Mordell-Weil lattices and Galois representation. II
type: Item
instance of: scholarly article
title: Mordell-Weil lattices and Galois representation. II
zbMATH Open document ID: 0715.14016
author: Tetsuji Shioda
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 4177255
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: elliptic curves over function fields
zbMATH Keywords: generators of Mordell-Weil group
zbMATH Keywords: Kodaira-Néron model
zbMATH Keywords: number of minimal sections
zbMATH Keywords: specialization homomorphisms
reviewed by: Rolf-Peter Holzapfel
MaRDI profile type: Publication
full work available at URL: pjaa.65.296
OpenAlex ID: W4244742735
cites work: Mordell-Weil lattices and Galois representation. I
DOI: 10.3792/PJAA.65.296
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description: scientific article; zbMATH DE number 4177255