Theory of Mordell-Weil Lattices Tetsuji Shioda Department of Mathematics, Faculty of Science, Rikkyo University Nishi-Ikebukuro 3-chome, Tokyo 171, Japan

0. Introduction The basic idea of Mordell-Weil lattices is to view the Mordell-Weil group ( = the group of rational points on an elliptic curve or an abelian variety defined over some "global" field) as a Euclidean lattice by means of suitable inner product. The necessary setup for this has been known for some time. First of all, the finite generation of such a group was established by Mordell and Weil in 1920s (thus named after them) in the case of elliptic curves over Q [Mo] or Jacobian varieties over a number field [Wl]. After the general theory of abelian varieties was founded by Weil [W2], it was extended by Néron and Lang in 50s to a more general situation [L]. Second the notion of the canonical height on abelian varieties was developed by Néron, Tate and Manin in 60s [N2, TI, T3, Ml; cf. L, Se2]. A more geometric method, based on the theory of elliptic surfaces [K], was used by the author [SI] and Cox-Zucker [CZ] in 70s. Certainly the lattice-theoretic feature of Mordell-Weil groups has appeared in various works, e.g. in the statement of the Birch-Swinnerton-Dyer conjecture. But the idea to view them as lattices seems relatively new, and a systematic study of Mordell-Weil lattices has been done only very recently by the author [S3, S4, S5, S6, S7] and independently by N. Elkies [E], in the case of elliptic curves over function fields, or equivalently, in the case of elliptic surfaces. Actually the general scope of the theory of Mordell-Weil lattices should cover also the case of algebraic surfaces with higher genus fibration and the case of arithmetic surfaces as well (cf. [F, H] for the latter). But, in this talk, we shall restrict our attention to the case of elliptic surfaces. In Part I, we review the definition and the basic results on Mordell-Weil lattices, and in Part II, we consider the Galois representations and algebraic equations arising from them, and discuss some applications. In the case of rational elliptic surfaces, we obtain some new insight into various problems related to F 6 , F 7 , F 8 —a mysteriously rich subject in Mathematics. In particular, we can answer some questions raised by Weil [W3] and Manin [M2] concerning the Galois representation arising from the 27 lines on a cubic surface.

Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 © The Mathematical Society of Japan, 1991

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Tetsuji Shioda

Acknowledgement. I would like to thank Professor J-P. Serre for many valuable comments on my work. I have had helpful discussions or correspondences with T. Ekedahl, N. Elkies, D. Gross, E. Horikawa, K. Oguiso, K. Saito, F. Sato, I. Shimada and P. Slodowy, and I thank them all.

Parti 1. Definition of Mordell-Weil Lattices Let K = k(C) be the function field of a smooth projective curve C over an algebraically closed field k. Let F be an elliptic curve defined over K, and let E(K) denote the group of X-rational points of F, with the origin O e E(K). For what follows, the main reference is [S7]. Our basic tool is the associated elliptic surface f:S->C (the Kodaira-Néron model of E/K) and the intersection theory on S. Recall [K, NI, T2] that S is a smooth projective surface over k and / is a relatively minimal fibration with the generic fibre F. We can naturally identify the global sections of / : S A C with the X-rational points of E, and so E(K) denotes the group of sections. For P e E(K), (P) will denote the image curve of P : C -> S. We assume throughout that (*) / is not smooth, i.e., there is at least one singular fibre. Then E(K) is finitely generated by the Mordell-Weil theorem. On the other hand, let N = NS(S) be the Néron-Severi group of S, i.e. the group of divisors modulo algebraic equivalence. It is a free module of finite rank under the assumption (*), and it becomes an indefinite integral lattice with respect to the intersection pairing (D • D'). Let T be the sublattice of N generated by the zero section (O), a fibre F and the irreducible components of fibres. Then T is a direct sum: T= C. Observe that 2x

if P or P' e E(K)°,

for any P e E(K)°, P ^ 0.

(1.7) (1.8)

The former shows that E(K)/E(K)i0T is contained in the dual lattice M* of M = E(K)°. Theorem 1.3. Assume further that NS(S) is unimodular. Then E(K)/E(K)lOT is equal to the dual lattice M* of M = E(K)°. Moreover we have [M* : M ] = det M = (det T)/\E(K\or\2 (see [S7, Th.9.1]).

2. Basic Invariants of MWL By the very definition, the Mordell-Weil lattices (abbreviated henceforth as MWL) are of Diophantine nature and have quite rich structure, as will be seen later. But, apart from that, they provide a purely algebraic method for constructing lattices of some interest. In general, given a lattice L, one would like to know its basic invariants such as the rank rk(L), the determinant det(L) and, if L is positive-definite

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Tetsuji Shioda

(which we assume now), the minimal norm p(L); recall (cf. [CS] for what follows) that det(L) = |det( 0; the proof uses supersingular Fermât curves [ShT] or supersingular Fermât surfaces [S2]. From the viewpoint of MWL, Elkies and the author have independently studied these cases and obtained, among others, the following (see [E] and [S4], cf. [G]): Theorem 2.2. Let E be the elliptic curve y2 = x 3 + tq+1 + 1 over K = k^(t), k1 any extension of the finite field of q2 elements with q = pe = — 1 (mod 6). Then E(K) = E(K)° is an even integral lattice with the invariants: r = 2q-2,

det = q^^/b,

ti

= 2X

(x = (q+ l)/6),

(2.4)

where b is a positive integer (an even power of p) such that b = \(e

= l),

b>

^C-D(CP-5)A5)- ( ß

>

j)

(2.5)

in which equality holds if e = 3. The integer b is equal to the order of the ShafarevichTate group of E/K for k± = ¥q2orofthe Brauer group of S/kx (cf. [TI]). Inparticular, we have ô = y/b-((q + l ) / 1 2 r V ^ 5 " 6 .

(2.6)

N.B. This gives denser sphere packings than previously known ones in certain dimensions; for instance, for q = 41, we haver = 80andlog 2 C is now defined over fe0. Obviously the Galois group G acts on E(K), and E(K0) coincides with E(K)G, the subgroup of G-invariants. First we note: Lemma 4.1. (i) The map cp defined by (1.3) is G-equivariant. (ii) The height pairing (1.6) on E(K) is stable under G. (Cf. [S7, Prop.8.13]). Therefore we get a Galois representation on the MW group Q : G = Gal(fe//c0) -» Aut(F(X), < , » = a finite group

(4.1)

and its variant on the MWL or the narrow MWL of E/K, say M, Q':G-+

Aut(M) c GLr(Z)

(r = rk(M)).

(4.2)

Let Jf/k0 be the extension corresponding to Ker(#); equivalently, Jf is the smallest extension of fc0 such that F(jf (C)) = E(K). By definition, Jf /k0 is a finite Galois extension such that Gal(Jf /k0) = Imte). (4.3) The basic problem on the Galois representation (4.1) is this: Problem 4.2. Determine the image of Q. In particular, we ask: (i) How big or (ii) how small can Im(g) be? To study this will be a main theme in the subsequent sections. Remark 4.3. The Galois representations Q or Q' arising from MWL are quite different from those arising from the torsion points of an elliptic curve or an abelian variety (e.g. the Tate modules), because we are dealing with poiiits of infinite order here! Next we note the connection to the Hasse zeta functions (cf. [M2, Sel]). We state the result in the simplest situation, which is related to the questions of Weil [W3] and Manin [M2]; see Theorem 8.4. Proposition 4.4. Let k0 = Q. Assume that S is a rational elliptic surface (over Pl>