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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.

##### MSC:
 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
##### Keywords:
Iwasawa theory; Selmer groups; big Galois representations