Simple algebras, base change, and the advanced theory of the trace formula. (English) Zbl 0682.10022

Arthur, James; Clozel, Laurent

Annals of Mathematics Studies, 120. Princeton, NY: Princeton University Press. xiii, 230 p. $ 55.00/hbk; $ 19.95/pbk (1989).

The subject of this book is an application of the theory of the invariant trace formula. Let E/F be a cyclic extension of number fields and \(G=R_{E/F} GL(n)\). Then G has GL(n) as an endoscopic group (in fact, a more general type of groups G is considered, comprising also the multiplicative group of central simple algebras over F; they all have GL(n) as an endoscopic group). The trace formula for G is compared with that for GL(n). Essential is the correspondence \(f\to f'\) between (Hecke) functions on G and functions on GL(n) giving the transfer of orbital integrals. On the other side of the trace formulas we have then a correspondence between tempered representations of \(G(F_ v)\) and \(GL(n,F_ v)\). The comparison consists of an identification of terms in the geometric side of both trace formulas and the same for the representation theoretic side. Descent is an important tool here.
As an application global base change for GL(n) is established for automorphic representations induced from cuspidal (and E/F of prime degree). Another part of the Langlands conjectures is the “induction” of automorphic representations from \(GL(n,A_ E)\) to \(GL(n\ell,A_ F)\), where \(\ell =[E : F]\), which should correspond to induction of representations of the (still conjectural) Tannaka group. This correspondence of automorphic representations is derived here from the base change lifting.
Finally the strong Artin conjecture is proved for an extension of number fields with nilpotent Galois group, i.e. to any irreducible complex representation of degree n of the Galois group corresponds a cuspidal representation of \(GL(n,A_ F)\) with the same local L-factors at almost all places.

Reviewer: J.G.M.Mars

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11-02	Research exposition (monographs, survey articles) pertaining to number theory
11R39	Langlands-Weil conjectures, nonabelian class field theory
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings
12-02	Research exposition (monographs, survey articles) pertaining to field theory
22-02	Research exposition (monographs, survey articles) pertaining to topological groups
11R32	Galois theory

Keywords:

Hecke functions; application; invariant trace formula; endoscopic group; orbital integrals; correspondence between tempered representations; Descent; global base change for GL(n); automorphic representations; Langlands conjectures; induction of representations; Tannaka group; base change lifting; strong Artin conjecture; extension of number fields with nilpotent Galois group; cuspidal representation; local L-factors