Statement of the main results Sample Clauses

Statement of the main results. As anticipated, our main results are not only concerned with the existence of the limits in (4.1.8) and (4.1.9), but also with their universality. In other words, for a Xxxxxx matrix Γ (ω) and a given multiplier τ, we show that under some mild assumptions on τ, see (A)-(C) below, the functions‌ LD (t; Γ (ω)) and LD±(t; Γ (ω)) are independent of the choice of τ. τ ^ τ ^‌ Assumptions 4.1.1. Let us state the following assumptions on τ:

Statement of the main results. Just as in Chapter 4, the main results of this Chapter not only show the existence of the limits in (5.1.8) and (5.1.9), but also, given some mild assumptions on the couple τ, their universality. Let us now state the following assumptions on the τ:‌ Assumptions 5.1.2. (A) τ induces a uniformly bounded sequence of Xxxxx-Xxxxxxxx multipliers, i.e. (5.1.21) holds; (B) the limits a0 = lim ϕ0(t, s), b0 = lim ϕ∞(t, s), t,s→0 a = lim exist and are finite; t,s→0 ϕ0(t, s), b = lim ϕ∞(t, s), (C) there exist α > 1/2 and some positive constants Cα, such that for some ε > 0 one has |ϕ0(t, s) − a0| ≤ Cα |log (s + t)|—α , (5.1.26) |ϕ∞(t, s) − b0| ≤ Cα |log (s + t)|—α , (5.1.27) for all s, t ∈ [0, ε]; (D) we can find β > 1/2 and some positive constants Cβ, such that for all s, t > 0 one has |ϕ0(t, s) − a∞| ≤ Cβ |log (s + t + 2)|—β , (5.1.28) |ϕ∞(t, s) − b∞| ≤ Cβ |log (s + t + 2)|—β ; (5.1.29) Then (5.1.16) is a particular case of the following:

Statement of the main results. We recall that for any number field E we write rE for the number of its complex places and δE for its p-adic ‘Leopoldt defect’. We recall also that δE is a non-negative integer which vanishes if and only if Xxxxxxxx’s Conjecture is valid for E at p and that there are equalities rkZ (BΣ) = rkZ (AΣ) − 1 = rE + δE (4.1) (cf. [33, (10.3.7) Corollary]). Theorem 4.1.1. Let L/K and Σ be as above and set G := Gal(L/K). Then the isomorphism class of the Zp[G]-module AΣ is determined (in a sense to be made precise in Theorem 4.2.1 below) by rK, the Leopoldt defects δLi for each i with 0 ≤ i ≤ n and the diagrams of finite groups Σ L/L1 fL1,L2 L/L1 ← − gL2,L1 Σ L/L2 fL2,L3 L/L2 ← − gL3,L2

Statement of the main results. ‌ For an A-lattice X and an idempotent e of A we set Xe := {x ∈ X : e · x = x in F ⊗R X} (so that Xe = e · X if e belongs to A). For any A-module X we write Xtor for the submodule comprising all R-torsion elements. For any element x of X we denote the A-submodule that x generates by ⟨x⟩. The main algebraic result in this section is the following. 3.3.1. Assume that A is Gorenstein. Fix data (C, λ, L, X ) as in Definition 3.1.3. Set a := |X |, ea := eC,a, e(a) := eC,(a) and Ar := Ae(a) and write η in place of η(C,λ,L,F). Then for any elements x of A ∩ Ar and y of XxxX(Ext2 ′ (Ar ⊗A H2(C), Ar)) such that both xea and yea are invertible in Aea, there exists a canonical isomorphism of (finite) A-modules XxxX Xx H1(C) , A ∼= A , ⟨xya · η⟩ tor A xya · I(η) tor and hence also a canonical short exact sequence of A-modules Fita (H2(C)) 0 → → XxxX Xx H1(C) xya · I(η) tor ⟨xya · η⟩ tor A FitA(H2(C))ea tor