Mordell-Weil lattices and Galois representation. I
In a series of notes the author announces new results on the Mordell-Weil group \(E(K)\), where \(K=k(C)\) is the function field of a smooth projective curve C over an algebraically closed field k and E an elliptic curve over K. The K-rational points of E are identified with the global sections of the associated elliptic surface \(f: S\to C\), the Kodaira-Néron model of E/K. Under the assumption that f is not smooth the Néron-Severi group NS(S) of S is torsion-free, hence a lattice with respect to the intersection product of curves on S. Let T be the sublattice generated by the zero section and all fibre components of f. It holds that \(E(K)\cong NS(S)/T\) [theorem 1.1, proved by the author, J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013)]. A finer investigation yields a commutative diagram \[ \begin{alignedat}{3} 0\rightarrow E(K)_{\text{tor}} \rightarrow & E(K) && \rightarrow && L^* \\ & \cup &&&& \cup \\ & E(K)^ 0 && \tilde\rightarrow && L \end{alignedat} \] with exact rows (theorem 1.2). here L is the orthogonal complement of T in \(NS(S)\), \(L^*\) its dual lattice and \(E(K)^ 0\) the subgroup of E(K) consisting of all sections passing through the same fibre components as the zero section. Changing the sign \(E(K)/E(K)_{tor}\) is endowed with a positive-definite lattice structure. It is called the Mordell-Weil lattice of E/K. If NS(S) is unimodular, then \(E(K)/E(K)_{tor}\cong L^*\) (up to sign, theorem 1.4). Up to a trivial factor the product \(<.,.>\) of the Mordell-Weil lattice above is the same as the canonical height pairing due to Néron, Tate and Manin. From the explicit knowledge of the special fibres of f the author deduces explicit formulas for the height pairing and for the relation between \(\rho (S)=rk(NS(S))\) and the rank \(r=rk(E(K))\) of the Mordell-Weil group. [See also part II and III of this paper, see the following reviews.]
scientific article; zbMATH DE number 4177254
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