Mordell-Weil lattices and Galois representation. II (Q752110)

The basic results of part I of this paper (cf. the preceding review) are applied to elliptic curves E over function fields \(K=k(C)\) with rational Kodaira-Néron model \(f:S\to C\cong{\mathbb{P}}^ 1\). One has \(K=k(t)\), \(r=rk(E(K))\leq 8\). The knowledge of positive lattices of small rank allows to determine the structure of the Mordell-Weil group in special cases (theorem 2.1): (i) If f has no reducible fibre, then \(r=8\) and \(E(K)=E(K)^ 0\cong E_ 8.\) (ii) If there is precisely one reducible fibre and it has two components, then \(r=7\), \(E(K)\cong E^*_ 7\), \(E(K)^ 0\cong E_ 7.\) (iii) If the only reducible fibre has three components, then \(r=6\), \(E(K)\cong E^*_ 6\), \(E(K)^ 0\cong E_ 6.\) (iv) If f has precisely two reducible fibres, both with two components, then \(r=6\), \(E(K)\cong D^*_ 6\), \(E(K)^ 0\cong D_ 6.\) Here, \(E_ i, D_ 6\) or \(E_ i^*, D^*_ 6\) are the standard root lattices or their duals, respectively. For \(r=8,7,6\) the number of minimal sections of E(K) is calculated, E(K) is minimally generated by some of them in the cases (i), (ii), (iii) (theorem 2.2) and generated by all \(P\in E(K)\) with \(<P,P>\leq 2\) in all cases \(r\geq 6\) (corollary 2.3). The orders of \(E(K)_{tor}\) for all rational cases are: 1,2,3,4,5,6,8,9 (proposition 2.5). In section 3 more precise informations about generators of E(K) are given in connection with the minimal Weierstrass equation of E over \(K=k(t)\), \(char(k)\neq 2,3:\) \(y^ 2=x^ 3+p(t)x+q(t),\) \(p(t),q(t)\in k[t],\) \(\deg(p)\leq 4,\) \(\deg q)\leq 6.\) The additional tool of specialization homomorphisms \(sp_ v\) of E(K) into the smooth part of special fibres \(f^{-1}(v)\) is applied to special examples in section 4 in order to identify the structure of E(K) with additive subgroups of number fields of the form \({\mathbb{Z}}[\zeta](p/G)^{1/n}\), \(\zeta\) a unit root, \(p\in k\), \(G\in {\mathbb{Q}}(\zeta)\). Minimal sections, hence generators of E(K), or equivalently the exceptional curves on S, can be written down explicitly in these number theoretic terms. [See also part III of this paper, cf. the following review.]