THE MODULARITY CONJECTURE FOR RIGID CALABI–YAU THREEFOLDS OVER ℚ

MASA-HIKO SAITO AND NORIKO YUI

Abstract. : We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field ℚ of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A_{3}. Our proof is based on geometric analysis.

1. The L–series of Calabi–Yau threefolds

Let ℚ be the field of rational numbers, and let ℚ̄ be its algebraic closure with Galois group G := Gal(ℚ̄/ℚ). Let X be a smooth projective threefold defined over ℚ or more generally over a number field. Definition 1.1. X is a Calabi–Yau threefold if it satisfies the following two conditions:

(a) H^{1}(X, O_{X}) = H^{2}(X, O_{X}) = 0, and

(b) The canonical bundle is trivial, i.e., K_{X} ≃ O_{X}.

The numerical invariants of Calabi–Yau threefolds.

Let X be a Calabi–Yau threefold defined over ℚ, and let X̄ = X ×_{ℚ} ℚ̄.

(1) The (i, j)-th Hodge number h^{i,j}(X) of X is defined by

h^{i,j}(X) = dimℚ̄H^{j}(X̄, Ωi ; X̄).

There are symmetries among Hodge numbers: h^{i,j}(X) = h^{j,i}(X), and h^{i,j}(X) = h^{3−j,3−i}(X).

The first identity follows from the complex conjugation operation and the latter from Serre duality on the Hodge cohomology groups. The condition (a) implies that h^{1,0}(X) = h^{2,0}(X) = 0, and the condition (b) that h^{3,0}(X) = h^{0,3}(X) = 1. The number h^{2,1}(X) represents the number of deformations of complex structures on X, and h^{1,1}(X) is the number of Hodge (1, 1)-cycles on X. The Hodge numbers are concocted to define the Hodge diamond:

Date: September 14, 2000.

1991 Mathematics Subject Classification. 14J32,11F11, 11F80, 11G18, 14K30.

Key words and phrases. Modularity Conjecture, Rigid Calabi–Yau Threefolds, Fiber product of elliptic modular surfaces, L-functions.

M.-H. Saito was partially supported by JSPS and Grant-in Aid for Scientific Research (B-09440015), (B-12440008), the Ministry of Education, Science and Culture, Japan. N. Yui was partially supported by a Research Grant from NSERC, Canada.

1 0 0

0  h^{1,1}  0
1  h^{1,2} h^{1,2}  1
0  h^{2,2}  0

0 0 1

(2) The n-th Betti number of X is defined by

B_{n}(X) = dim_{ℚ_{ℓ}}H_{et}^{n}(X̄, ℚ_{ℓ}) = dim_{ℂ}H^{n}(X ⊗ ℂ, ℂ)

where the cohomology groups are, respectively, the ℓ-adic étale and singular cohomology groups. The Hodge decomposition H^{n}(X ⊗ℂ, ℂ) = ⊕_{i+j=n}H^{j}(X ⊗ℂ, Ω_{X⊗ℂ}^{i}) ensures that

B_{n}(X) = ∑_{i+j=n} h^{i,j}(X) = ∑_{i=0}^{n} h^{i,n−i}(X).

Using the above formula and the Poincaré duality B_{n}(X) = B_{2dim(X)−n}, we can compute the Betti numbers of Calabi–Yau threefolds:

B_{0}(X) = 1 = B_{6}(X), B_{1}(X) = 0 = B_{5}(X),
B_{2}(X) = h^{1,1}(X) = h^{2,2}(X) = B_{4}(X) and B_{3}(X) = 2(1 + h^{2,1}(X)).

The (topological) Euler characteristic is χ(X) = 2(h^{1,1}(X) − h^{2,1}(X)).

Definition 1.2. A smooth projective Calabi–Yau threefold X over ℚ (or over any field) is called rigid if h^{2,1}(X) = h^{1,2}(X) = 0 so that B_{3}(X) = 2. Definition 1.3. Let X be a Calabi–Yau threefold defined over ℚ. Assume that X has a suitable integral model. The L–series of X is defined to be the L–series of the (semisimplification of the) Galois representation on H_{et}^{3}(X̄, ℚ_{ℓ}). That is,

L(X, s) := L(H_{et}^{3}(X̄, ℚ_{ℓ}), s).

Digression: Let X be a Calabi–Yau threefold defined over ℚ with a suitable integral model. Let p be a good prime for X, that is, the reduction X (mod p) defines a smooth projective variety over the prime field 𝔽_{p}. Let Frob_{p} denote the (geometric) Frobenius morphism of X at p. We consider the action of Frob_{p} on the ℓ-adic etale cohomology group H_{et}^{3}(X̄, ℚ_{ℓ}), and let t_{3}(p) denote the trace of the Frob_{p} on H_{et}^{3}(X̄, ℚ_{ℓ}). By the Lefschetz fixed point formula, t_{3}(p) can be determined by counting the number of 𝔽_{p}–rational points on X:

t_{3}(p) = 1 + p^{3} + (1 + p)t_{2}(p) − #X(𝔽_{p}).

Here t_{2}(p) = p Trace(ρ(Frob_{p})) where ρ is an Artin representation of dimension h^{1,1}(X) so in particular its trace is an integer which is a sum of h^{1,1}(X) roots of unity. This implies that |t_{2}(p)| ≤ p h^{1,1}(X), and for a set of primes of positive density, we get t_{2}(p) = ph^{1,1}(X). Define

P_{3,p}(T) := det(1 − Frob_{p} T | H_{et}^{3}(X̄, ℚ_{ℓ})).

If X is a rigid Calabi–Yau threefold, then P_{p,3}(T) is an integral polynomial of degree deg(P_{3,p}) = 2; it is of the form:

P_{3,p}(T) = 1 − t_{3}(p)T + p^{3}T^{2} ∈ 1 + T ℤ[T]

where t_{3}(p) is subject to the Riemann Hypothesis: |t_{3}(p)| ≤ 2p^{3/2}. The L–series L(H_{et}^{3}(X̄, ℚ_{ℓ}), s) is then given by

L(H_{et}^{3}(X̄, ℚ_{ℓ}), s) = (∗) ∏_{p} P_{3,p}(p^{−s})^{−1}

where p runs over good primes, and (∗) is the Euler factor corresponding to bad primes.

2. The modularity conjecture for rigid Calabi–Yau threefolds over ℚ

Let k ≥ 1 be an integer. Let Γ be an arithmetic subgroup of SL_{2}(ℤ) of finite index. We denote by S_{k}(Γ) the complex vector space of all cusp forms of weight k with respect to Γ. We now formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over ℚ.

Conjecture 2.1. The modularity conjecture: Any rigid Calabi–Yau threefold X defined over ℚ is modular in the sense that its L–series of X coincides with the Mellin transform of the L–series of a modular (cusp) form f of weight 4 on Γ_{0}(N). Here N is a positive integer divisible by the primes of bad reduction. More precisely, we have, up to a finite Euler factors,

L(X, s) = L(f, s) for f ∈ S_{4}(Γ_{0}(N)).

Here are some justifications for formulating the modularity conjecture for rigid Calabi– Yau threefolds over ℚ.

Remark 2.1. The conjecture of Taniyama–Shimura–Weil that every elliptic curve over ℚ is modular, has been established by Wiles and his former students in totality (see Breuil, Conrad, Diamond and Taylor [BCDT]). Noting that elliptic curves are dimension one Calabi–Yau varieties, our modularity conjecture (2.1) may be regarded a dimension three generalization of the Taniyama–Shimura–Weil conjecture to rigid Calabi–Yau threefolds over ℚ.

Remark 2.2. Livné [Li2] considered a rank 2 motive M over ℚ with Hodge numbers h^{p,q}(M) = h^{q,p}(M) = 1 where p > q, which respects an orthogonal form up to similitudes. Livnè showed how to express the L–series of M in terms of Hecke characters.

The examples of rank 2 motives M considered by Livnè arose from an elliptic curve E over ℚ or from a singular K3 surface X, i.e., M = H^{1}(E) ⊗ ℚ or H^{2}(X, ℤ)/Pic(X) ⊗ ℚ.

In the latter case, Shioda and Inose [SI] determined the L–series of a singular K3 surface X (up to a finite numbers of Euler factors) passing to some finite extension of ℚ:

L(X, s) = L(ψ_{1}^{2}, s)L(ψ̄_{1}^{2}, s)

where ψ_{1} is the Hecke character associated to an elliptic curve with complex multiplication.

Remark 2.3. Fontaine and Mazur [FM] have conjectured that all irreducible odd 2– dimensional Galois representations “coming from geometry” should be modular, up to a Tate twist. Our modularity conjecture (2.1) may be regarded as a concrete realization of the Fontaine–Mazur conjecture. For a recent progress on a conjecture of Fontaine–Mazur, the reader is referred to a paper of Richard Taylor [T].

Remark 2.4. Let X be a projective variety of odd dimension m over ℚ such that the m-th Betti number B_{m} = dim H^{m}(X ⊗ ℂ, ℂ) = 2 and H^{m}(X ⊗ ℂ, ℂ) has the Hodge decomposition of type (m, 0) + (0, m). Serre [Sr] has formulated a modularity conjecture for the residual mod p 2-dimensional Galois representation attached to X for all primes. In particular, our modularity conjecture 2.1 is a special case of the conjectural Theorem 6 of Serre [Sr] for m = 3. Serre has informed us about this in his email dated June 23, 2000. We are thankful to him for pointing this out.

3. A rigid Calabi–Yau threefold arising from the root lattice A_{3}

This example of a rigid Calabi–Yau threefold is constructed from the root lattice A_{3} via a toric construction.

(3.1) Toric construction. For the general backgrounds on toric varieties, the reader is referred to Batyrev [Bat], Fulton [Fu] and Fulton and Harris [FH].

We consider a root system R of rank r. Let L_{R} be the root lattice generated by R, and let L_{R}^{∗} be its dual lattice. Let Σ_{R} be the fan in L_{R}^{∗} ⊗ ℚ. The fan Σ_{R} gives rise to a toric variety, which is denoted by X(Σ_{R}). Let Δ_{R} be a polyhedron in L_{R} ⊗ ℚ with vertices in R, and let L(Δ_{R}) denote the space of Laurent polynomials with support in Δ_{R}. For each root r ∈ R, assign a monomial e^{r}, and define a Laurent polynomial

χ_{R} := ∑_{r∈R} e^{r} ∈ L(Δ_{R}).

Then we have a rational function χ_{R} : X(Σ_{R}) → ℙ^{1}. For each λ ∈ ℙ^{1}, define

X_{λ} := \overline{χ_{R}^{−1}(λ)} ⊂ X(Σ_{R}).

There is a base locus B. Let B : X_{R} → X (Σ_{R}) be the blow up of the base locus. Then we obtain a pencil of varieties

X_{R} := \{(λ, x) ∈ ℙ^{1} × X(Σ_{R}) | \overline{χ_{R}(x) = λ}\}.

We are interested in root lattices R which give rise to Calabi–Yau threefolds X_{R}.

(3.2) The root lattice A_{3} and the construction of a Calabi–Yau threefold. Verrill[V] constructed a Calabi–Yau threefold associated to the root lattice A_{3}. We will briefly describe Verrill’s construction. Let {E_{i} | i = 1, 2, 3, 4} be the standard basis for ℝ^{4}. The root lattice A_{3} is a sublattice of ℝ^{4} of rank 3 generated by v_{1} := E_{1} − E_{2}, v_{2} := E_{2} − E_{3}, v_{3} := E_{3} − E_{4} and the collection of all roots is given by the set

{E_{i} − E_{j} | 1 ≤ i, j ≤ 4, i ≠ j}.

To a root E_{i} − E_{j}, we associate a monomial X_{i}X_{j}^{−1} by putting X_{i} = e^{E_{i}}. Then the character of the adjoint representation is given by

χ_{A_{3}} = X_{1}X_{2}^{−1} + X_{1}X_{3}^{−1} + X_{1}X_{4}^{−1} + X_{2}X_{3}^{−1} + X_{2}X_{4}^{−1} + X_{3}X_{4}^{−1}
+X_{2}X_{1}^{−1} + X_{3}X_{1}^{−1} + X_{4}X_{1}^{−1} + X_{3}X_{2}^{−1} + X_{4}X_{2}^{−1} + X_{4}X_{3}^{−1}
= (X_{1} + X_{2} + X_{3} + X_{4})(X_{1}^{−1} + X_{2}^{−1} + X_{3}^{−1} + X_{4}^{−1}) − 4.

Let X(Σ_{A_{3}}) → ℙ^{1} denote the toric variety defined by χ_{A_{3}} = λ ∈ ℙ^{1}. It is a rational variety. Now we take a double cover of X (Σ_{A_{3}}) by putting λ = \frac{(t−1)^{2}}{t} with t ∈ ℙ^{1}. Then a desingularization X of the double cover is a Calabi–Yau threefold defined over ℚ whose generic fiber X_{λ} is a K3 surface.

Lemma 3.1. The numerical characters of X are given as follows:

h^{1,2} = 0, h^{3,0} = 1, h^{1,1} = 50, χ(X) = 100

and hence X is a rigid Calabi–Yau threefold.

We can see from the defining equation of X that 2 and 3 are bad primes. For this rigid Calabi–Yau threefold X over ℚ, we can establish the modularity conjecture. We will give a proof of the following theorem in the section 6 below. Verrill [V] also proved the theorem with totally different method (à la Livné [Li1]) from ours.

Theorem 3.1. Let X be the rigid Calabi–Yau threefold constructed by Verrill. Then L– series of X coincides, up to the Euler factors corresponding to primes 2 and 3, with the Mellin transform of a weight 4 newform on Γ_{0}(6). That is,

L(X, s) = L(f, s) for f ∈ S_{4}(Γ_{0}(6)).

The cusp form f has the q–expansion of the form:

f(q) = η(q)^{2}η(q^{2})^{2}η(q^{3})^{2}η(q^{6})^{2}.

4. The self-products of elliptic modular surfaces.

Let Γ be an arithmetic subgroup of SL_{2}(ℤ) with no torsion elements and let H be the upper half complex plane. Then we can define the modular curve C_{Γ}^{0} := H/Γ and its compactification C_{Γ} := \overline{H/Γ} = H/Γ ∪ \{cusps of Γ\}. It is known (cf. [D]) that the universal family of elliptic curves

f^{0} : S_{Γ}^{0} ⟶ C_{Γ}^{0} = H/Γ

exists, and that its Néron model has the unique smooth minimal compactification

f := f_{Γ} : S_{Γ} ⟶ C_{Γ} = \overline{H/Γ}.

The fibration f : S_{Γ} ⟶ C_{Γ} is called the elliptic modular surface associated to Γ.

Let us now recall some basic facts about elliptic modular surfaces, which are relevant to our subsequent discussions. Consider the following diagram

S_{Γ}^{0} ↪ S_{Γ}
f^{0} ↓  f ↓
C_{Γ}^{0}  ↪^{j} C_{Γ}.

Making use of the invariant cycle theorem, we obtain an isomorphism of sheaves

R^{1}f_{∗}ℚ_{S_{Γ}} ≃ j_{∗}R^{1}f_{∗}^{0}ℚ_{S_{Γ}^{0}}.

In fact, we have the natural map

R^{1}f_{∗}ℚ_{S_{Γ}} ⟶ j_{∗}R^{1}f_{∗}^{0}ℚ_{S_{Γ}^{0}}  (1)

which is an isomorphim over C_{Γ}^{0} = C_{Γ} \ Σ where Σ = {p_{1}, ⋯, p_{r}} is the set of cusps of Γ. Looking at the stalks at a cusp p_{i}, the morphism

(R^{1}f_{∗}ℚ_{S_{Γ}})_{p_{i}} ⟶ (j_{∗}R^{1}f_{∗}^{0}ℚ_{S_{Γ}^{0}})_{p_{i}}  (2)

is surjective by local invariant cycle theorem (cf. e.g. [Proposition 15.12, [Z]]). On the other hand, it is easy to see that both of the stalks in (2) are isomorphic to a onedimensional ℚ-vector space. Hence the linear map (2) is an isomorphism.

Let V denote the natural representation of SL_{2}(ℚ), and let S^{k}(V) be the k-th symmetric tensor representation of V. Then it is known ([Z], Proposition 12.5) that for each k ≥ 0 there exists an isomorphism

H̃^{1}(Γ, S^{k}(V)) ≃ H^{1}(C_{Γ}, j_{∗}(S^{k}(R^{1}f_{∗}^{0}ℚ_{S_{Γ}^{0}}))) for each k ≥ 0.  (3)

Here H̃^{1}(Γ, S^{k}(V)) is the parabolic cohomology group associated to the representation of Γ on S^{k}(V). Let ω_{S_{Γ}/C_{Γ}} = K_{S_{Γ}} ⊗ f^{∗}(K_{C_{Γ}}^{−1}) be the relative canonical bundle of f, and we set

ω = f_{∗}(ω_{S_{Γ}/C_{Γ}}).

Then we see that ω is an invertible sheaf on C_{Γ}.

Now we are ready to describe an isomorphism, the so-called Shimura isomorphism and its far-reaching consequences. Proposition 4.1. ([Sh], [D], [Z].) There exists a natural isomorphism (the Shimura isomorphism)

H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1} ⊗ ω^{k}) ≃ S_{k+2}(Γ)  (4)

and it gives rise to the following commutative diagram:

H̃^{1}(Γ, S^{k}(V)) ⊗_{ℚ} ℂ  ≃  S_{k+2}(Γ) ⊕ \overline{S_{k+2}(Γ)}
↓  ↓
H^{1}(C_{Γ}, j_{∗}(S^{k}(R^{1}f_{∗}^{0}ℂ_{S_{Γ}^{0}}))) ≃ H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1} ⊗ ω^{k}) ⊕ \overline{H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1} ⊗ ω^{k})}

The first vertical arrow is the isomorphism induced by (3).

With the Shimura isomorphism at our disposal, we may now characterize rational elliptic modular surfaces. Proposition 4.2. The elliptic modular surface associated to Γ, f : S_{Γ} ⟶ C_{Γ} is a rational elliptic surface, if and only if

dim S_{2}(Γ) = dim S_{3}(Γ) = 0.

Proof. From the formula for the canonical bundle of the elliptic fibration f : S_{Γ} ⟶ C_{Γ}, we see that

K_{S_{Γ}} = f^{∗}(Ω_{C_{Γ}}^{1} ⊗ ω).  (5)

The formula (5) combined with the Shimura isomorphism (4) then give rise to the isomorphisms

H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1}) ≃ S_{2}(Γ), H^{0}(S_{Γ}, K_{S_{Γ}}) ≃ H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1} ⊗ ω) ≃ S_{3}(Γ).  (6)

Suppose that S_{Γ} is a rational surface. Then C_{Γ} is also a rational curve, so that dim S_{2}(Γ) = 0. We also have dim H^{0}(S_{Γ}, K_{S_{Γ}}) = 0, and hence dim S_{3}(Γ) = 0.

Conversely, assume that dim S_{k}(Γ) = 0 for k = 2, 3. Then again the Shimura isomorphism (4) implies that H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1}) = 0. Hence C_{Γ} ≃ ℙ^{1}. Since f is not isotrivial, the formula for the canonical bundle asserts that we may write ω as ω = O_{ℙ^{1}}(a) and K_{S_{Γ}} = f^{∗}(O_{ℙ^{1}}(a − 2)) with a suitable positive integer a. Note that the positivity of a follows from the positivity of the direct image of the relative canonical sheaf ω = f_{∗}(ω_{S_{Γ}/C_{Γ}}) = ω.

Under this situation, dim S_{3}(Γ) = dim H^{0}(S_{Γ}, K_{S_{Γ}}) = dim H^{0}(ℙ^{1}, O_{ℙ^{1}}(a − 2)) = 0 implies that a = 1. We also have the isomorphisms

f_{∗}O_{S_{Γ}} ≃ O_{ℙ^{1}}, R^{1}f_{∗}O_{S_{Γ}} ≃ ω^{∨} ≃ O_{ℙ^{1}}(−1).

Further the Leray spectral sequence shows that H^{1}(S_{Γ}, O_{S_{Γ}}) = 0. Moreover, for every k > 0, K_{S_{Γ}}^{k} = f^{∗}(O_{ℙ^{1}}(−k)) has no section. These facts finally imply that S_{Γ} is a rational surface. q.e.d

Next, let us consider the self-fiber product of the elliptic modular surface associated to Γ

f^{2} = f × f : S_{Γ}^{2} = S_{Γ} ×_{C_{Γ}} S_{Γ} ⟶ C_{Γ}

In general, S_{Γ}^{2} has singularities which arise from the critical points of the map f : S_{Γ} ⟶ C_{Γ}. However, if f has only semistable fibers (that is, the singular fibers are all reduced and have only nodal singularities), then the fiber product S_{Γ}^{2} has only ordinary double points as its singularities.

Proposition 4.3. Suppose that f : S_{Γ} ⟶ C_{Γ} has only semistable fibers. Then there exists a small projective resolution π : S̃_{Γ}^{2} ⟶ S_{Γ}^{2}. Moreover, if we set h = f^{2} ∘ π, then the canonical bundle of S̃_{Γ}^{2} can be written as

KS̃_{Γ}^{2} ≃ (h)^{∗}(Ω_{C_{Γ}}^{1} ⊗ ω^{2}).  (7)

Proof. The first assertion follows from Lemma 3.1 [Sch2]. Since π is a small resolution, KS̃_{Γ}^{2} = π^{∗}(K_{S_{Γ}^{2}}). On the other hand, since S_{Γ}^{2} is a hypersurface in S_{Γ} × S_{Γ} (i.e., it is the pull-back of the diagonal △ : C_{Γ} ↪ C_{Γ} × C_{Γ}), we can derive from the adjunction formula that

K_{S_{Γ}^{2}} = (K_{S_{Γ}×S_{Γ}} + f^{−1}(△(C_{Γ})))_{|f^{−1}(△(C_{Γ}))}
= (f^{2})^{∗}((Ω_{C_{Γ}}^{1})^{⊗2} ⊗ ω^{2} ⊗ (Ω_{C_{Γ}}^{1})^{−1})
= (f^{2})^{∗}(Ω_{Γ}^{1} ⊗ ω^{2}).

This gives rise to the formula (7). q.e.d.

Corollary 4.1. (cf. [Sok].) Under the assumption of Proposition 4.3, we have a canonical isomorphism

H^{0}(S̃_{Γ}^{2}, KS̃_{Γ}^{2}) ≃ H^{0}(C_{Γ}, Ω_{C_{Γ}}^{1} ⊗ ω^{2}) ≃ S_{4}(Γ).

For the full cohomology group H^{3}(S̃_{Γ}^{2}, ℚ), the following fact holds (cf. §1, [Sch1]). Lemma 4.1. (Lemma 1.7, [Sch1].) The rational Hodge structure of H^{3}(S̃_{Γ}^{2}, ℚ) is isomorphic to the direct sum of three copies of H^{1}(C_{Γ}, ℚ)[−1] and a piece of pure type (3, 0), (0, 3). In particular, if C_{Γ} is a rational curve, then H^{3}(S̃_{Γ}^{2}, ℚ) is of pure type (3, 0), (0, 3).

Corollary 4.2. Assume that S_{Γ} ⟶ C_{Γ} is a rational elliptic modular surface associated to Γ with only semistable singular fibers. Then any small resolution S̃_{Γ}^{2} of the self-product S_{Γ}^{2} is a rigid Calabi–Yau threefold.

Proof. Since f : S_{Γ} ⟶ C_{Γ} is a rational surface, it is immediate that C_{Γ} ≃ ℙ^{1}. Let h : S̃_{Γ}^{2} ⟶ ℙ^{1} be the natural fibration. We may write ω as ω = O_{ℙ^{1}}(1) using the formula (7). Then we see that

KS̃_{Γ}^{2} = h^{∗}(Ω_{ℙ^{1}}^{1} ⊗ ω^{2}) = h^{∗}(O_{ℙ^{1}}(−2 + 2)) = OS̃_{Γ}^{2}.

Recall from Lemma (4.1) that the rational Hodge structure H^{3}(S̃_{Γ}^{2}, ℚ) is of pure type (3, 0), (0, 3). So it follows that H^{1}(S̃_{Γ}^{2}, Ω2 ; S̃_{Γ}^{2}) = H^{1}(ΘS̃_{Γ}^{2}) = 0. We also see (cf. [Sch1]) that

h^{1}(OS̃_{Γ}^{2}) = h^{2}(OS̃_{Γ}^{2}) = 0. These facts then imply that S̃_{Γ}^{2} is a rigid Calabi–Yau threefold. q.e.d.

Remark 4.1. There are in total six arithmetic subgroups Γ ⊂ SL_{2}(ℤ) which give rise to rational elliptic modular surfaces with only semistable fibers. The list of such groups coincides with the list of Beauville [B]. (Beauville classified elliptic surfaces with exactly four semistable fibers (i.e., of type I_{b}, b ≥ 1) over ℙ^{1}.)

In Table 1, groups Γ are listed in the first column, 4 singular fibers of type I_{b}, b ≥ 1 in the second column and h^{1,1} (resp. the Euler characteristic χ) of the corresponding Calabi– Yau threefolds S̃_{Γ}^{2} in the third (resp. fourth) column. Note that the Euler characteristic χ(S̃_{Γ}^{2}) is exactly equal to 2h^{1,1}. This is because S̃_{Γ}^{2} is rigid. Also note that for any group Γ in Table 1, dim S_{4}(Γ) = 1. This follows from the isomorphism S_{4}(Γ) ≃ H^{3,0}(S̃_{Γ}^{2}) ≃ ℂ.

Table 1

Γ  h^{1,1}(S̃_{Γ}^{2}) χ(S̃_{Γ}^{2})
Γ(3)  I_{3} I_{3} I_{3} I_{3}  36  72
Γ_{1}(4) ∩ Γ(2) I_{4} I_{4} I_{2} I_{2}  40  80
Γ_{1}(5)  I_{5} I_{5} I_{1} I_{1}  52  104
Γ_{1}(6)  I_{6} I_{3} I_{2} I_{1}  50  100
Γ_{0}(8) ∩ Γ_{1}(4) I_{8} I_{2} I_{1} I_{1}  70  140
Γ_{0}(9) ∩ Γ_{1}(3) I_{9} I_{1} I_{1} I_{1}  84  168

Now we quote a theorem which is a special case of the fundamental result due to Sato– Kuga–Shimura ([KS]; cf. [D]). We make use the argument of [D], where Deligne worked over the case when Γ = Γ(N) (full modular case). However his argument works for other cases like Γ = Γ_{1}(N) as far as the universal family of generalized elliptic curves exists. Theorem 4.1. Let Γ be an arithmetic subgroup of SL_{2}(ℤ) listed in Table 1, and let f : S_{Γ} ⟶ ℙ^{1} the rational elliptic surface associated to Γ. Let f^{2} : S̃_{Γ}^{2} ⟶ ℙ^{1} be the smooth rigid Calabi–Yau threefold arising from the self-product S_{Γ}^{2}. Then S̃_{Γ}^{2} is modular, that is, the L-series of S̃_{Γ}^{2} coincides with the Mellin transform of a weight 4 newform of Γ up to finite Euler factors coming from the bad primes:

L(S̃_{Γ}^{2}, s) = L(f, s) for f ∈ S_{4}(Γ).

Remark 4.2. The bad primes for S̃_{Γ}^{2} depends on the level of the discrete group Γ. For example, we can easily see that only 2 and 3 are the bad primes for Γ_{1}(6).

5. The geometry of Verrill’s Calabi–Yau threefold

In this section, we will look into the geometric structure of Verrill’s rigid Calabi– Yau threefold X constructed in Section 3. Note that from now on all varieties are defined over ℚ.

We denote by [x : y : z : w] the homogeneous coordinate of ℙ^{3} and by t the inhomogeneous coordinate of ℙ^{1}. Then Verrill’s rigid Calabi–Yau threefold X is a desingularization of the hypersurface in ℙ^{3} × ℙ^{1}:

X′ = \{(x + y + z + w)(x^{−1} + y^{−1} + z^{−1} + w^{−1}) − \frac{(t − 1)^{2}}{t} + 4 = 0\} ⊂ ℙ^{3} × ℙ^{1}. (8)

It is easily seen that this equation is equivalent to the following equation:

F (x, y, z, w, t) = (x + y + z + w) · [(yz + zx + xy)w + xyz] · t  (9)
−(t + 1)^{2} · xyzw = 0.

The elliptic modular surface associated to Γ_{1}(6).

Let us consider the hypersurface S in ℙ^{1} × ℙ^{2} defined by

H(x, y, z, s) = (s + 1)xyz − (x + y + z)(yz + zx + xy) = 0.  (10)

The hypersurface S has three singular points at

[s, [x : y : z]] = [∞, [1 : 0 : 0]], [∞, [0 : 1 : 0]], [∞, [0 : 0 : 1]].

Let

π : S̃ ⟶ S

be the blowing up of these three points. Then it is easy to see that S̃ is the minimal resolution of S and f = p_{1} ∘ π : S̃ ⟶ ℙ^{1} induces the structure of the elliptic surface. Moreover the rational elliptic surface f : S̃ ⟶ ℙ^{1} has only four semistable singular fibers of types I_{1}, I_{2}, I_{3}, I_{6} (see Table 2):

Table 2

s  8 −1 0 ∞
singular fiber I_{1} I_{2} I_{3} I_{6}

By [B], f : S̃ ⟶ ℙ^{1} is an integral model of the elliptic modular surface

f : S_{Γ_{1}(6)} ⟶ \overline{H/Γ_{1}(6)} ≃ ℙ^{1}

of Γ_{1}(6).

Theorem 5.1. Verrill’s Calabi–Yau threefold X defined in (8) is birationally equivalent over ℚ to a crepant resolution S̃_{Γ_{1}(6)}^{2} of the self-product of the elliptic modular surface S_{Γ_{1}(6)}. More precisely, there exists a birational map defined over ℚ between X and S̃_{Γ_{1}(6)}^{2}.

Remark 5.1. There are two fibrations, one is the fibration of X ⟶ ℙ^{1} of K3 surfaces and the other is the fibration f^{2} : S̃_{Γ_{1}(6)}^{2} ⟶ ℙ^{1} of abelian surfaces. We will show below that these two fibrations are not equal. (However, we do not know any relation (apart from being non-equal) between these two fibrations.)

Proof of Theorem 5.1: Birational transformations.

To prove Theorem 5.1, we will construct a birational map over ℚ between the two varieties explicitly. We define the birational map:

\begin{matrix}π : ℙ^{2} × ℙ^{2} & ⟶ & ℙ^{3} × ℙ^{1} \\ [x : y : z] × [T : W : U] & ↦ & [x : y : z : w] × (t) \end{matrix}

by putting

x = x, y = y, z = z, t = T/U, w = W · (x + y + z)/U.

Pulling back the equation F (x, y, z, w, t) in (9) via π, we obtain the equation

F̃ : = (x + y + z)^{2}(yz + xz + xy)T W^{2}
−(x + y + z) · [(x + y + z)(yz + xz + xy) − xyz]T W U
+xyz(x + y + z)T U^{2} − (x + y + z)xyz[T^{2} + U^{2}]W = 0

Consider the hypersurface X′′ ⊂ ℙ^{2} × ℙ^{2} defined by F̃ = 0. Of course, X′′ is birationally equivalent to the original hypersurface X′ in (8). We have the following commutative diagram:

X′′  ↪^{ι} ℙ^{2} × ℙ^{2}
f_{1} ↓ ↙ p_{1}
ℙ^{2}

Now recall the hypersurface S ⊂ ℙ^{1} × ℙ^{2} defined in (10). Considering the birational map μ = (p_{2})_{|S} : S ⟶ ℙ^{2} and taking the pull-back morphism f_{1} by μ, we obtain the following commutative diagram:

ℙ^{1} × ℙ^{2} × ℙ^{2} ⊃ X′′′  ⟶^{ν} X′′
↓ p_{12}  f′_{1} ↓  f_{1} ↓
ℙ^{1} × ℙ^{2} ⊃ S ⟶^{μ} ℙ^{2}

Here we set X′′′ := S ×_{ℙ^{2}} X′′. Note that the natural induced map ν : X′′′ ⟶ X′′ is a birational morphism. Now we see that X′′′ is a complete intersection defined by the equations

\begin{aligned}F̃ & = & 0 \\ H(x, y, z) & = & (s + 1)(xyz) − (x + y + z)(yz + zx + xy) = 0. \end{aligned}

Now using the second equation, F̃ = 0 can be transformed into the following equation:

F̃ : = (x + y + z) · (s + 1)xyz · T W^{2}
+(x + y + z) · [(s + 1)xyz − xyz]T W U
+xyz(x + y + z)T U^{2} − (x + y + z)[T^{2} + U^{2}]xyzW
= (x + y + z)xyz × {(s + 1) · T W^{2} − sT W U + T U^{2} − T^{2}W − W U^{2}}

Forgetting the factor (x + y + z)xyz from F̃, we obtain the equation

G(T, W, U) = (s + 1) · T W^{2} + sT W U + T U^{2} − T^{2}W − W U^{2} = 0.

Therefore we see that the subvariety X′′′ is isomorphic to the complete intersection in ℙ^{1} × ℙ^{2} × ℙ^{2} defined by the equations

\begin{aligned}G(T, W, U) & = & (s + 1)T W^{2} + sT W U + T U^{2} − T^{2}W − W U^{2} = 0, \\ H(x, y, z) & = & (s + 1)xyz − (x + y + z)(yz + zx + xy) = 0. \end{aligned}

Next, considering the the birational transformation of ℙ^{2} given by

T = Z · (X + Y + Z), W = −XY, U = Y (X + Y + Z).

Then the polynomial G(T, W, U) above can be transformed into the polynomial

G̃(X, Y, Z) = (s + 1)XY Z − (X + Y + Z)(Y Z + ZX + XY),

which is nothing but H(X, Y, Z). Hence the complete intersection X′′′ ⊂ ℙ^{1} × ℙ^{2} × ℙ^{2} is isomorphic to the fiber product S ×_{ℙ^{1}} S ⟶ ℙ^{1} of the elliptic surface f : S ⟶ ℙ^{1} defined by the equation H(x, y, z) = 0. Hence, X′ defined in (8) is birationally equivalent to the self-product S ×_{ℙ^{1}} S of S. Since the minimal resolution of S is isomorphic to the elliptic modular surface S_{Γ_{1}(6)}, X is birationally equivalent to the self product S̃_{Γ_{1}(6)}^{2}. Observing that the birational maps discussed above are all defined over ℚ, this completes the proof of Theorem 5.1. q.e.d.

6. Proof for the modularity of X (Theorem 3.1)

Since the birational map between Verrill’s example X and S̃_{Γ_{1}(6)}^{2} constructed in Section 5 is defined over ℚ, we obtain the following Lemma. Lemma 6.1. The birational map constructed in Section 5 gives rise to the following isomorphism compatible with the Galois action up to bad primes 2 and 3

H_{et}^{3}(X̄, ℚ_{ℓ}) ≃ H_{et}^{3}(S̄_{Γ_{1}(6)}^{2}, ℚ_{ℓ}).

Consequently, we have

L(X, s) = L(S̃_{Γ_{1}(6)}^{2}, s).

up to Euler factors at bad primes 2 and 3.

For a subgroup Γ ⊂ SL_{2}(ℤ), its projectivization Γ/{±I_{2}} will be denoted by P Γ. By

virtue of Theorem 4.1, Theorem 5.1, Remark 4.2 and Lemma 6.1, the proof of Theorem 3.1 (the modularity of Verrill’s Calabi–Yau threefold X) is reduced to the proof of the following lemma. Lemma 6.2. The projective image of Γ_{1}(N) and Γ_{0}(N) are the same in PSL_{2}(ℤ) if and only if N is a divisor of 4 or a divisor of 6. In particular, ℙΓ_{1}(6) = ℙΓ_{0}(6) and so it follows that S_{4}(ℙΓ_{1}(6)) ≃ S_{4}(ℙΓ_{0}(6)).

Proof. Let (\begin{matrix}a & b \\ c & d \end{matrix}) be a matrix in Γ_{0}(N) or Γ_{1}(N). Then ad ≡ 1 (mod N). The

first assertion follows from the fact that the only positive integers N for which ad ≡ 1 (mod N) implies that a ≡ d ≡ ±1 (mod N) are the divisors of 4 and 6. q.e.d.

Remark 6.1. Verrill [V] has an alternative proof of the modularity for X. Her proof is along the lines of Livné’s paper [Li1]. It makes use of the Serre criterion [Sr] and Faltings results [F] to prove equality of two L–series. The main point was to show that finitely many Euler factors of two L–series coincide.

Remark 6.2. Our geometric proof of the modularity conjecture for X guarantees that the 2–dimensional Galois representations associated to X and that attached to the modular form are indeed isomorphic (i.e., have the same semi-simplification). This provides a confirmation to the conjecture of Fontaine and Mazur [FM].

We close this section by posing an open problem. Problem 6.1. May one use the method of Wiles to establish the modularity of X? More concretely, find a single good prime ℓ and establish the modularity for the residual mod ℓ Galois representation associated to the rigid Calabi–Yau threefold X in question.

7. The intermediate Jacobians of rigid Calabi–Yau threefolds

In this section, we shall define the intermediate Jacobian for Calabi–Yau threefolds, following the exposition of Bloch [Bl].

Definition 7.1. Let X be a smooth projective Calabi–Yau threefold defined over ℂ. There is a Hodge filtration

H^{3}(X, ℂ) = F^{0} ⊃ F^{1} ⊃ F^{2} ⊃ F^{3} ⊃ (0)

where

F^{1} = H^{3,0} ⊕ H^{2,1} ⊕ H^{1,2}
F^{2} = H^{3,0} ⊕ H^{2,1}
F^{3} = H^{3,0}.

This complex vector space also has a ℤ–structure defined by the image H^{3}(X, ℤ) → H^{3}(X, ℂ).

Now we recall some key properties of Hodge filtrations not only for H^{3} but for more general cases. We write F for generic filtrations. We have

F^{i} ⊕ \overline{F^{2r−i}} ≃ H^{2r−1}(X, ℂ), F^{i} ∩ \overline{F^{2r−i−1}} ≃ H^{i,2r−i−1}

where H^{i,j} ≃ H^{j}(X, Ω_{X}^{i}) and r ∈ {1, 2, 3}. From this, one can compute that

dim_{ℂ}F^{r} = \frac{1}{2}dim_{ℂ}H^{2r−1}(X, ℂ)

and

F^{r} ∩ Image(H^{2r−1}(X, ℤ) → H^{2r−1}(X, ℂ)) = (0).

Therefore, the quotient

J^{r}(X) = H^{2r−1}(X, ℂ)/(F^{r} + H^{2r−1}(X, ℤ))

is a compact complex torus, called the intermediate Jacobian. By the Poincaré duality, we have

H^{2r−1}(X, ℂ)/F^{r} ≃ F^{4−r} H^{7−2r}(X, ℂ)^{∗}

where ∗ denotes ℂ–linear dual. It then follows that

J^{r}(X) ≃ F^{4−r} H^{7−2r}(X, ℂ)^{∗}/H_{7−2r}(X, ℤ).

Digression. Now assume that X is rigid, and take r = 2 and F = F. Then we have

F^{1} = H^{3,0} = F^{2} = F^{3}

and

J^{2}(X) ≃ H^{3}(X, ℂ)/(F^{2} + H^{3}(X, ℤ)) ≃ H^{3,0}(X)^{∗}/H_{3}(X, ℤ)

is a complex torus of dimension one. This means that there is an elliptic curve E such that E(ℂ) ≃ J^{2}(X). We will formulate the following question.

Question 7.1. Is it true that a rigid Calabi–Yau threefold defined over ℚ is modular if and only if the intermediate Jacobian is an elliptic curve defined over ℚ?

Acknowledgments

We thank several people who read through the earlier versions of this paper, and gave us constructive comments and suggestions. These include I. Chen, K. Khuri-Makdisi, R. Livné, A. Sebbar, and J.-P. Serre.

At the early stages of this work, M.-H. Saito was a visiting Professor at Queen’s University in the summer of 1998. The visit was made possible by the grant from JSPS.

During the course of this work, Yui was a visiting Professor at various universities and research institutes, e.g., CRM Montréal, The Fields Institute, MSRI Berkeley, Max-Planck-Institut Bonn, Chiba University, Tsuda College, Kyoto University and Tôhoku University. She is grateful for hospitality and supports of all these institutions.

References

[Bat] Batyrev, V., Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994), pp. 493–533.

[B] Beauville, A., Les familles stables de courbes elliptiques sur ℙ^{1} admettant quatre fibres singulières, C. R. Acad. Sc. Paris, Série I, t. 294 (1982), pp. 657–660.

[Bl] Bloch, S., Lectures on Algebraic Cycles, Duke University Mathematical Series IV, 1980.

[BCDT] Brueil, C., Conrad, B., Diamond, F., and Taylor, R., On the modularity of elliptic curves over ℚ, preprint 1999.

[D] Deligne, P., Formes modulaires et représentations ℓ-adiques, Sem. Bourbaki 355 (1968/69), Lecture Notes in Math. 179, Springer–Verlag 1979.

[F] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), pp.

[Fu] Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton University Press, Princeton, NJ, 1993.

[FH] Fulton, W., and Harris, J., Representation Theory: A First Course, Graduate Texts in Math. 129, Springer–Verlag 1991.

[FM] Fontaine, M., and Mazur, B., Geometric Galois representations, in “Elliptic Curves, Modular Forms, and Fermat’s Last Theorem” (Hong Kong 1993), Ser. Number Theory 1, Internat. Press, Cambridge, MA 1995, pp. 41–78.

[KS] Kuga, M., and Shimura, G., On the zeta-function of a fiber variety whose fibers are abelian varieties, Ann. of Math. 82 (2) (1965), pp. 478–539.

[Li1] Livné, R., Cubic exponential sums and Galois representations, in “Current Trends in Arithmetic Algebraic Geometry”, Contemporary Mathematics 67, 1987, Amer. Math. Soc., pp. 187–201.

[Li2] Livné, R., Motivic orthogonal two-dimensional representations of Gal(ℚ̄/ℚ), Israel J. Math. 92 (1995), pp. 149–156.

[Sch1] Schoen, C., Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 No.3, (1986), pp. 771–794.

[Sch2] Schoen, C., On fiber products of rational elliptic surfaces with section, Math. Z. 197 (1988) No.2, pp. 177–199.

[Scl] Scholl, A., Motives for modular forms, Invent. Math. 100, (1990), pp. 419–430

[Sr] Serre, J.-P., Sur les représentations modulaires de degré 2 de Gal(ℚ̄/ℚ), Duke Math. J. 54 (1987), pp. 179–230.

[SI] Shioda, T., and Inose, H., On singular K3 surfaces, in Complex and Algebraic Geometry, Cambridge University Press, 1977, pp. 117–136.

[Sh] Shimura, G., Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11, 1959, pp.291–311.

[Sok] Sokurov, V., Holomorphic differential forms of highest degree on Kuga’s modular varieties, Mat. Sb. (N.S) 101 (143) (1976), No.1, pp. 131–157 (160).

[T] Taylor, R., Remarks on a conjecture of Fontaine and Mazur, preprint May 2000.

[V] Verrill, H., The L–series of certain rigid Calabi–Yau threefolds, J. Number Theory 81 (2000), pp. 310–334.

[Z] Zucker, S. Hodge theory with degenerating coefficients. L_{2} cohomology in the Poincaré metric, Ann. of Math. (2) 109 No.3, (1979), pp. 415–476.

Department of Mathematics, Faculty of Science, Kobe University, Rokko, Kobe, 657-8501, Japan

  E-mail address: mhsaito@math.kobe-u.ac.jp

Department of Mathematics, Queen’s University, Kingston. Ontario Canada K7L 3N6

  E-mail address: yui@mast.queensu.ca; yui@fields.utoronto.ca