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We consider the p-adic distributions derived from Eisenstein series studied by Gelbart, Miller, Panchishkin, and Shahidi, whose Mellin transforms are reciprocals of the Kubota-Leopoldt p-adic L-function. These distributions were shown there to be measures when p is regular. They fail to be measures when p is irregular; in this paper, we give quantitative estimates that describe their behavior more precisely.

We prove a coarse lower bound for L-functions of Langlands-Shahidi type of generic cuspidal automorphic representations on the line Re (s) = 1. We follow the path suggested by Sarnak using Eisenstein series and the Maass-Selberg relations. The bounds are weaker than what the method of de la Vallee Poussin gives for the standard L-functions of GLn, but are applicable to more general automorphic L-functions. Our Theorem answers in a strong form a conjecture posed by Gelbart and Shahidi (J. Amer. Math. Soc. 14 (2001)), and sharpens and considerably simplifies the proof of the main result of that paper.

Let L(s,π,r) be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for L(s,π,r) when R(s)=1 using Eisenstein series. This method is applicable even when L(s,π,r) is not known to be absolutely convergent for R(s)>1. To cite this article: S.S. Gelbart et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).

In recent yearsLL-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations ofLL-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties ofLL-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties ofLL-functions.

We show that for the quasi-split unitary group in three variables every tempered packet of cuspidal automorphic representations contains a globally generic representation.

The purpose of this paper is to prove the boundedness in vertical strips of finite width for all the L-functions that appear in constant terms of Eisenstein series, under a certain natural assumption on local normalized intertwining operators. As a corollary, we prove the boundedness for a number of important L-functions, among them the symmetic cube and triple product L-functions attached to cusp forms on GL(2), as well as several Rankin-Selberg product L-functions, where the local assumption was alredy proved to be valid in each case. Under the same assumption, the finiteness of poles on all of C for all of these L-functions is also proved, a necessary step in the proof of boundedness in vertical strips.

Resume On demontre que pour le groupe unitaire quasi-deploye a trois variables, tout paquet tempere de representations automorphes cuspidales contient une representation globalement generique. © 1999 Academie des Sciences/Editions scientifiques et medicales Elsevier SAS

In this paper we discuss relations between the following types of conditions on a representationπ in a cuspidalL-packet ofU(3): (1)L(s, π×ξ) has a pole ats=1 for someξ; (2) aperiod ofπ over some algebraic cycle inU(3) (coming from a unitary group in two variables) is non-zero; and (3) π is atheta-series lifting from some unitary group in two variables. As an application of our analysis, we show that the algebraic cycles on theU(3) Shimura variety arenot spanned (over the Hecke algebra) by the modular and Shimura curves coming from unitary subgroups.

Preface.- E. Kowalski - Elementary Theory of L-Functions I.- E. Kowalski - Elementary Theory of L-Functions II.- E. Kowalski - Classical Automorphic Forms.- E. DeShalit - Artin L-Functions.- E. DeShalit - L-Functions of Elliptic Curves and Modular Forms.- S. Kudla - Tate's Thesis.- S. Kudla - From Modular Forms to Automorphic Representations.- D. Bump - Spectral Theory and the Trace Formula.- J. Cogdell - Analytic Theory of L-Functions for GLn.- J. Cogdell - Langlands Conjectures for GLn.- J. Cogdell - Dual Groups and Langlands Functoriality.- D. Gaitsgory - Informal Introduction to Geometric Langlands.

PREFACE This article follows the format of five lectures that we gave on automorphic Lfunctions. The lectures were intended to be a brief introduction for number theorists to some of the main ideas in the subject. Three of the lectures concerned the general properties of automorphic L-functions, with particular reference to questions of spectral decomposition. We have grouped these together as Part I. While many of the expected properties of automorphic Lfunctions remain conjectural, a significant number have now been established. The remaining two lectures were focused on the techniques which have been used to establish such properties. These lectures form Part I1 of the article.

L-packets for the quasi-split unitary group in three variables U(3). We shall give an explicit parametrization of these L-packets using theta liftings and describe some relations between the structure of L-packets and certain period integrals. We also prove that an endoscopic L-packet contains a unique generic representation in local and global cases. Suppose that G is a reductive algebraic group over a local or global field

We introduce the notion of Whittaker models for representations of a metaplectic covering group of GL (2) and establish the uniqueness and existence of such models. Our results generalize corresponding results of Jacquet-Langlands, but the methods are new.

The simple relation between representations of the covering groups of SL2 and GL2 makes it possible to fuse and extend the recent metaplectic results of Shimura, Waldspurger, Flicker, and ourselves. By giving a new (purely local andL-function theoretic) treatment of the Waldspurger-Shintani correspondence, we also simplify some of Waldspurger’s original results.

The purpose of this paper is to construct examples of automorphic cuspidal representations which possess a ψ-Whittaker model even though their ψ-Fourier coefficients vanish identically. This phenomenon was known to be impossible for the groupGL(n), but in general remained an open problem. Our examples concern the metaplectic group and rely heavily upon J L Waldspurger’s earlier analysis of cusp forms on this group.

An explicit theory of special functions is developed for the homogeneous space S O ( n ) / S O ( n − m ) SO(n)/SO(n - m) generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on n × m n \times m matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of S O ( n ) SO(n) and the polynomial representations of G L ( m , C ) GL(m,{\mathbf {C}}) .

Let ρ n denote the standard n-dimensional representation of GL(n ,C) and ρ n 2 its symmetric square. For each automorphic cuspidal representation π of GL (2,A) we introduce an Euler product L ( s ,π,ρ 2 2 ) of degree 3 which we prove is entire. We also prove that there exists an automorphic representation II of GL (3)—“the lift of π”—with the property that L ( s ,II,ρ 3 ) = L ( s ,π,ρ 2 2 ). Our results confirm conjectures described in a more general context by R. P. Langlands [(1970) Lecture Notes in Mathematics , no. 170 (Springer-Verlag, Berlin-Heidelberg-New York)].

Publisher Summary This chapter describes the results concerning the analytic continuation of Langlands' automorphic L-functions. It presents the methods that are used to establish the functional equation and meromorphic properties of these L-functions: the method of zeta-integrals and the method of Eisenstein series and local coefficients. Automorphic L-functions attached to the exceptional groups cannot be analyzed via the method of Langlands-Shahidi. These groups cannot be embedded as Levi components of any larger reductive group. A new development related to the L-functions is the generalization to arbitrary groups of the “classical” theory of Poincare series. However, the strategy of directly establishing Langlands' conjecture by way of Poincare series seems to be a job for the distant future.

This chapter presents an analysis of the method of Jacquet–Langlands. By an automorphic cuspidal representation π of GL (2) over Φ, one understands a collection of irreducible representations {π ρ }, one for each prime ρ of Φ, including the infinite prime ρ = ∞. The chapter discusses global zeta-integrals and their factorization. It explains how local and global zeta-integrals are used to prove the analytic continuation and functional equations of the automorphic L -functions attached to GL (2). It analyzes the analytic properties of the local zeta-integrals Z ( W , s ). The key missing ingredients in the local theory are at present the functional equation and the unramified analysis. The chapter explains how the local and global theory yields the required results for L ( s , π ) = II p L ( s , π p ). It presents the steps taken to describe the analytic continuation and functional equation of L ( s , π) = L ( s , π , r ). The chapter presents the five step procedure as the (local-global) L -function machine. It further explains how Langlands' theory of Eisenstein series establishes the meromorphic continuation of is L S ( s , π , r ) for at least one non-trivial r of L G .