Arithmetic E_8 lattices with maximal Galois action

TL;DR

This paper constructs explicit examples of E_8 lattices in arithmetic with maximal Galois action, including elliptic curves over Q(t) and del Pezzo surfaces, demonstrating the full automorphism group action.

Contribution

It provides explicit constructions of E_8 lattices with maximal Galois action in arithmetic geometry, including elliptic curves and del Pezzo surfaces.

Findings

Explicit elliptic curves over Q(t) with E_8 Mordell-Weil lattices and maximal Galois action.

Construction of del Pezzo surfaces of degree 1 with maximal Galois action on Picard group.

Verification of the full automorphism group acting on the lattices.

Abstract

We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.