The Galois representation of type \(E_8\) arising from certain Mordell-Weil groups (Q920161)

The author determines explicitly the Galois lattice structure of the Mordell-Weil group \(E_{\gamma}(\overline{\mathbb{Q}}(t))\) of all elliptic curves over the function field \(\overline{\mathbb{Q}}(t)\) with equation \(E_{\gamma}:\;y^2=x^3+\gamma x+t\) \((\gamma\in\overline{\mathbb{Q}}^{\times})\) with respect to the height pairing and the Galois group \(H\) of the field extension \(\mathbb{Q}(\zeta_{20})(\sqrt[20]{\gamma /G})/{\mathbb{Q}}(\gamma)\), \(G\) a well-determined real number in \(\mathbb{Q}(\zeta_{20})\). The lattice is isometric to the unique negative-definite quadratic even unimodular lattice \(E_8\) of rank 8. As \(H\)-module over \(\mathbb{Z}\) it is isomorphic to \(\mathbb{Z}[\zeta_{20}]\cdot \sqrt[20]{\gamma /G}\) of degree 8 of \(H\). Two applications to the corresponding Artin (respectively Hecke) \(L\)-function and Hasse zeta function are announced. Computer calculations have been used. Details and more general accounts are promised to be published elsewhere.