TAME PRO-p GALOIS GROUPS: A SURVEY OF RECENT WORK

Fix a prime p, a number field K, and a finite set S of places of K none of which has residue characteristic p. Fix an algebraic closure K of K and let KS be the maximal p-extension of K inside K which is unramified outside S; it is the compositum of all finite p-power degree extensions of K unramified outside S. We assume that real places of K not contained in S do not complexify in the extension KS/K. Put GK,S = Gal(KS/K) for its (pro-p) Galois group. Very little is known about this “tame arithmetic fundamental group.” Before Shafarevich’s pioneering work [Sh], a few examples where it was possible to determine GK,S explicitly (and show that it was finite), were known, and it was in fact generally believed that all such GK,S are finite. That this is not so was first demonstrated in [GS] by Golod and Shafarevich. As was noted by Artin and Shafarevich, the mere existence of infinite GK,S (with S finite) has an arithmetic application to the estimation of discriminants because the discriminants of successive fields in a tamely and finitely ramified tower grow as slowly as possible. For a more detailed discussion of this topic (and the analogy with curves over finite fields with many rational points) see, for example, [HM1] and the references therein. Infinite GK,S satisfy a number of interesting group-theoretic properties (stemming from class field theory) which we will discuss below, but little attention was focussed on the group-theoretical structure of these infinite groups in the decades following their discovery. In the 1990s, through an important and influential work of Fontaine and Mazur [FM] on p-adic Galois representations, to this list of properties was added a conjectural one. This development is concurrent with a revitalization of the study of tame arithmetic fundamental groups. In this brief survey, I sketch two recent contributions to this subject, the first, due to Khare, Larsen, and Ramakrishna concerning the case where S is infinite, and the second, due to Boston, suggesting a purely group-theoretical approach to the Fontaine-Mazur conjecture. I would like to thank all of these researchers for making preprints of their work available; it should be clear that the present article is merely a summary of some of their beautiful ideas. I am grateful to R. Ramkrishna and N. Boston for helpful remarks on earlier drafts of this article. Finally, I would like to thank Y. Aubry, G. Lachaud and M. Tsfasman (the organizers of AGCT-9), as well as the staff of CIRM at Luminy, for making possible a wonderful conference and inviting me to it.