Organizing matrices for arithmetic complexes (Q2929668)

In this paper the authors generalize the theory of organizing matrices of \textit{B. Mazur} and \textit{K. Rubin} [Adv. Math. 198, No. 2, 504-546 (2005; Zbl 1122.11038)]. The idea is that a single matrix encapsulates a lot of arithmetic information.NEWLINENEWLINEMore precisely, the authors study two classes of perfect complexes (``admissible complexes'' and ``weakly admissible complexes'') of \(\mathfrak A\)-modules, where \(\mathfrak A\) is a direct factor of a group ring \(R[G]\) of a finite group \(G\) over a Dedekind domain \(R\). Among other things ``admissible'' (respectively ``weakly admissible'') means that a complex is acyclic outside degrees \(1\), \(2\) (and \(3\)).NEWLINENEWLINEAssume that \(\mathfrak A = \mathbb Z_p[G]\). Given a weakly admissible complex \(C\) there is a ``weakly organizing matrix'' \(\Phi \in M_d(\mathbb Z_p[G])\) for some natural number \(d\) such that the kernel and cokernel of \(\Phi\) are almost \(H^1(C)\) and \(H^2(C)\), respectively. When \(H^3(C)\) is finite, then a certain explicit number of columns of \(\Phi\) have entries in the augmentation ideal (the kernel of the epimorphism \(\mathbb Z_p[G] \rightarrow \mathbb Z_p\)) and, finally, the reduced norm of \(\Phi\) is strongly related to a characteristic element \(\mathcal L\) of \(C\) (any invertible element in the centre of \(\mathbb C_p[G]\) that maps under the extended boundary homomorphism to the inverse of a refined Euler characteristic of \(C\)).NEWLINENEWLINEWhen \(C\) is admissible and \(\pi: H^2(C) \to \Pi\) is an epimorphism of \(\mathbb Z_p[G]\)-modules, where \(\Pi\) is a permutation module, then there is a ``organizing matrix'' \(\Phi\) fulfilling even more conditions which are too cumbersome to be stated here explicitly. The most important additional condition is that the columns of \(\Phi\) compute certain natural height pairings.NEWLINENEWLINEThe authors give many natural examples of (weakly) admissible complexes arising in the context of the equivariant Tamagawa number conjecture (ETNC). Assuming the (local and global) ETNC their theory predicts new constraints on the Galois module structure of ideal class groups, wild kernels in higher algebraic \(K\)-theory and Selmer groups of abelian varieties over finite (not necessarily abelian) Galois extensions of number fields. In particular, this leads to generalizations of the theory of annihilation of Selmer modules due to \textit{J. Barrett} and \textit{D. Burns} [J. Reine Angew. Math. 675, 191--222 (2013; Zbl 1276.11173)] and of the theory of congruences for values of motivic \(L\)-functions by \textit{D. Burns} [Pure Appl. Math. Q. 6, No. 1, 83--172 (2010; Zbl 1227.11118)]. Moreover, it is closely related (and partially recovers) the results of \textit{D. Burns} [Invent. Math. 186, No. 2, 291--371 (2011; Zbl 1239.11128)] and of the reviewer [Math. Proc. Camb. Philos. Soc. 151, No. 1, 1--22 (2011; Zbl 1254.11096)] on the annihilation of class groups and higher algebraic \(K\)-groups, respectively.