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.

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 p E p E rkZ (BΣ) = rkZ (AΣ) − 1 = rE + δE (4.1) (cf. [33, (10.3.7) Corollary]). We now state the main result that will be proved in this chapter. Theorem 4.1.1. Let L/K and Σ be as above and set G := Gal(L/K). L 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 T Σ L/L1 T Σ fL1,L2 −→ L/L1 ← − gL2,L1 Σ T L/L2 T Σ 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.