Liftings of Projective 2-Dimensional Galois Representations and Embedding Problems

Abstract

In this paper we study liftings of 2-dimensional projective representations of the absolute Galois group of a field k. These liftings are related to solutions of certain Galois embedding problems. For representations with image isomorphic to one of the groups PSL(2, q) or PGL(2, q) with q ≡ ±3 (mod 8) or to one of the groups PSL(2, 7) or PSL(2, 9), and k of characteristic different from 2, we compute the obstruction to the existence of a lifting of index 2 (Theorems 3.6 and 3.7). To make this computation we consider the natural representations of these groups as permutation groups and then apply Serre′s formula on the Witt invariant of Tr(x2) given by Serre (Comment. Math. Helv. 59, 1984, 651-676). We also obtain from Crespo (C. R. Acad. Sci. Paris315, 1992, 625-628; C4 extensions of Sn as Galois groups, preprint, Barcelona, 1993; Central extensions of the alternating group as Galois groups, preprint, Barcelona, 1993; Galois realization of central extensions of the symmetric group with kernel a cyclic 2-group, preprint, Barcelona, 1993), a criterion for the existence of liftings of index 4 and 8 (Theorem 3.8). In the case k = Q, we exploit a result by Tate to obtain an easy computable criterion for the existence of a lifting with any given index (Proposition 4.1 and corollaries). The same kind of criterion can be used for other embedding problems over Q; in Theorem 4.4 we give a criterion for the existence of a solution to some embedding problems over the symmetric group. Section 1 contains preliminaries on projective representations and their liftings. In Section 2 we classify the cyclic central extensions of the groups PSL(2, q) and PGL(2, q), and establish the relation to liftings of the projective representations we are interested in. In Section 3 we find which degree 2 extensions of the symmetric and alternating group correspond to each degree 2 extension of PSL(2, q) and PGL(2, q) under the natural permutation representations, and then apply Serre′s formula. Section 4 is devoted to the case k = Q. In Section 5 some numerical examples are given.