On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface

Abstract

Let X denote a K3 surface over an arbitrary field k. Let ks denote a separable closure of k and let Xs denote the base change of X to ks. Let O(PicX) and O(PicXs) denote the group of isometries of the lattices PicX and PicXs, respectively. Let RX denote the Galois invariant part of the Weyl group of PicXs. One can show that each element in RX can be restricted to an element of O(PicX). The following question arises: Is the image of the restriction mapRX→O(PicX)a normal subgroup ofO(PicX)for every K3 surfaceX? We show that the answer is negative by giving counterexamples over k=Q.