* uschian groups of the first kind * Automorphic forms and functions * Hecke operators and the zeta-functions associated with modular forms * Elliptic curves * Abelian extensions of imaginary quadratic fields and complex multiplication of elliptic curves * Modular functions of higher level * Zeta-functions of algebraic curves and abelian varieties * The cohomology group assoicated with cusp forms * Arithmetic Fuschian groups

Introduction to the arithmetic theory of automorphic functions

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The Geometry and Cohomology of Some Simple Shimura Varieties.

analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

A proof of Langland’s conjecture on Plancherel measures; Complementary series of $p$-adic groups

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Functoriality for the exterior square of ₄ and the symmetric fourth of ₂

Table of Contents:* Parameters * Local transfer * Global stabilization * The standard model * A study of critical cases * The local classification * Local nontempered representations * The global classification * Inner forms * Bibliography * Index

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The Endoscopic Classification of Representations

A general principle, discovered by Robert Langlands and named by him the "functoriality principle," predicts relations between automorphic forms on arithmetic subgroups of different reductive groups. Langlands functoriality relates the eigenvalues of Hecke operators acting on the automorphic forms on two groups (or the local factors of the "automorphic representations" generated by them). In the few instances where such relations have been probed, they have led to deep arithmetic consequences. This book studies one of the simplest general problems in the theory, that of relating automorphic forms on arithmetic subgroups of GL(n,E) and GL(n,F) when E/F is a cyclic extension of number fields. (This is known as the base change problem for GL(n).) The problem is attacked and solved by means of the trace formula. The book relies on deep and technical results obtained by several authors during the last twenty years. It could not serve as an introduction to them, but, by giving complete references to the published literature, the authors have made the work useful to a reader who does not know all the aspects of the theory of automorphic forms.

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

A trace formula for reductive groups I terms associated to classes in G(Q)

Part I. The Unrefined Trace Formula 7 1. The Selberg trace formula for compact quotient 7 2. Algebraic groups and adeles 11 3. Simple examples 15 4. Noncompact quotient and parabolic subgroups 20 5. Roots and weights 24 6. Statement and discussion of a theorem 29 7. Eisenstein series 31 8. On the proof of the theorem 37 9. Qualitative behaviour of J (f) 46 10. The coarse geometric expansion 53 11. Weighted orbital integrals 56 12. Cuspidal automorphic data 64 13. A truncation operator 68 14. The coarse spectral expansion 74 15. Weighted characters 81

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An Introduction to the Trace Formula

Introduction The trace formula for GL2 has yielded a number of deep results on automorphic forms. The same results ought to hold for general groups, but so far, little progress has been made. One of the reasons has been the lack of a suitable trace formula. In [l(d)] and [l(e)] we presented a formula or, as we wrote it in [l(e), $51, G is a reductive group defined over Q, and f is any function in C;(G(A)l). The left hand side of (I*) is the trace of the convolution operator off on the space of cusp forms on G(Q) \\ G(A)'. It is a distribution which is of great importance in the study of automorphic representations. One would hope to study it through the distributions o E O} and { 1 ; : x E X \\ X(G)}. Unfortunately, these distributions depend on a number of unpleasant things. There is the parameter T , as well as a choice of maximal compact subgroup of G(A)' and a choice of minimal parabolic subgroup. What is worse, they are not invariant; their values change when f is replaced by a conjugate of itself. In any generalization of the applications of the trace formula for GL2, we would not be handed the function f. We could only expect to be given a function such as +(f 1: 77-t r d f 1 7 whose values are invariant in f. Here TT ranges over the irreducible tempered representations of G(A)'. The decomposition of trRcusp(f) into the right hand side of (l*) would then be of uncertain value, for the individual terms actually depend on f and not just +(f). a 1981 by Princeton University (Mathematics Department) For copying information, see inside back cover. The purpose of this paper is to modify the terms in (1) so that they are invariant. Under certain assumptions on the local groups G(Q_), we will obtain a formula in which the individual terms are invariant distributions. The definitions will be such that I = 1: if x belongs to X(G). We will therefore also have the analogue of (I*). The main assumptions on the local groups G (Q) are set forth in Section 5. One expects them to hold for all groups, but they are a little beyond the present state of harmonic analysis. They are, essentially, that any invariant distribution, I, on G(A)' can be identified with a …

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James Arthur

Papers

Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula. (AM-120), Volume 120

The Endoscopic Classification of Representations

A trace formula for reductive groups I terms associated to classes in G(Q)

An Introduction to the Trace Formula

The trace formula in invariant form