On the Mordell-Weil lattices. (English) Zbl 0725.14017

In some notes [see “Mordell-Weil lattices and Galois representation”, I-III, Proc. Japan Acad., Ser. A 65, No.7, 268-271; No.8, 296-299 and 300-303 (1989; Zbl 0715.14015-14017)], the author announced new results on the Mordell-Weil group E(K), \(K=k(C)\) the function field of a smooth projective curve C over an algebraically closed field k and E an elliptic curve over K. It is understood as general fibre of an elliptic surface S/C. The present article is dedicated to basic proofs. It is subdivided into two parts.
Part I. The Mordell-Weil group and the Néron-Severi group of an elliptic surface (1. The basic theorems; 2. Intersection theory on an elliptic surface; 3. Algebraic and numerical equivalence; 4. The Picard variety of an elliptic surface; 5. Proof of theorem 1.3).
Three fundamental theorems are proved by geometric meethods based on the intersection theory of algebraic surfaces: 1.1 \(E(K)\) is finitely generated; 1.2 The Néron-Severi group NS(S) is finitely generated and torsionfree; 1.3 \(E(K)=NS(S)/T,\) T the subgroup generated by the algebraic equivalence classes of vertical curves and the 0-section.
Part II. The Mordell-Weil lattice of an elliptic surface (6. Preliminaries on lattices; 7. The Néron-Severi lattice; 8. The Mordell- Weil lattices; 9. The unimodular case; 10. Rational elliptic surfaces).
The lattice structure of \(E(K)/E(K)_{tor}\) is introduced coming from the intersection product on NS(S). An explicit formula for the height pairing is proved (theorem 8.6). Looking at the fibres the Mordell-Weil lattices are complementary to sublattices of sums of some of the standard lattices \((E_ 6, E_ 7, E_ 8, A_ n, D_ n\) and their duals). This is the starting point for finer classifications. For example, the determination of the Mordell-Weil groups of rational elliptic surfaces is reduced to the study of root sublattices of \(E_ 8\).


14G05 Rational points
14H52 Elliptic curves
14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
11G05 Elliptic curves over global fields
14G40 Arithmetic varieties and schemes; Arakelov theory; heights