Selmer complexes. (English) Zbl 1211.11120

Astérisque 310. Paris: Société Mathématique de France (ISBN 978-2-85629-227-3/pbk). viii, 559 p. (2007).
Summary: This book builds new foundations of Iwasawa theory, based on a systematic study of cohomological invariants of big Galois representations in the framework of derived categories. A new duality formalism is developed, which leads to generalized Cassels-Tate pairings and generalized \(p\)-adic height pairings. One of the applications is a parity result for Selmer groups associated to Hilbert modular forms.
Contents: Introduction; Homological algebra: Products and signs; Local duality; Continuous cohomology; Continuous cohomology of pro-finite groups; Duality theorems for Galois cohomology revisited; Selmer complexes; Unramified cohomology; Iwasawa theory; Classical Iwasawa theory; Generalized Cassels-Tate pairings; \(R\)-valued height pairings; Parity of ranks of Selmer groups.


11R23 Iwasawa theory
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11R34 Galois cohomology
22E41 Continuous cohomology of Lie groups