊕ 1_{S}^{⊕2} ⊕ V_{S}(2,0,...),

where 1_{S} is the trivial irreducible representation. Let V_{X}(2,0,...,0) denote the highest-weight representation associated to Sym^{2}(H^{2}(X)) so that

Sym^{2}(H^{2}(X)) = V_{X}(2,0,...) ⊕ 1_{X}.

The branching rule gives

V_{X}(1,0,...) = V_{S}(1,0,...) ⊕ 1_{S}

and

V_{X}(2,0,...) = V_{S}(2,0,...) ⊕ V_{S}(1,0,...) ⊕ 1_{S}.

Therefore we obtain

\begin{aligned}H^{2}(X) & = & V_{X}(1,0,...) \\ H^{4}(X) & = & V_{X}(2,0,...) ⊕ 1_{X}. \end{aligned}

Now consider X = S^{[3]}. We have

H^{∗}(S^{[3]}) = A ⊕ (A ⊗ A) ⊕ Sym^{3}(A)

inducing the following decompositions (as described in [15, Example 2.9]):

H^{2}(S^{[3]}) = (H^{0}(S)^{⊗2} ) ⊕ (H^{2}(S) ⊗ H^{0}(S)^{⊗2} )
= 1_{S}⊕ V_{S}(1,0...)
H^{4}(S^{[3]}) = H^{0}(S) ⊕ (H^{0}(S) ⊗ H^{2}(S))^{⊕2}
⊕(Sym^{2}(H^{2}(S)) ⊗ H^{0}(S)) ⊕ (H^{4}(S) ⊗ H^{0}(S)^{⊗2} )
= 1_{S}^{⊕3} ⊕ V_{S}(1,0,...)^{⊕2} ⊕ V_{S}(2,0,...)
H^{6}(S^{[3]}) = H^{2}(S) ⊕ (H^{2}(S) ⊗ H^{2}(S)) ⊕ (H^{0}(S) ⊗ H^{4}(S))^{⊕2}
⊕Sym^{3}(H^{2}(S)) ⊕ (H^{4}(S) ⊗ H^{2}(S) ⊗ H^{0}(S))
= 1_{S}^{⊕3} ⊕ V_{S}(1,0,...)^{⊕3} ⊕ V_{S}(1,1,0,...)
⊕V_{S}(2,0,...) ⊕ V_{S}(3,0,...).

Note that two of the trivial summands in H^{4}(S^{[3]}) lie in the decomposable classes, i.e., the image of Sym^{2}(H^{2}(S^{[2]})).

Let V_{X}(1,1,0,...) = ⋀^{2}V_{X}(1,0,...) and V_{X}(3,0,...) denote the highest weight representation in Sym^{3}(V_{X}(1,0,...)) so that

Sym^{3}(V_{X}(1,0,...)) = V_{X}(3,0,...) ⊕ V_{X}(1,0,...).

Therefore we obtain

\begin{aligned}H^{2}(X) & = & V_{X}(1,0,...) \\ H^{4}(X) & = & V_{X}(2,0,...) ⊕ V_{X}(1,0,...) ⊕ 1_{X} \\ H^{6}(X) & = & V_{X}(3,0,...) ⊕ V_{X}(1,1,0...) ⊕ V_{X}(1,0,...) ⊕ 1_{X}. \end{aligned}

The trivial factor in H^{4}(X) corresponds to the Chern class c_{2}(X); indeed, c_{2}(X) is a Hodge class and is nonzero because c_{2}(X)^{3} > 0 [5, Remark 5.5]. Note that c_{2}(X) is proportional to the trivial summand in Sym^{2}(H^{2}(X)) associated with the Beauville-Bogomolov form, regarded as an object in the symmetric algebra over H^{2}(X).

Our main task is to analyze the trivial summand in H^{6}(X).

5 Cohomology computations for length-three subschemes

The general rule for multiplication in A{𝔖_{n}} is fairly complicated, so we will only give a formula in the case (n = 3) we need. The fact that A only has terms of even degree simplifies the expressions of [15, §2.17]:

\begin{aligned}(α_{\{1,2\}}⊗ β_{\{3\}})(12) · (γ_{\{1,3\}}⊗ δ_{\{2\}})(13) & = & αβγδ(132) \\ (α_{\{1,2\}}⊗ β_{\{3\}})(12) · (γ_{\{1,2\}}⊗ δ_{\{3\}})(12) & = & Δ_{∗}(αγ) ⊗ (βδ)(id) \\ α_{\{1,2,3\}}(123) · β_{\{1,2,3\}}(123) & = & (αβe)(132) \\ α_{\{1,2,3\}}(123) · β_{\{1,2,3\}}(132) & = & (Δ_{∗}(αβ))_{\{1,2,3\}}(id), \end{aligned}

where Δ_{∗} in the last line is the adjoint of the threefold multiplication A⊗A⊗A → A.

The remaining products can be deduced as formal consequences using the associativity of the multiplication, e.g.,

(α_{{1,2}}⊗ β_{{3}})(12) · γ_{{1,2,3}}(132)
= (α_{{1,2}}⊗ β_{{3}})(12) · (γ_{{1,2}}⊗ 1_{{3}})(12) · (13)
= (Δ_{∗}(αγ)_{{1,2}}⊗ β_{{3}})(id) · (1_{{1,3}}⊗ 1_{{2}})(13)
= αβγ(Δ_{∗}(1))_{{1,3},{2}}(13),

where α,β, and γ act on the diagonal via either the first or second variable. Thus in particular

(12) · (132) = (Δ_{∗}(1))_{{1,3},{2}}(13).

We compute intersections among the absolute Hodge classes for S^{[3]}, i.e., classes that are Hodge for general K3 surfaces S. From now on, to condense notation we omit factors of the form 1_{{i}},1_{{i,j}}, etc. from our expressions.

Based on the representation-theoretic analysis in Section 4, we expect one independent classes in codimension one, three in codimension two, and three in codimension three. We have the unique divisor

δ = (12) + (13) + (23).

In codimension two, we have

\begin{aligned}P & = & [pt]_{\{1\}}+ [pt]_{\{2\}}+ [pt]_{\{3\}} \\ Q & = & ∑ _{j=1}^{22} e_{j}_{\{1\}}⊗ e_{j}^{∨}_{\{2\}}+ e_{j}_{\{1\}}⊗ e_{j}^{∨}_{\{3\}}+ e_{j}_{\{2\}}⊗ e_{j}^{∨}_{\{3\}} \\ R & = & (132) + (123). \end{aligned}

We identify the linear combinations of P,Q, and R that are decomposable, i.e., lie in the symmetric algebra generated by H^{2}(S^{[3]}). First, we have

δ^{2} = (Δ_{∗}1)_{{1,2}}(12) + (Δ_{∗}1)_{{1,3}}(13) + (Δ_{∗}1)_{{2,3}}(23)

+ 3((132) + (123))

= −2P − Q + 3R.

For the second decomposable class, we extract the intersection form on H^{2}(S) interpreted as an element in Sym^{2}(H^{2}(S)):

∑ _{j=1}^{22} (e_{j}_{{1}}+ e_{j}_{{2}}+ e_{j}_{{3}}) · (e_{j}^{∨}_{{1}}+ e_{j}^{∨}_{{2}}+ e_{j}^{∨}_{{3}})
= ∑ _{j=1}^{22} (e_{j} · e_{j}^{∨}_{{1}}+ e_{j} · e_{j}^{∨}_{{2}}+ e_{j} · e_{j}^{∨}_{{3}})
+2∑ _{j=1}^{22} e_{j}_{{1}}⊗ e_{j}^{∨}_{{2}}+ e_{j}_{{1}}⊗ e_{j}^{∨}_{{3}}+ e_{j}_{{2}}⊗ e_{j}^{∨}_{{3}}
= 22P + 2Q.

We now isolate the class in Sym^{2}(H^{2}(S^{[3]})) ⊂ H^{4}(S^{[3]}) corresponding to the Beauville-Bogomolov quadratic form (,). Recall this generates the unique trivial factor in the fourth cohomology, regarded as a representation of G_{X}. As we saw

12 HARVEY, HASSETT, AND TSCHINKEL

in Section 2, the Beauville-Bogomolov form agrees with the intersection form on H^{2}(S) ⊂ H^{2}(S^{[3]}) and (δ,δ) = −4. Thus we obtain

(5.1)  22P + 2Q − \frac{1}{4}δ^{2}.

In codimension three, we have

U = [pt]_{{1,2}}(12) + [pt]_{{1,3}}(13) + [pt]_{{2,3}}(23)
V = [pt]_{{3}}(12) + [pt]_{{2}}(13) + [pt]_{{1}}(23)
W = ∑ _{j=1}^{22} e_{j}_{{1,2}}⊗ e_{j}^{∨}_{{3}}(12) + e_{j}_{{1,3}}⊗ e_{j}^{∨}_{{2}}(13) + e_{j}_{{2,3}}⊗ e_{j}^{∨}_{{1}}(23).

Furthermore, we have

δ · P = ((12) + (13) + (23)) · ([pt]_{{1}}+ [pt]_{{2}}+ [pt]_{{3}})
= 2U + V
δ · Q = ((12) + (13) + (23)) · (∑ _{j=1}^{22} e_{j}_{{1}}⊗ e_{j}^{∨}_{{2}}+ e_{j}_{{1}}⊗ e_{j}^{∨}_{{3}}
+ e_{j}_{{2}}⊗ e_{j}^{∨}_{{3}})
= 22([pt]_{{1,2}}(12) + [pt]_{{1,3}}(13) + [pt]_{{2,3}}(23))
+2(∑ _{j=1}^{22} e_{j}_{{1,2}}⊗ e_{j}^{∨}_{{3}}+ e_{j}_{{1,3}}⊗ e_{j}^{∨}_{{2}}+ e_{j}_{{2,3}}⊗ e_{j}^{∨}_{{1}})
= 22U + 2W
δ · R = ((12) + (13) + (23))((132) + (123))
= 2(Δ_{∗}1_{{1,2},{3}}(12) + Δ_{∗}1_{{1,3},{2}}+ Δ_{∗}1_{{2,3},{1}})
= −2(U + V + W).

We deduce then that

δ^{3} = δ · (−2P − Q + 3R) = −32U − 8V − 8W

and

δ · (22P + 2Q) = 88U + 22V + 4W.

Finally, we compute the intersection pairing on the subspace of the middle cohomology spanned by U,V, and W. Dimensional considerations give vanishing

U^{2} = V^{2} = U · W = V · W = 0.

For the remaining numbers, we get

U · V = ([pt]_{{1,2}}(12) + [pt]_{{1,3}}(13) + [pt]_{{2,3}}(23))
·([pt]_{{3}}(12) + [pt]_{{2}}(13) + [pt]_{{1}}(23))
= −3[pt]_{{1}}⊗ [pt]_{{2}}⊗ [pt]_{{3}}id

and

W^{2} = (∑ _{j=1}^{22} e_{j}_{{1,2}}⊗ e_{j}^{∨}_{{3}}(12) + e_{j}_{{1,3}}⊗ e_{j}^{∨}_{{2}}(13) + e_{j}_{{2,3}}⊗ e_{j}^{∨}_{{1}}(23))^{2}
= −3 · 22 · [pt]_{{1}}⊗ [pt]_{{2}}⊗ [pt]_{{3}}id.

Remark 5.1. As a consistency check, we evaluate

δ^{6} = (−32U − 8V − 8W)^{2} = 2^{6}(8UV + W^{2})
= 2^{6}(−24 − 66)[pt]_{{1}}⊗ [pt]_{{2}}⊗ [pt]_{{3}}id.

Using the formula for the point class (Equation 3.2), we obtain

δ^{6} = −\frac{2^{7}· 3^{2}· 5}{2 · 3}= −2^{6}· 3 · 5.

This is compatible with the Fujiki-type identity

D^{6} = 15(D,D)^{3}, D ∈ H^{2}(S^{[3]},ℚ),

as (δ,δ) = −4.

6 Evaluation of the distinguished absolute Hodge class

Let S be a general K3 surface and X a general deformation of S^{[3]}. The computations above show that the middle cohomology of X admits one Hodge class

H^{6}(X,ℚ) ∩ H^{3,3}(X) = ℚη

and the middle cohomology of S^{[3]}admits three Hodge classes

H^{6}(S^{[3]},ℚ) ∩ H^{3,3}(S^{[3]}) = ℚη ⊕ ℚδ^{3}⊕ ℚc_{2}(X)δ.

Our goal is to compute the self-intersection of η, at least up to the square of a rational number. Note that η is orthogonal to δ^{3} and δc_{2}(X) under the intersection form, by the analysis in Section 4. The analysis here gives the one structure constant left open in [16, Ex. 14].

Proposition 6.1. Let X be deformation equivalent to S^{[3]}, for S a K3 surface. Let η ∈ H^{6}(X,ℚ) denote the unique (up to scalar) absolute Hodge class. Then η^{2} = 4.

Proof. The argument relies heavily on the analysis in Section 5. Consider the decomposable classes in H^{6}(S^{[3]}), i.e., those coming from Sym^{3}(H^{2}(S^{[3]})). We have computed

δ^{3} = −32U − 8V − 8W, δ · (22P + 2Q) = 88U + 22V + 4W,

hence the subspace

span{32U + 8V + 8W,88U + 22V + 4W } = span{4U + V,W }

is spanned by decomposable classes. It has orthogonal complement spanned by 4U − V. Thus we have

η = 4U − V

and

η^{2} = −8UV
= 24([pt]_{{1}}⊗ [pt]_{{2}}⊗ [pt]_{{3}})id
= 4.
□

7 Proof of the main theorem

We compute the cohomology class of a Lagrangian subspace ℙ^{3} ⊂ X, where X is deformation equivalent to the Hilbert scheme of length-three subschemes. As we shall see, the formula for [ℙ^{3}] involves only decomposable classes, and not the absolute Hodge class η:

Lemma 7.1. Let ℙ^{n} ⊂ X be embedded in a general irreducible holomorphic symplectic variety of dimension 2n. Then we have

c_{2j}(T_{X}|ℙ^{n}) = (−1)^{j}h^{2j} \binom{n + 1}{j} ,

where h is the hyperplane class.

This is proved using the exact sequence

0 → T_{ℙ^{n}} → T_{X}|ℙ^{n} → N_{ℙ^{n}/X} → 0

and

N_{ℙ^{n}/X} ≃ T_{ℙ^{n}}^{∨},

reflecting the fact that ℙ^{n} is a Lagrangian subvariety of X.

Regarding

H^{2}(X,ℤ) ⊂ H_{2}(X,ℤ)

as a subgroup of index four, we can express ℓ = λ/4 for some divisor class λ ∈ H^{2}(X,ℤ). (This might not be primitive.)

Given a deformation of X such that λ remains algebraic, the subvariety ℙ^{3} deforms as well [10]. Without loss of generality, we can deform X to a variety containing a ℙ^{3}, but otherwise having a general Hodge structure. In particular, we have an injection

Sym(H^{2}(X,ℚ)) ↪ H^{∗}(X,ℚ).

We expect to be able to write

[ℙ^{3}] = aλc_{2}(X) + bλ^{3}+ dη

for some a,b,d ∈ ℚ.

Furthermore, the Fujiki relations [6] imply that for each f ∈ H^{2}(X,ℤ),

f^{6} = e_{0}(f,f) ^{3}, c_{2}(X)f^{4} = e_{2}(f,f) ^{2}, c_{4}(X)f^{2} = e_{4}(f,f)

for suitable rational constants e_{0},e_{2},e_{4}. Precisely, we have [5]

c_{2}^{2}(X)f^{2} = \frac{5}{2}c_{4}(X)f^{2}.

The Riemann-Roch formula gives

χ(O_{X}(f)) = \frac{f^{6}}{6!} + \frac{c_{2}(X)f^{4}}{12 · 4!} + \frac{f^{2}(3c_{2}^{2}− c_{4})}{720 · 2!} + 4.

On the other hand, we know that

χ(O_{X}(f)) = \frac{1}{3!2^{3}} ((f,f) + 8)((f,f) + 6)((f,f) + 4).

Perhaps the quickest way to check this formula is to observe that if X = S^{[3]}and f is a very ample divisor on S with no higher cohomology then the induced sheaf O_{X}(f) has no higher cohomology and

dimΓ(O_{X}(f)) = dimSym^{3}(Γ(O_{S}(f))) = \binom{χ(O_{S}(f)) + 2}{3} .

Equating coefficients, we find

\begin{aligned}f^{6} & = & 15(f,f) ^{3} \\ f^{4}c_{2} & = & 108(f,f) ^{2} \\ f^{2}c_{4} & = & 480(f,f) \\ f^{2}c_{2}^{2} & = & 1200(f,f) . \end{aligned}

Remark 7.2. These structural constants allow us to normalize expression (5.1) to get the second Chern class

c_{2}(T_{X}) = \frac{4}{3}(22P + 2Q − \frac{1}{4}δ^{2}).

We generate Diophantine equations for a,b,(λ,λ) and eventually, d. First, observe that

(λ,ℓ)= λ · ℓ = degλ|ℙ^{3}

so that λ|ℙ^{3} is (λ,λ) /4 times the hyperplane class. Thus we have

[ℙ^{3}]λ^{3} = ((λ,λ) /4)^{3}

and

[ℙ^{3}]λ^{3} = aλ^{4}c_{2}(X) + bλ^{6}.

Equating these expressions and evaluating the terms, we find

(λ,λ) (15b − 1/64) + 108a = 0.

We have divided out by (λ,λ); the solution (λ,λ) = 0 is not possible for geometric reasons, and we shall exclude it algebraically below.

Second, the Lemma on restrictions of Chern classes implies

[ℙ^{3}]λc_{2}(X) = −(λ,λ)

whereas the formula for the class of ℙ^{3} yields

[ℙ^{3}]λc_{2}(X) = aλ^{2}c_{2}(X)^{2}+ bλ^{4}c_{2}(X).

Thus we obtain

108b(λ,λ) + (1200a + 1) = 0.

Remark 7.3. The cup product of H^{∗}(X) is compatible with the G_{X}-action, so the subring generated by Chern classes and elements of H^{2}(X) is orthogonal to η. Thus even if the decomposition of [ℙ^{3}] were to involve η, the computations up to this point would not reflect this.

Finally, the fact that

[ℙ^{3}]^{2} = c_{3}(N_{ℙ^{3}/X}) = c_{3}(T_{ℙ^{3}}^{∨}) = −4

yields the cubic equation

15b^{2}(λ,λ)^{3}+ 216ab(λ,λ)^{2}+ 1200(λ,λ) a^{2}+ d^{2}η · η = −4.

The proof of Proposition 6.1 implies that η · η = 4. In particular, (λ,λ) = 0 is excluded.

Eliminating a and b from these equations and setting L = (λ,λ), we obtain

(7.1)  −2^{16}· 3 · 11d^{2} = 5^{2}L^{3}+ 2^{5}· 3^{2}L^{2}+ 2^{8}· 5L + 2^{16}· 3 · 11.

We know, a priori, that L ∈ ℤ and d ∈ ℚ.

Proposition 7.4. The only solution to (7.1) with L ∈ ℤ and d ∈ ℚ is d = 0 and L = −48.

We assume this for the moment; its proof can be found in Section 8.

Back-substitution yields

a = 1/96, b = 1/384, (ℓ,ℓ)= −3.

We claim that λ/2 ∈ H^{2}(X,ℤ), i.e., λ is not primitive. Using the isomorphism

H_{2}(X,ℤ) = H_{2}(S,ℤ) ⊕_{⊥} ℤδ^{∨}, (δ^{∨},δ^{∨}) = −1/4

we can express

ℓ = D + mδ^{∨}, D ∈ H_{2}(S,ℤ),m ∈ ℤ.

If λ were primitive then m would have to be odd and

−3 = (ℓ,ℓ)= (D,D)− m^{2}/4.

Since (D,D)∈ 2ℤ, we have a contradiction.

8 Diophantine analysis

Theorem 8.1. The only solution to

−2^{16}·3·11d^{2} = 5^{2}L^{3}+ 2^{5}·3^{2}L^{2}+ 2^{8}·5L + 2^{16}·3·11

with L ∈ ℤ and d ∈ ℚ is L = −48, d = 0.

Proof. Put x = −2^{−4} ·3·5^{2}·11(L + 48) and y = 2^{2}·3^{2}·5^{2}·11^{2}d. The equation then takes the form

(8.1)  E : y^{2} = x^{3}+ ax^{2}+ bx

where

a = 3^{3}·11·23, b = 2^{2}·3^{2}·5^{2}·11^{3}·13.

It suffices to prove the stronger statement that there are no solutions to (8.1) with x,y ∈ ℤ[\frac{1}{2}], apart from x = y = 0.

The proof is given in two steps. Proposition 8.2 below determines explicitly the structure of the Mordell–Weil group E(ℚ). Proposition 8.3 then identifies the integral points. □

Algorithms for both of these steps are implemented in computer algebra systems such as Sage [22] and Magma [2], and the theorem may be verified this way. To avoid depending on the correctness of these systems, we give alternative proofs that use as little machine assistance as possible.

We first set notation and briefly recall some facts about point multiplication on elliptic curves given in the form (8.1). Let O denote the zero element of E(ℚ) (the point at infinity). For nonzero R ∈ E(ℚ) we write

R = (x(R),y(R)) = ( \frac{α(R)}{e(R)^{2}}, \frac{β(R)}{e(R)^{3}}) ,

where α,β,e ∈ ℤ, e ≥ 1 and (α,e) = (β,e) = 1. If p is a prime, then p|e(R) if and only if R reduces to the identity in E(𝔽_{p}). If m ≥ 1 and mR ≠ O, then e(R) | e(mR) [14, Ch. III, Thm. 1.2]. The point Q = (0,0) has order two, and addition with Q is given by the formula

(8.2)  R + Q = ( \frac{b}{x(R)}, \frac{−b·y(R)}{x(R)^{2}})  (R ≠ O,Q).

Proposition 8.2. We have E(ℚ) ≅ ℤ×(ℤ/2ℤ), where the torsion part is generated by Q, and the free part by

P = ( \frac{2^{2}·3·5·277^{2}}{53^{2}}, \frac{2^{3}·3^{2}·5·29·277·11311}{53^{3}}) .

Proof. The discriminant of the Weierstrass equation (8.1) is given by

Δ = 16b^{2}(a^{2}− 4b) = −2^{8}·3^{6}·5^{4}·11^{8}·13^{2}·113·127,

so the model is minimal, and the primes of bad reduction are 2, 3, 5, 11, 13, 113 and 127.

We first check that the torsion subgroup is as described. For ℓ prime, by [20, Prop. VII.3.1] we see that E(ℚ)[ℓ] injects into E(𝔽_{17}) = ℤ/18ℤ for ℓ ≠ 17 and into E(𝔽_{19}) = ℤ/14ℤ for ℓ ≠ 19. These facts force E(ℚ)[2] = ℤ/2ℤ and E(ℚ)[ℓ] = 0 for ℓ ≠ 2. Hence E_{tors}(ℚ) = ⟨Q⟩.

Now we consider the free part. The point P was found using Cremona’s mwrank library [3] via the Sage computer algebra system [22]. We may check that P ∈ E(ℚ) using a computer (or by hand with some patience); this shows that rankE ≥ 1.

To show that rankE ≤ 1 we use a standard 2-descent strategy (see for example [21, Ch. III]). Consider the auxiliary curve

E′: y^{2} = x^{3}− 2ax^{2}+ (a^{2}− 4b)x.

There are isogenies ϕ : E → E′and ϕ̂: E′→ E of degree 2, and injections

E(ℚ)/ϕ̂(E′(ℚ)) ↪^{ψ} S ⊂ ℚ^{∗}/(ℚ^{∗})^{2},

E′(ℚ)/ϕ(E(ℚ)) ↪^{ψ′} S′⊂ ℚ^{∗}/(ℚ^{∗})^{2},

where S consists of the cosets δ(ℚ^{∗})^{2} for δ | 2·3·5·11·13, and S′of the cosets for δ | 3·11·113·127 (these are the primes dividing b and a^{2}− 4b respectively).

We must determine which elements of S and S′arise from points in E(ℚ) and E′(ℚ). This is achieved by testing for the existence of ℚ-rational points on the two families of quartic curves

C_{δ} : δw^{2} = δ^{2}z^{4}+ δaz^{2}+ b, δ ∈ S,
C′_{δ} : δw^{2} = δ^{2}z^{4}− 2δaz^{2}+ (a^{2}− 4b), δ ∈ S′.

First consider C_{δ}, which may be rewritten as

δw^{2} = δ^{2}z^{4}+ 3^{3}·11·23δz^{2}+ 2^{2}·3^{2}·5^{2}·11^{3}·13,

or

4δw^{2} = (2δz^{2}+ 3^{3}·11·23)^{2}+ 3^{2}·11^{2}·113·127,

for δ | 2·3·5·11·13.

For δ = 11·13 there is a trivial rational point z = 0, w = 2·3·5·11, corresponding to the class of Q in E(ℚ)/ϕ̂(E′(ℚ)). For δ = 3·5 there is a nontrivial rational point corresponding to P, namely z = 2·277/53, w = 2^{2}·3·29·11311/53^{2}. Rational points are automatic for δ = 1 and 3·5·11·13 since the image of ψ is a subgroup of S. We will show that C_{δ}(ℚ) = ∅ for all other δ, by identifying local obstructions. We only summarize the obstructions encountered, omitting detailed arguments (which are straightforward, but in some cases lengthy).

Consider the conic 4δw^{2} = u^{2} + 3^{2}·11^{2}·113·127. If δ < 0 then the conic has no points over ℝ. If (δ/113) = −1 then the conic has no points over ℚ_{113}. Since (2/113) = (11/113) = (13/113) = 1 and (3/113) = (5/113) = −1, this shows that 3 | δ if and only if 5 | δ. No further information can be obtained from local analysis of the conic, so we proceed to C_{δ} itself. If 2 | δ then C_{δ} has no points over ℚ_{2}. If (δ/11) = −1, or if δ = 11ε with (ε/11) = 1, then C_{δ} has no points over ℚ_{11}. These conditions rule out all but the four values for δ noted in the previous paragraph.

Next we give a similar analysis for C′_{δ}, which becomes

δw^{2} = δ^{2}z^{4}− 2·3^{3}·11·23δz^{2}− 3^{2}·11^{2}·113·127,

or

δw^{2} = (δz^{2}− 3^{3}·11·23)^{2}− 2^{4}·3^{2}·5^{2}·11^{3}·13,

for δ |3·11·113·127. The values δ = 1 and δ = −113·127 correspond to the classes in E′(ℚ)/ϕ(E(ℚ)) of O and the unique two-torsion point of E′(ℚ); both have trivial image in ϕ̂(E′(ℚ))/2E(ℚ). We will show that C′_{δ}(ℚ) = ∅ for all other possible δ.

If 3 | δ, then the conic δw^{2} = u^{2} − 2^{4}·3^{2}·5^{2}·11^{3}·13 has no points over ℚ_{3}. If 11 | δ, then (δ/13) = −1 (because (−1/13) = (113/13) = (127/13) = 1) and the conic has no points over ℚ_{13}. If (δ/11) = −1 the conic has no points over ℚ_{11}. If (δ/5) = −1, then C′_{δ} has no points over ℚ_{5}. These conditions rule out all but δ = 1 and δ = −113·127.

This completes the 2-descent. In particular, we have found that

|E(ℚ)/2E(ℚ)| = |E(ℚ)/ϕ̂(E′(ℚ))| · |ϕ̂(E′(ℚ))/2E(ℚ)| = 4 · 1 = 4,

and that E(ℚ)/2E(ℚ) is generated by the classes of P and Q. Moreover, for R ≠ O,Q the image of x(R) in ℚ^{∗}/(ℚ^{∗})^{2} is one of {1,3·5,11·13,3·5·11·13}.

At this stage we know that ⟨P,Q⟩ is of finite index in E(ℚ); we must still check that it exhausts E(ℚ). Suppose not; then for some odd prime ℓ and some R ∈ E(ℚ) we have ℓR = ±P or ℓR = ±P + Q. In the latter case we replace R by R + Q, and we may also switch the sign of R if necessary, so now may assume that ℓR = P and ℓ(R + Q) = P + Q.

In this case e(R)|e(P) = 53 and e(R+Q)|e(P +Q) = 277. Moreover, by (8.2) we have α(R)α(R + Q) = b·e(R)^{2}e(R + Q)^{2}. Since

(α(R),e(R)) = (α(R + Q),e(R + Q)) = 1

this implies that α(R) = b_{1}e(R + Q)^{2} and α(R + Q) = (b/b_{1})e(R)^{2} for some b_{1} | b. But also P = R and P + Q = R + Q in E(ℚ)/2E(ℚ), so comparing with the result of the 2-descent shows the only possibilities are b_{1} = 3·5, 2^{2}·3·5, 3·5·11^{2}, or 2^{2}·3·5·11^{2}. This leaves only 16 choices for x(R), and it is easy to check that only one of them defines a point on the curve (namely for R = P). □

Proposition 8.3. The only solution to (8.1) with x,y ∈ ℤ[\frac{1}{2}] is x = y = 0.

Proof. First consider the points nP for n ≠ 0. Since 53 | e(±P), also 53 | e(nP), so x(nP) ∉ ℤ[\frac{1}{2}].

Now consider nP +Q. Write n = 2^{i}r for some i ≥ 0 and odd r. We may assume r is positive. Then nP + Q = 2^{i}rP + Q = r(2^{i}P + Q), so it suffices to show that e(2^{i}P + Q) is divisible by some prime q ≠ 2. By (8.2), it suffices to show that α(2^{i}P) is divisible by some prime q ≠ 2,3,5. For i = 0 we may take q = 277.

To establish the result for i ≥ 1, we prove the following series of claims. Let N_{p} denote the set of points R ∈ E(ℚ) such that R reduces to a nonsingular point in E(𝔽_{p}); it is a subgroup of E(ℚ). For p = 2,3,5,11,13 the only singular point on E(𝔽_{p}) is (0,0), so for these primes we have N_{p} = {P ∈ E(ℚ) : p | α(P)}.

Claim 1: P ∈ N_{p} for all p except p = 2,3,5.

Claim 2: 2P ∈ N_{p} for all p except p = 5.

Claim 3: 4P ∈ N_{p} for all p, and hence 2^{i}P ∈ N_{p} for all p and all i ≥ 2.

Claim 4: α(2P) = 5^{2}U, where U = 4 (mod 11) and 2,3,5 ∤ U.

Claim 5: α(4P) = 5 (mod 11).

Claim 6: For i ≥ 2, α(2^{i}P) = \{ \begin{matrix}5 (mod 11) & i even, \\ 9 (mod 11) & i odd. \end{matrix}

Claims 1–5 can easily be proved with a computer, by directly computing the coordinates of 2P and 4P. This approach is infeasible by hand, as the coordinates are rather large (for example α(4P) has almost 100 decimal digits). In Remark 8.4 we sketch an indirect method that is amenable to manual verification.

For Claim 6, we will need the following doubling formula, valid for R ∈ E(ℚ), 2R ≠ O:

(8.3)  x(2R) = ( \frac{x(R)^{2}− b}{2y(R)}) ^{2} = \frac{(α(R)^{2}− b·e(R)^{4})^{2}}{(2e(R)β(R))^{2}} .

Moreover, if R ∈ N_{p} then p cannot divide both the numerator and denominator of the fraction on the right side of (8.3). In other words, there is no cancellation locally at p. One proof of this is given in [25, Prop. IV.2]; as pointed out in that paper, it can also be proved from properties of real-valued non-archimedean local heights.

Using this non-cancellation result, if R ∈ N_{p} for all p, we obtain α(2R) = (α(R)^{2}− b·e(R)^{4})^{2}. Claim 6 follows by induction, the base case being Claim 5.

The main result for i = 1 follows from Claim 4, since U must be divisible by a suitable prime. (In fact the only primes dividing U are 159319 and 709141!) For i ≥ 2 it follows from Claim 6. □

Remark 8.4. Claims 1–5 can all be checked by using low-precision p-adic approximations, together with the non-cancellation result for (8.3), without needing to compute the full coordinates of 2P or 4P. For example, one calculates that

(α(P)^{2}− b·e(P)^{4})^{2} = 2^{4} mod2^{5}, 3^{4} mod3^{5}, 5^{4} mod5^{5},
(2e(P)β(P))^{2}= 2^{8} mod2^{9}, 3^{4} mod3^{5}, 4·5^{2} mod5^{3}.

Thus the cancellation between the numerator and denominator of (8.3) is exactly 2^{4}3^{4}5^{2}. It follows that

α(2P)= 2^{−4} 3^{−4} 5^{−2} (α(P)^{2}− b·e(P)^{4})^{2},
e(2P)^{2}= 2^{−4} 3^{−4} 5^{−2} (2e(P)β(P))^{2},

and Claims 2 and 4 follow easily. A similar but more involved calculation establishes Claim 5.

Acknowledgment. We are grateful to Noam Elkies, Lothar Göttsche, Manfred Lehn, Eyal Markman, and Christoph Sorger for useful conversations, to Letao Zhang for suggesting a number of improvements to our manuscript, to Benjamin Bakker and Andrei Jorza for pointing out errors in an earlier version, and to Jay Pottharst for helping to streamline the proof in Section 8. The second author was supported by National Science Foundation Grant 0554491 and 0901645; the third author was supported by National Science Foundation Grants 0554280 and 0602333. We appreciate the hospitality of the American Institute of Mathematics, where some of this work was done.

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