Statement of the main result Sample Clauses

Statement of the main result. Following Movahhedi and Xxxxxx Xxxxx Xx [32], a number field is said to be ‘p-rational’ if the Galois group of its maximal pro-p extension that is unramified outside p is a free pro-p group. It is known that a number field K has this property if and only if it both validates Xxxxxxxx’s Conjecture at p and is such that Ap is torsion-free (see Xxxxxxx and Xxxxxx Xxxxx Do [24, Theorem 1.2]). We further recall that any K for which both K(ζp) has a unique p-adic place and class number coprime to p has these properties. In particular, if p is regular, then any abelian extension of Q of p-power conductor is p-rational (see [24, Corollary 1.3(ii)]). As another application of our methods, in this chapter we will prove the following result (which does not assume that G is cyclic). In the following we write Z(G) for the centre of a group G. 1.1. Assume that K is p-rational, let L be any finite p-power degree Galois extension of K and set G := Gal(L/K). Let Σ be a finite set of places of K containing all archimedean places, places above p and all those that ramify in L/K, and is also such that the group AΣ is torsion-free. Then the following claims are valid. (i) L validates Xxxxxxxx’s Conjecture at p. (ii) The Qp[G]-module spanned by BΣ is free of rank rK. The Zp[G]-module BΣ is free if and only if one of the following conditions is satisfied: (a) L is contained in Kcyc; (b) G is cyclic, L is disjoint from Kcyc and for every proper subfield F of L that contains K the maximal abelian extension of F cyc in M Σ is equal to M Σ. (iii) Fix a cyclic subgroup C of G and set F := LC. (a) If γ is any element of Gal(M Σ/F ) of infinite order that projects to give a generator of C, then γ|C| ∈ Z(Gal(M Σ/F )) ∩ AΣ and the Zp[C]-module AΣ/⟨γ|C|⟩ is free of rank rF = [G : C] · rK. (b) The Zp[C]-module AΣ is isomorphic to Zp[C][G:C]·rK ⊕ Zp. (c) If rK = 0, then BΣ vanishes. If rK > 0, then there exists an exact sequence of Zp[C]-modules 0 → Z1—nC → BΣ → Zp[C][G:C]·rK → Zp /(Zp ·nC[L : L ∩ F cyc]) → 0 where nC is equal to 0 if L ∩ F cyc = F and equal to 1 otherwise.

Statement of the main result. In order to state the main result of this section we fix a topological generator γQ of ΓQ. Noting that Gn/Hn can be identified with the (unique) quotient of ΓQ of order pn—1 we also fix a generating element γn of Gn that projects to the same element of Gn/Hn as γQ. γQ We write χQ for the cyclotomic character of ΓQ and define an element Σ := n γQ ψ∈G∗n ϵψ eψ of Qp[Gn]× by setting for each ψ in G∗n ψ γQ := (1 — ψ(γ  ))—1, if ψ(H ) = 1 and ψ(γ ) /= 1, logp(χQ(γQ))—1, if ψ = 1G , Finally we set 1, if ψ(Hn) /= 1. Gn θn∗ (1) := (1 — p—1)e1 Σ + ψ∈G∗n\{1Gn} L(ψ, 1) · eψ ∈ Cp[Gn]× where L(ψ, 1) is the value at z = 1 of the Dirichlet L-series Σ n≥1 ψ(n) nz .