Abstract. In the first, mostly expository, part of this paper, a graded Lie algebra is associated to every group G given with an N-series of subgroups. The asymptotics of the Poincaré series of this algebra give estimates on the growth of the group G. This establishes the existence of a gap between polynomial growth and growth of type e √ n in the class of residually–p groups, and gives examples of finitely generated p–groups of uniformly exponential growth. In the second part, we produce two examples of groups of finite width and describe their Lie algebras, introducing a notion of Cayley graph for graded Lie algebras. We compute explicitly their lower central and dimensional series, and outline a general method applicable to some other groups from the class of branch groups. These examples produce counterexamples to a conjecture on the structure of just-infinite groups of finite width. 1.

Abstract. Let E/K be an abelian extension of number fields, with E ̸ = Q. Let ∆ and n denote the absolute discriminant and degree of E. Letσdenote an element of the Galois group of E/K. Weprovethefollowingtheorems, assuming the Extended Riemann Hypothesis: () (1) There is a degree-1 prime p of K such that p = σ, satis-

Abstract. We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], Golod and Shafarevich showed in 1964 [8] that there exists a number field with an infinite, everywhere unramified pro-p extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure

Abstract. We introduce a class of noncommutatative algebras called representation complete intersections (RCI). A graded associative algebra A is said to be RCI provided there exist arbitrarily large positive integers n such that the scheme Repn A, of n-dimensional representations of A, is a complete intersection. We discuss examples of RCI algebras, including those arising from quivers. There is another interesting class of associative algebras called noncommutative complete intersections (NCCI). We prove that any graded RCI algebra is NCCI. We also obtain explicit formulas for the Hilbert series of each nonvanishing cyclic and Hochschild homology group of an RCI algebra. The proof involves a noncommutative cyclic Koszul complex, K cyc � A, and a matrix integral similar to the one arising in quiver gauge theory. Table of Contents 1.

To our teacher Yu.I.Manin on the occasion of his 65th birthday Abstract. The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of Q or of Fr(t). We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant we prove generalizations of the Odlyzko–Serre bounds and of the Brauer–Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio log hR / log √ |D | valid without the standard assumption n / log √ |D | → 0, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer–Siegel theorem to hold. As an easy consequence we ameliorate on existing bounds for regulators. 2000 Math. Subj. Class. 11G20, 11R37, 11R42, 14G05, 14G15, 14H05 Key words and phrases. Global field, number field, curve over a finite field, class number, regulator, discriminant bound, explicit formulae, infinite global field, Brauer–Siegel theorem 1

this paper are: 142 Newman, Sauerbier, Wisliceny For every prime p there is a pro-p-presentation with 5 generators and 7 relations which defines a finite pro-p-group with generator number 5. (Theorem 4.2 and Remark 4.3)

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

We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension. 2000 Mathematics subject classification: 16N, 16P90.

with an appendix on

It is well known that many famous Burnside-type problems have positive solutions for PI-groups and PI-algebras. In the present article we also consider various Burnside-type problems for PI-groups and PIrepresentations