Abstract: 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.
