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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.

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 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 .

We describe a theory of Whittaker models and L -functions for irreducible representations of a metaplectic covering group of GL (2). We explain how to use these L -functions to establish an arithmetical correspondence between “genuine” cuspidal representations of the metaplectic group and cuspidal representations of GL (2). When our ground field is Q, this correspondence generalizes and reformulates recent results of G. Shimura [(1973) Ann. Math. 97, 440-481]. A crucial role in our theory is played by what we call “exceptional” cusp forms—those that are completely determined by just one Fourier coefficient.

Publisher Summary This chapter describes the most recent results available for the L -functions L ( s , π, r ). The L-function L ( s , π, r ), defined by a convergent Euler product in the half-plane Re ( s ) > 2, extends to a meromorphic function. A tentative solution to the problem of defining L -functions at the ramified primes is by way of suitably normalized gamma factors for the local functional equation. More precisely, Whittaker functional axe introduced to define some local coefficients these local coefficients are then used to normalize the intertwining operators and gamma factors in the functional equation so that the resulting normalized objects satisfy equations of the type M* ° M* = I. In the chapter, it is shown how to extend the partial L -functions L S ( s , π, r i ) to include local factors at every place υ , to prove an exact functional equation for these completed L -functions and to prove that these L -functions have only a finite number of poles.

The local results announced in this paper concern the analytic continuation of interwining operators for the metaplectic covering group of SL (2) over an arbitrary local field and the decomposition of Weil's representation attached to the quadratic form q ( x ) = x 2 . The global results concern Eisenstein series on the metaplectic group and a Siegel-Weil type formula relating the residues of these series to certain generalized theta functions.

This chapter presents the developments and refinements happened regarding the expression of L -functions. It discusses several methods and theories introduced to this subject and their applications. Local and global zeta-integrals are used to prove the analytic continuation and functional equations of the automorphic L -functions attached to GL (2) following five some steps. In the decade immediately following the work of Jacquet and Langlands, similar strategies were developed for GL ( n ), groups closely related to GL ( n ), and representations r of L G closely related to the standard representation. In connection to these steps, the chapter focuses on the main new ideas introduced in each of the special cases. It discusses zeta-integrals for GL ( n ) and beyond, and presents an analysis and reformulation of the method of Rankin–Selberg–Jacquet for GL (2) X GL (2).