The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R 0 (2m) be the minimal root discriminant for totally complex number fields of degree 2m, and put α 0 = lim infm R 0 (2m). Define R 1 (m) to be the minimal root discriminant of totally real number fields of degree m and put α 1 = lim infm R 1 (m). Assuming the Generalized Riemann Hypothesis, α 0 ≥ 8πe γ ≈ 44.7, and, α 1 ≥ 8πe γ+π/2 ≈ 215.3. By constructing number fields of degree 12 with suitable properties, we give the best known upper estimates for α 0 and α 1 : α 0 < 82.2, α 1 < 954.3.

Abstract. We carry out the computation of the Iwasawa invariants ρ T S , µ T S , λ T S associated to abelian T-ramified S-decomposed ℓ-extensions over the finite steps K n of the cyclotomic Z ℓ -extension K ∞ /K of a number field of CM-type.Reçu par la rédaction le 2 juillet, 2001. Classification (AMS) par sujet: principale: 11R23; secondaire: 11R37.

Downloaded from856 Wayne Aitken et al.In this work, we discuss a method for studying finitely ramified extensions of number fields via arithmetic dynamical systems on P 1 . At least conjecturally, this method provides a vista on a part of G K,S invisible to p-adic representations. We now sketch the construction, which is quite elementary. Suppose ϕ ∈ K[x] is a polynomial of degree d ≥ 1. 1 For each n ≥ 0, let ϕ •n be the n-fold iterate of ϕ, that is, ϕ •0 (x) = x and ϕ •n+1 (x) = ϕ(ϕ •n (x)) = ϕ •n (ϕ(x)) for n ≥ 0. Let t be a parameter for P 1 /K with function field F = K(t). We are interested in the tower of branched covers of P 1 given by(1.1)as well as extensions of K obtained by adjoining roots of its specializations at arbitraryFix an algebraic closure F of F, and let K be the algebraic closure of K determined by this choice, that is, the subfield of F consisting of elements algebraic over K. For n ≥ 0, let T ϕ,n be the set of roots in F of Φ n (x, t); it has cardinality d n . We denote by T ϕ the d-regular rooted tree whose vertex set is ∪ n≥0 T ϕ,n , and whose edges point from v to w exactly when ϕ(v) = w; its root (at ground level) is t.We choose and fix an end ξ = (ξ 0 , ξ 1 , ξ 2 , . . .) of this tree; in other words, we choose a compatible system of preimages of t under the iterates of ϕ: ϕ(ξ 1 ) = ξ 0 = t and ϕ(ξ n+1 ) = ξ n for n ≥ 1. For each n ≥ 1, we consider the field F n = F(ξ n ) F[x]/(Φ n ) andits Galois closure F n = F(T ϕ,n ) over F. Let O Fn be the integral closure of K[t] in F n . Corresponding to each t 0 ∈ K, we may fix compatible specialization maps σ n,t 0 : O Fn → K with image K n,t 0 , a normal extension field of K, and put ξ n | t 0 = σ n,t 0 (ξ n ) for the corresponding compatible system of roots of Φ n (x, t 0 ). We denote by K n,t 0 the image of the restriction of σ n,t 0 to O Fn . We refer the reader to Section 2.2 for more details, but we should emphasize here that Φ n (x, t 0 ) is not necessarily irreducible over K; hence, although K n,t 0 depends only on ϕ, n, and t 0 , the isomorphism class of K n,t 0 depends a priori on the choice of ξ as well as on the choice of compatible σ n,t 0 . Also, the Galois closure of K n,t 0 /K is contained in, but possibly distinct from, K n,t 0 .Taking the compositum over all n ≥ 1, we obtain the iterated extension F ϕ = ∪ n F n attached to ϕ, with Galois closure F ϕ = ∪ n F n over F. Similarly for each t 0 ∈ K, we obtain a specialized iterated extension K ϕ,t 0 = ∪ n K n,t 0 with Galois closure over K contained in K ϕ,t 0 = ∪ n K n,t 0 . We put M ϕ = Gal(F ϕ /F) for the iterated monodromy group of ϕ and for t 0 ∈ K, we denote by M ϕ,t 0 = Gal(K ϕ,t 0 /K) its specialization at t 0 .The group M ϕ has a natural and faithful action on the tree T ϕ , hence comes equipped 1 This construction actually works for any perfect K as long as the derivative ϕ is not identically zero in K[x].

For Γ = Z p , Iwasawa was the first to construct Γ-extensions over number fields with arbitrarily large µ-invariants. In this work, we investigate other uniform pro-p groups which are realizable as Galois groups of towers of number fields with arbitrarily large µ-invariant. For instance, we prove that this is the case if p is a regular prime and Γ is a uniform pro-p group admitting a fixed-point-free automorphism of odd order dividing p − 1. Both in Iwasawa's work, and in the present one, the size of the µ-invariant appears to be intimately related to the existence of primes that split completely in the tower. A(F),where the limit is taken over all number fields F in L/K with respect the norm map. Then X is a Z p [[Γ]]-module and, thanks to a structure theorem (see section 1.1.2), one attaches a µ-invariant to X , generalizing the well-known µ-invariant introduced by Iwasawa in the classical case Γ ≃ Z p . Iwasawa showed that the size of the µ-invariant is related to the rate of growth of p-ranks of p-class groups in the tower. For the simplest Z p -extensions, i.e. the cyclotomic ones, he conjectured that µ = 0; this was verified for base fields which are abelian over Q by Ferrero and Washington [9] but remains an outstanding problem for more general base fields. Iwasawa initially suspected that his µ-invariant vanishes for all Z p -extensions, but later was the first to construct Z p -extensions with non-zero (indeed arbitrarily large) µ-invariants. It is natural to ask what other p-adic groups enjoy this property. Our present work leads us to the following conjecture:Conjecture 0.1. -Let Γ be a uniform pro-p group having a non-trivial fixed-point-free automorphism σ of order m co-prime to p (in particular if m = ℓ is prime, Γ is nilpotent). Then Γ has arithmetic realizations with arbitrarily large µ-invariant, i.e. for all n ≥ 0, there exists a number field K and an extension L/K with Galois group isomorphic to Γ such that µ L/K ≥ n.Our approach for realizing Γ as a Galois group is to make use of the existence of socalled p-rational fields. See below for the definition, but for now let us just say that the critical property of p-rational fields is that in terms of certain maximal p-extensions with restricted ramification, they behave especially well, almost as well as the base field of rational numbers. As we will show, Conjecture 0.1 can be reduced to finding a p-rational field with a fixed-point-free automorphism of order m co-prime to p. These considerations lead us to formulate the following conjecture about p-rational fields.Conjecture 0.2. -Given a prime p and an integer m ≥ 1 co-prime to p, there exist a totally imaginary field K 0 and a degree m cyclic extension K/K 0 such that K is p-rational.Although we will not need it, we believe K 0 in the conjecture may be taken to be imaginary quadratic; see Conjecture 4.16 below. Our key result is: Theorem 0.3. -Conjecture 0.2 for the pair (p, m) implies Conjecture 0.1 for any uniform pro-p group Γ having a fixed-point-free automorphism of order m.One knows tha...

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