Organizing Matrices for Arithmetic Complexes

Abstract

We extend and refine the theory of 'organising modules' of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (non-abelian) Galois extensions of number fields.

Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

“…Now, if we denote by [f ] the pre-image in Ext 2Zp[G] (A(F ) * p , A t (F ) p ) under the isomorphism (13) of the class of an endomorphism f of A t (F ) p , then Lemma 3.3 implies that [γ] − [−φ −1 ] = [ψ] = 0 and hence also that [γ] = [−φ −1 ].In addition, since both γ and −φ −1 are bijective, this class in Ext 2 Zp[G] (A(F ) * p , A t (F ) p ) may be represented by both of the exact sequences obtained by replacing φ −1 by γ −1 or by −φ in the exact sequence(15). By the general result[7, Lem. 4.7] there exist automorphisms κ 1 and κ 2 of X with the property that the (exact) diagram (24)…”

On Higher Special Elements ofp-adic Representations

“…In this case the module Y F vanishes and so the complex C(1) φ constructed in §5.3.1 coincides with C(1). The idempotent e * in Proposition 5.10 therefore coincides with the idempotent e F/k,1 that occurs in [7,Th. 5.4].…”

“…Then the following result is proved in §5.3.2 by combining several of the technical results that we obtain concerning admissible complexes with previous work of Macias Castillo and the first author in [7] and [8].…”

“…Then we shall first prove several technical results in homological algebra concerning the theory of 'admissible complexes' of A-modules that was introduced by Barrett and the first author in [2] and then subsequently used by Macias Castillo and the first author in [7] (but see Remark 2.3 for details of terminology differences). These results will play a key role in several of our later arguments and, although we shall not pursue this point any further here, they can also be used to obtain improvements in the theory of 'organising matrices' that is developed [7].…”

Annihilation of Selmer groups for monomial CM-extensions

“…Then by [BC,Theorem 5.1], Problem 1.1 has the affirmative answer replacing L A,F/Q,S by "L S,β " which is defined by the values that are interpolated by the p-adic L-function constructed in [KF]. The element L S,β is a p-adic element and L A,F/Q,S can be regarded as a complex version of L S,β .…”