Arithmetic E8 lattices with maximal Galois action

A Várilly-Alvarado, D Zywina - LMS Journal of Computation and …, 2009 - cambridge.org
LMS Journal of Computation and Mathematics, 2009cambridge.org
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, ie, 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 …
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.
Cambridge University Press
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