LIFTINGS OF PROJECTIVE 2-DIMENSIONAL GALOIS REPRESENTATIONS AND EMBEDDING PROBLEMS

In this paper we study liftings of 2-dimensional projective representa tions of the absolute Galois group of a field k. These liftings are re lated to solutions of certain Galois embedding problems. For represent ations with image isomorphic to one of the groups PSL(2, q) or PGL(2, q) with q drop +/-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 obstruc tion 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 th ese groups as permutation groups and then apply Serre's formula on the Witt invariant of Tr(x(2)) given by Serre (Comment. Math. Helv. 59, 1 984, 651-676). We also obtain from Crespo (C. R. Acad. Sci. Paris 315, 1992, 625-628; C-4 extensions of S-n as Galois groups, preprint, Barc elona, 1993; Central extensions of the alternating group as Galois gro ups, preprint, Barcelona, 1993; Galois realization of central extensio ns of the symmetric group with kernel a cyclic 2-group, preprint, Barc elona, 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 ob tain an easy computable criterion for the existence of a lifting with any given index (Proposition 4.1 and corollaries). The same kind of cr iterion 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 embeddin g problems over the symmetric group. Section 1 contains preliminaries on projective representations and their liftings. In Section 2 we clas sify the cyclic central extensions of the groups PSL(2, q) and PGL(2, q), and establish the relation to liftings of the projective represent ations we are interested in. In Section 3 we find which degree 2 exten sions 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. (C) 1995 Academic Press, Inc.