Mordell-Weil lattices and Galois representation. III
type: Item
instance of: scholarly article
title: Mordell-Weil lattices and Galois representation. III
zbMATH Open document ID: 0715.14017
author: Tetsuji Shioda
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: [For part I and II of this paper [see the two preceding reviews.] This part III is dedicated to Galois representations arising from Mordell-Weil lattices of elliptic curves defined over \({\mathbb{Q}}(t)\). The Galois group Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\) acts on E(K), \(K={\bar {\mathbb{Q}}}(t)\), and isometrically on the lattices \(E(K)/E(K)_{tor}, E(K)^ 0\). Of great interest is the image of the representing homomorphism \(\rho\) : Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\to Aut(E(K)^ 0).\) Let \(\lambda =(p_ 0,...,p_ 3,q_ 0,...,q_ 3)\) be a general point of the affine space \({\mathbb{A}}^ 8/{\mathbb{Q}}\) and \(E_{\lambda}\) the elliptic curve defined by the Weierstrass equation \[ y^ 2=x^ 3+(\sum^{3}_{i=0}p_ it^ i)x+(\sum^{3}_{i=0}q_ it^ i+t^ 5). \] Via Mordell-Weil lattice and specialization E(\(\overline{{\mathbb{Q}}(\lambda)}(t))\to \overline{{\mathbb{Q}}(\lambda)}\) one gets a surjective homomorphism \(\rho_{\lambda}:\;Gal(\overline{{\mathbb{Q}}(\lambda)}/{\mathbb{Q}}(\lambda))\to W(E_ 8).\) The theorem (6.1) allows to construct field extensions with Galois group \(W(E_ 8)\). The proof needs deformation theory of isolated singularities (monodromy) due to Brieskorn and others. By means of Hilbert's irreducibility theorem and specialization to \(\lambda \in {\mathbb{Q}}^ 8\) one gets elliptic curves \(E_{\lambda}/{\mathbb{Q}}(t)\) with Mordell-Weil group \(E_ 8\) together with explicit generators (theorems 7.1, 7.2). Moreover the Hasse zeta function of the corresponding rational elliptic surfaces \(S_{\lambda}\) can be determined: \(\zeta (S_{\lambda}/{\mathbb{Q}})=\zeta (s)\zeta (s- 1)L(\rho_{\lambda},s-1)\) with Artin L-function \(L(\rho_{\lambda},s)\) of (essentially) nonabelian type; \(\zeta\) is Riemann's zeta function. Further specializations to \(t\in {\mathbb{Q}}\) allow to construct elliptic curves \(E^{(t)}\) with \({\mathbb{Q}}\)-rational points \(P_ 1^{(t)},...,P_ 8^{(t)}\), such that \(\lim_{t\to \infty} \det (<P_ i^{(t)},P_ j^{(t)}>)/h…
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14J27
Mathematics Subject Classification ID: 11E20
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 4177256
zbMATH Keywords: Galois representations
zbMATH Keywords: Mordell-Weil lattices
zbMATH Keywords: elliptic curves
zbMATH Keywords: deformation theory of isolated singularities
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: Hasse zeta function
zbMATH Keywords: elliptic surfaces
zbMATH Keywords: Artin L-function
zbMATH Keywords: Weil height
zbMATH Keywords: del Pezzo surfaces
zbMATH Keywords: cubic forms
reviewed by: Rolf-Peter Holzapfel
MaRDI profile type: Publication
full work available at URL: pjaa.65.300
OpenAlex ID: W4251418011
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cites work: Q3237900
DOI: 10.3792/PJAA.65.300
description: scientific article; zbMATH DE number 4177256