an:01295617
Zbl 1024.11034
Labesse, Jean-Pierre
Cohomology, stabilization and base change
FR
Ast??risque. 257. Paris: Soci??t?? Math??matique de France, 161 p. (1999).
00384822
1999
b
11F70 22E55 18G30 11-02
crossed set; Galois cohomology; twisted trace formula; existence of stable transfer; cyclic base change; abelianized Galois cohomology; stable conjugacy; orbital integrals; norm maps; local transfer
From the abstract: ``The authors introduce the concept of a ``crossed set'' (generating the notion of a crossed module) and study the Galois cohomology of these objects. This is crucially used in the stabilisation of all elliptic terms in the twisted trace formula. The authors then prove the existence of stable transfer for cyclic base change.''
In the first chapter preliminaries on Galois cohomology with values in crossed sets, and relations with abelianized Galois cohomology (Breen, Borovoi, Deligne, Kottwitz) are given. In the second chapter he studies stable conjugacy, orbital integrals and relations with norm maps. In Chapter 3 the local transfer is determined in several cases, and finally in Chapter 4 the stabilization of the trace formula is exposed with applications. Two appendices (the first one co-authored by L. Clozel) give a corrected proof of an earlier result of Clozel concerning that for certain unitary groups, one can lift a given automorphic representation by base change and (the second one, written by L. Breen) crossed sets are given in a simplicial algebra framework.
Tyakal Venkataramana (Bombay)