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Arithmetic E8 Lattices with Maximal Galois Action

Published online by Cambridge University Press:  01 February 2010

Anthony Várilly-Alvarado  and
David Zywina
Anthony Várilly-Alvarado
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA, varilly@math.berkeley.edu
David Zywina
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA, zywina@math.upenn.edu

Abstract

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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.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 144 - 165
Copyright
Copyright © London Mathematical Society 2009

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