This graduate-level textbook provides an elementary exposition of the theory of automorphic representations and L-functions for the general linear group in an adelic setting. Definitions are kept to a minimum and repeated when reintroduced so that the book is accessible from any entry point, and with no prior knowledge of representation theory. The book includes concrete examples of global and local representations of GL(n), and presents their associated L-functions. In Volume 1, the theory is developed from first principles for GL(1), then carefully extended to GL(2) with complete detailed proofs of key theorems. Several proofs are presented for the first time, including Jacquet's simple and elegant proof of the tensor product theorem. In Volume 2, the higher rank situation of GL(n) is given a detailed treatment. Containing numerous exercises by Xander Faber, this book will motivate students and researchers to begin working in this fertile field of research.
In this paper we provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of essentially self-dual representations, that is representations which are isomorphic to the twist of their own contragredient by some Hecke character. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin2n to GL2n.
Abstract. This paper studies the Fourier expansion of Hecke-Maass eigenforms for GL(2, Q) of arbitrary weight, level, and character at various cusps. It is shown that the Fourier coefficients at a cusp satisfy certain very explicit multiplicativity relations. As an application, it is proved that a local representation of GL(2, Q p ) which is isomorphic to a local factor of a global cuspidal automorphic representation generated by the adelic lift of a newform of arbitrary weight, level N , and character χ (mod N ) cannot be supercuspidal if χ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width m and cusp parameter = 0) the mp Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers . In the last part of this paper a three term identity involving the Fourier expansion at three different cusps is derived. §1. IntroductionConsider the groupThen Γ 0 (N ) acts on the upper half-plane h := {x + iy | x ∈ R, y > 0} by linear fractional transformations. We fix an integer k (called the weight), an integer N ≥ 1 (called the level), and a Dirichlet character χ : (Z/N Z) → C × . For any function f : h → C, and any matrix γ ∈ GL(2, R) of positive determinant, define the slash operatorand the character χ :An automorphic function of weight k, and character χ for Γ 0 (N ) is a smooth function f : h → C which satisfies the automorphy relation1991 Mathematics Subject Classification. Fourier coefficients of automorphic forms, Hecke-Petersson operators, automorphic representations over local and global fields.Typeset by A M S-T E X 2 DORIAN GOLDFELD JOSEPH HUNDLEY MIN LEE for all γ ∈ Γ 0 (N ). Note that this forces f to be identically zero unless k and χ satisfy the compatibility condition χ(−1) = (−1) k . Clearly, an automorphic function of weight k and character χ for Γ 0 (N ) is also an automorphic function of weight k and character χ for Γ 0 (M ) whenever N divides M and χ is the Dirichlet character (mod M ) obtained by pulling χ back via the natural projection (Z/M Z) × → (Z/N Z) × . Such automorphic functions are said to be "old". An automorphic function is said to be of level N if it is an automorphic function of weight k and level χ for Γ 0 (N ), for some k, χ, and it is not an automorphic function of level M for any M dividing N. Now fix ν ∈ C. A Maass form of weight k, type ν, level N and character χ (mod N ) is an automorphic function of weight k, level N and character χ which has moderate growth and which is also an eigenfunction of the weight k Laplace operator (see §2) with eigenvalue ν(1 − ν). A Maass form is said to be a "new form" if it lies in the orthogonal complement (with respect to the Petersson inner product) of the space of old forms (see [2]). A Hecke newform is a newform which is an eigenfunction of all the Hecke operators. The main result of Atkin-Lehner theory [2] is that the space of newforms has a basis where each basis element is an eigenfunction of all the Hecke operators.Cusps are defined to be elements of Q ∪ {∞}. We define an action o...
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