Mordell-Weil lattices and Galois representation. III (Q752111)

[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 \(\lim_{t\to \infty} \det (<P_ i^{(t)},P_ j^{(t)}>)/h(t)=1,\) where h(t) is the Weil height of \(t\in {\mathbb{P}}^ 1({\mathbb{Q}})\) (corollary 7.3). Via del Pezzo surfaces there are interesting connections with the arithmetic theory of cubic forms due to Manin.