Mordell-Weil lattices and certain Calabi-Yau threefolds

Masa-Hiko Saito

Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-01, Japan

1 Mordell-Weil lattices for Jacobians

Let K = k(C) be the rational function field of a smooth projective curve C defined over an algebraically closed field k. Let Γ be a smooth projective curve defined over K of genus g ≥ 1 and let J_{Γ} be the Jacobian variety of Γ, which is an abelian variety defined over K. We denote by Γ(K) (resp. J_{Γ}(K)) the set of K-rational points of Γ (resp. of J_{Γ}). The group structure of J_{Γ} induces the structure of an abelian group on J_{Γ}(K), which is called the Mordell-Weil group of J_{Γ} (or the Mordell-Weil group of Γ/K by abuse of language).

The Mordell-Weil group can be considered geometrically as follows. By theory of smooth minimal models of algebraic surfaces, there exists a proper surjective morphism

(1.1)  f : X ⟶ C

from a smooth projective surface X to the curve C whose generic fiber X_{η} is isomorphic to Γ (over K). Moreover we can assume that there are no exceptional curves E of the first kind in any closed fiber of f. Such a model is unique up to isomorphism. By using this model, a K-rational point of Γ corresponds to a regular algebraic section σ : C ⟶ X of f. Since J_{Γ} is an abelian variety over K, we can also obtain the unique good model, that is, the Néron model of J_{Γ}

(1.2)  h : J ⟶ C,

which is a group scheme over C (and whose generic fiber is J_{Γ}). (Note that h : J ⟶ C is not necessarily proper.)

Let S(J /C) denote the group of sections of h : J ⟶ C. Then we have the canonical isomorphism

(1.3)  J_{Γ}(K) ≃ S(J /C).

It is known that the Mordell-Weil group J_{Γ}(K) is finitely generated if the K/k-trace of J_{Γ} is trivial.

Shioda ([Sh1], [Sh2]) described the Mordell-Weil group J_{Γ}(K) as follows. Let NS(X) be the Néron-Severi group of the surface X. Then it is known that NS(X) is a finitely generated abelian group with the intersection pairing

( , ) : NS(X) × NS(X) ⟶ Z.

From now on we assume that there exists a section σ_{0} : C ⟶ X.

We set O = σ_{0}(C) and denote by F a general closed fiber and consider them as elements in NS(X). We define subgroups U, T of NS(X) by

U =< O, F >, T =< U, all irreducible components of closed fibers > .

Clearly, we have U ⊂ T ⊂ NS(X). Moreover we set L = T^{⊥} ⊂ NS(X).

The following theorems are fundamental results of Shioda([Sh1], [Sh2]).

Theorem 1.1 Assume that the K/k-trace of J_{Γ} is trivial. Then there exists a group isomorphism

(1.4)  NS(X)/T ≃ J_{Γ}(K).

Theorem 1.2 Assume that the K/k-trace is trivial and NS(X) is torsion-free. Then we have the natural homomorphism

(1.5)  ϕ : J_{Γ}(K) ⟶ NS(X) ⊗_{Z} Q

such that ϕ(Q) ⊥ T. The kernel of ϕ is equal to the torsion part of J_{Γ}(K) and Imϕ ⊂ L^{∗} = Hom_{Z}(L, Z).

By using theorem 1.2, one can define a pairing on J_{Γ}(K) by < P, Q >= −(ϕ(P), ϕ(Q)) ∈ Q for P, Q ∈ J_{Γ}(K). This gives a positive-definite symmetric bilinear form on J_{Γ}(K)/J_{Γ}(K)_{tor}, and Shioda called the pair (J_{Γ}(K)/J_{Γ}(K)_{tor}, < , >) the Mordell-Weil lattice.

2 Upperbounds of Mordell-Weil rank

We denote by r the Mordell-Weil rank, i.e. r = dim_{Q} J_{Γ}(K) ⊗ Q. We have the following theorem which gives an upperbound of r. (See [Sa0], [Sa1].)

Theorem 2.1 Assume that char. k = 0. Let f : X ⟶ C be as above and assume that K/k-trace of J_{Γ} is trivial. Then we have

(2.1)  r ≤ (6 + 4/g)χ(X, O_{X}) + (1 − π)\{\frac{4g^{2} − 2g − 4}{g}\}.

Here we set π = genus of C.

We remark that if p_{g}(X) = dim H^{2}(X, O_{X}) > 0 it is rather difficult to check that the inequality (2.1) is sharp. On the other hand, if p_{g}(X) = q(X)(:= dim H^{1}(X, O_{X})) = 0 (e.g. X is a rational surface) we have the following theorem. ([Sa-Sak].)

Theorem 2.2 Let f : X ⟶ C be as above and assume that p_{g}(X) = q(X) = 0. Then we have

(2.2).  r ≤ 4g + 4

Moreover there exist examples of fibrations f : X ⟶ P^{1} with p_{g}(X) = q(X) = 0 and r = 4g + 4.

We can also determine the structure of the fibration of curves of genus g ≥ 2 with p_{g}(X) = q(X) = 0 and r = 4g + 4([Sa-Sak]).

Theorem 2.3 Let f : X ⟶ P^{1} be as in Theorem 2.2 and assume that r = 4g + 4 and g ≥ 2. Then there exists a finite double covering map f : X ⟶ P^{1} × P^{1} whose branch locus B ⊂ P^{1} × P^{1} is a smooth curve of bidegree (2, 2g + 2).

Remark 2.1 In [Sa-Sak], we assume that X is a rational surface to obtain the upperbounds r ≤ 4g + 4. However the upperbound of Mordell-Weil rank in Theorem 2.1 is a consequence of Xiao’s slope inequality ([Xiao]) and hence the assumption p_{g}(X) = q(X) = 0 is enough to obtain the upperbound in (2.2). However the structure theorem 2.3 says that if p_{g}(X) = q(X) = 0 and r = 4g + 4 then X must be a rational surface.

3 Maximal Mordell-Weil lattices D_{4g+4}^{+}

As far as the structure of maximal Mordell-Weil lattices is concerned, we can obtain the following theorem ([Sa-Sak]), which is a corollary of Theorem 2.3.

Theorem 3.1 Let f : X ⟶ P^{1} be the fibration of curves of genus g ≥ 2 with p_{g}(X) = q(X) = 0 and r = 4g + 4. Then the Mordell-Weil lattice J_{Γ}(K) is torsionfree and isometric to the positive-definite unimodular lattice D_{4g+4}^{+} (see bellow for the notation).

Let us explain about the lattice D_{4g+4}^{+}. The lattice D_{4g+4}^{+} is a positive-definite unimodular lattice which is an overlattice of the lattice D_{4g+4} such that [D_{4g+4}^{+} : D_{4g+4}] = 2. Following Conway-Sloane’s book (cf. [7, Ch. 4, C-S]), we will review the lattices D_{n} and D_{4m}^{+}.

For n ≥ 3, we can embed D_{n} into the Euclidean lattice Z^{n} as

(3.1)  D_{n} = {(x_{1}, ⋯, x_{n}) ∈ Z^{n} : x_{1} + ⋯ + x_{n} even}.

The standard integral basis is given as usual (see [7, Ch. 4, C-S]) and its intersection diagram is given by the Coxeter-Dynkin diagram of type D_{n}:

[figure: 1; 2; 2; n; 2; 2; 3; 2; 4; 2; ^{· · · · · ·}; n-2; 2; n-1; 2; D_{n}]

For n = 4m ≥ 4, we take a vector

[1] = (1/2, 1/2, ⋯, 1/2) ∈ Q^{4m},

and set

(3.2)  D_{4m}^{+} = D_{4m} ∪ (D_{4m} + [1]).

The lattice D_{4m}^{+} is a positive-definite integral unimodular lattice and has the standard integral basis with the Coxeter-Dynkin diagram:

[figure: 1; m; 2; 2; 3; 2; 4m; 2; 4; 2; ^{· · · · · ·}; 4m-2; 2; 4m-1; 2]

It should be noted here that D_{8}^{+} ≃ E_{8}, hence one may regard D_{4g+4}^{+}(g ≥ 2) as generalization of the lattice E_{8}. (One may recall that in Shioda’s theory of Mordell-Weil lattices for rational elliptic surfaces E_{8} arises as the frame lattice. )

In the connection with Mirror symmetry conjecture for related Calabi-Yau threefolds which will appear in the next section, it is interesting to consider the theta series of lattices. We also follow the notation in [4, Ch. 4, C-S].

Let L be a positive-definite lattice. For each positive integer m, we set

(3.3)  N_{L}(m) = #{x ∈ L| < x, x >= m}.

Then the theta series of L is defined by

(3.4)  θ_{L}(z) = ∑_{x∈L} q^{<x,x>} = ∑_{m=1}^{∞} N(m)q^{m},

where q = exp(πiz).

In order to write the theta series of D_{4g+4}^{+}, we introduce the following Jacobi’s theta functions:

θ_{2}(z) = 2q^{1/4} ∏_{m=1}^{∞}(1 − q^{2m})(1 + q^{2m})^{2},

θ_{3}(z) = ∏_{m=1}^{∞}(1 − q^{2m})(1 + q^{2m−1})^{2},

θ_{4}(z) = ∏_{m=1}^{∞}(1 − q^{2m})(1 − q^{2m−1})^{2}.

Then the theta series for D_{4g+4}^{+} can be written as (see 7.3, Ch. 4 in [C-S]):

(3.5)  θ_{D_{4g+4}^{+}}(z) = 1/2(θ_{2}^{4g+4}(z) + θ_{3}^{4g+4}(z) + θ_{4}^{4g+4}(z)).

Expanding the right hand side of (3.5) in the powers of q, we obtain the explicit number N_{4g+4}(m) = N_{D_{4g+4}^{+}}(m) of elements in D_{4g+4}^{+} with length m.

For example, if g = 2, then we have the following expansion up to order q^{15}: (3.6)

θ_{D_{12}^{+}}(z) = 1 + 264 q^{2} + 2048 q^{3} + 7944 q^{4} + 24576 q^{5} + 64416 q^{6} + 135168 q^{7}
+253704 q^{8} + 475136 q^{9} + 825264 q^{10} + 1284096 q^{11} + 1938336 q^{12}
+2973696 q^{13} + 4437312 q^{14} + 6107136 q^{15} + · · · .

This expansion gives us the number N_{12}(m) up to m = 15. (The above expansion was done by Mathematica.)

4 Certain Calabi-Yau threefolds

In this section, we will assume that k = C. Let X be a rational surface with fibration f : X ⟶ P^{1} of curves of genus 2. We assume that the fibration f is a Lefschetz pencil, that is, all singular fibers are reduced and have only one node. Then the Mordell-Weil lattice for such a fibration is isometric to D_{12}^{+}.

Theorem 4.1 ([Sa2].) Under the above notation and assumption, let J ⟶ P^{1} be the Néron model of J_{Γ}. Then there exists a smooth projective threefold Y with a fibration h : Y ⟶ P^{1} which gives a relative compactification of Néron model J /P^{1}. Moreover Y has a trivial canonical bundle and h^{2,0}(Y) = h^{1,0}(Y) = 0, that is, Y is a Calabi-Yau threefold. Other Hodge numbers are given as follows: h^{1,1}(Y) = h^{2,1}(Y) = 14, hence the Hodge diamond of Y is self-mirror, that is, invariant under the π/2 rotation.

Here we only remark that under the assumption of Lefschetz pencil f : X ⟶ P^{1} the smooth relative compactification h : Y ⟶ P^{1} of the Néron model J ⟶ P^{1} is constructed by Nakamura [N]. Moreover one can relatively embed X ⟶ P^{1} into h : Y ⟶ P^{1} as a relative principal theta divisor. Therefore X can be considered as a smooth divisor in Y.

In connection with Mirror symmetry conjecture for the above Calabi-Yau threefolds, the following theorem may be interesting.

Theorem 4.2 ([Sa2].) Let L = X + X^{−} + 2F be a divisor class on the Calabi-Yau threefold Y where X^{−} is the minus of X and F is a class of general closed fiber of h : Y ⟶ P^{1}. Then L is nef and big divisor on Y. Moreover for any section σ ∈ S(J /P^{1}) ≃ J_{Γ}(K), one has

(4.1)  deg L_{|σ(P^{1})} =< σ, σ > .

Here < σ, σ > denotes the height of the section σ with respect to the Mordell-Weil lattice. Hence the number of rational curves coming from sections of J /P^{1} with fixed degree m with respect to L is equal to N_{12}(m) in §3, hence can be calculated by (3.6).

The detail will be published in [Sa2].

References [C-S] J.H. Conway & N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer, Second Edition (1992).

[N] I. Nakamura, Relative Compactification of the Néron model and its application, Complex Analysis and Algebraic Geometry, edited by W.L. Baily, Jr. and T.Shioda, Iwanami Shoten and Cambridge University Press, 207-225, 1977. [Sh1]: T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39, 211-240, 1990. [Sh2]: T. Shioda, Mordell-Weil lattices for higher genus fibration, Proc. Japan Acad. 68A 247-250, 1992. [Sa-Sak]: M.-H. Saito and K.-I. Sakakibara, On Mordell-Weil lattices of higher genus fibrations on rational surfaces, Jour. of Math., Kyoto University, 34, No. 4, 859-871, 1994 [Sa0]: M.-H. Saito, On Upperbounds of Mordell-Weil ranks of higher genus fibrations, Proceeding of the workshop ”Hodge theory and Algebraic geometry”, Sapporo (1994), Juten Lecture Note, No. 10, 26-35, 1994. [Sa1]: M.-H. Saito, On Upperbounds of Mordell-Weil ranks of higher genus fibrations, in Algebraic Cycles and related topics, edited by Fumio Hazama, (World Scientific), (1995), 37-42. [Sa2]: M.-H. Saito, Mordell-Weil lattices and certain Calabi-Yau threefolds. (in preparation).

[Xiao] G. Xiao, Fibered algebraic surfaces with low slope Math. Ann. 276, 449-466, 1987