Organizing Matrices for Arithmetic Complexes

Abstract

We extend and refine the theory of “organizing 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 (nonabelian) Galois extensions of number fields.