Arithmetic E₈ lattices with maximal Galois action

Original · EN

We construct explicit examples of E₈ 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 E₈. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E₈ 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 E₈. We construct examples of such surfaces for which the action of Galois on the geometric Picard group is maximal.