DOCSLIB.ORG

Contributions in Analytic and Algebraic Number Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Contributions in Analytic and Algebraic Number Theory Springer Proceedings in Mathematics Volume 9 For further volumes: http://www.springer.com/series/8806 Springer Proceedings in Mathematics The book series will feature volumes of selected contributions from workshops and conferences in all areas of current research activity in mathematics. Besides an overall evaluation, at the hands of the publisher, of the interest, scientific quality, and timeliness of each proposal, every individual contribution will be refereed to standards comparable to those of leading mathematics journals. It is hoped that this series will thus propose to the research community well-edited and authoritative reports on newest developments in the most interesting and promising areas of mathematical research today. Valentin Blomer • Preda Mihailescu˘ Editors Contributions in Analytic and Algebraic Number Theory Festschrift for S. J. Patterson 123 Editors Valentin Blomer Preda Mihailescu˘ Mathematisches Institut Mathematisches Institut Universitat¨ Gottingen¨ Universitat¨ Gottingen¨ Bunsenstr. 3-5 Bunsenstr. 3-5 37073 Gottingen,¨ Germany 37073 Gottingen,¨ Germany [email protected] [email protected] ISSN 2190-5614 e-ISSN 2190-5622 ISBN 978-1-4614-1218-2 e-ISBN 978-1-4614-1219-9 DOI 10.1007/978-1-4614-1219-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011941120 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Foreword Few mathematicians in Germany in the last 30 years have influenced analytic number theory as much and as deeply as Samuel Patterson. His impressive genealogy is only one of the results of his active mathematical career. Many of the articles in this volume have been presented at the conference “Patterson 60++” at the University of Gottingen¨ in July 2009 on the occasion of his 60th birthday. The conference featured four generations of mathematicians including Patterson’s children and grandchildren as well as his own advisor Prof. Alan F. Beardon who did not miss the opportunity to contribute an enjoyable and lively talk to the many mathematical birthday presents. The articles in this volume reflect Patterson’s manifold interests ranging from classical analytic number theory and exponential sums via automorphic forms on metaplectic groups to measure-theoretic aspects of Fuchsian groups. The combina- tion of measure theory and spectral theory, envisioned by Patterson 30 years ago, is still a most fruitful instance of interdisciplinary mathematics. Besides purely mathematical topics, Samuel Patterson is also well known for his interest in the history of mathematics; a place like Gottingen¨ where he has been teaching and researching for the last 30 years seems to be an appropriate location in this respect. We hope that this book gives a lively image of all of these aspects. We would like to thank the staff of Springer for their cooperation, Michaela Wasmuth for preparing the genealogy, Stefan Baur for excellent and competent typesetting, the numerous referees for their valuable time and expertise, and first and foremost the authors for contributing to this volume. Some colourful and very personal recollections of Samuel Patterson have been compiled on the following pages. Gottingen,¨ Germany Valentin Blomer Summer 2011 Preda Mihailescu˘ vii viii Foreword Encounters with Samuel J. Patterson I came to Gottingen¨ in October 1982, as a first year student in mathematics. Samuel J. Patterson was lecturing on linear algebra. The course was very intense, never loosing a beat in the flow of ideas. His style was passionate, getting in touch with the subject even literally: he would wipe the board with his arm, or his right hand, and when it was over, after ninety minutes of hard work, there was chalk all over the place, including his pants and, sometimes, even his impressive beard. Needless to say, the girls liked it very much. At that time, I did not know much about mathematics and contemporary research, but it was clear to the audience that an enthusiastic scientist was at work here, and it was a significant encouragement to more than just a few to become serious about mathematics. Later, when I became his research student, he had gathered a larger group of younger people in the SFB 170. Those days, his efforts in number theory concentrated on Gauss sums and metaplectic forms, so it seemed. My reading list, however, reflected his heritage as a more classically oriented analytic number theorist, and it would send me off into a different direction. His approach to supervision is to guide the student toward independence as early as is possible. He would push the project as long as guidance is necessary, but get out of the way when there is enough momentum. Also, in the Oberseminar, we got a weekly treat of what he thought we should know, and that was quite a bit. It is not a surprise that his students work in different fields of mathematics. At heart, his love for the theory of numbers and classical mathematics is always present. Now he is soon to retire, whatever this means. Certainly, he will not retire as a mathematician, and we hope that he also continues to encourage the next generation. Jorg¨ Br¨udern I remember June 2001 in Oberwolfach, the conference commemorating the bicen- tennary of Carl Friedrich Gauss’s Disquisitiones Arithmeticae. We had brought together for the occasion mathematicians with historians and philosophers of mathematics. Fairly explicit mathematics dominated the first talks: composition of binary quadratic forms, irreducibility questions in Gauss’s Seventh Section on cyclotomy, nineteenth century reports on the state of the theory of numbers, etc. This changed when it was Patterson’s turn to talk. His subject was Gauss’s determination of the sign of (certain) quadratic Gauss sums, as they are called today. Not that he banned explicit mathematics from his talk – as the lecture went on, he indicated explicitly several proofs of Gauss’s relation, and generalisations thereof. But for some 20 min, he just talked about Gauss’s well-known letter to Olbers of 3 September 1805. This is the letter, where Gauss describes how long he had struggled with the proof – proof of a result, as Patterson duly pointed out, which Gauss had announced in the Disquisitiones on the sole basis of experimental evidence – and how suddenly, with no apparent connection to the ideas he had pursued before – wie der Blitz einschl¨agt – the crucial idea had finally struck him. Foreword ix A specialist of German literature commenting on this letter would be expected to point to the ambient discourse on genius at the time – der Geniebegriff bei Goethe (und anderen). At the Oberwolfach conference, it was precisely the most active research mathematician among all those present at the conference who struck this note. He explained how to read this letter as a self-portrait of Gauss working on his image, on the ethos reflected in it. Beethoven’s Eroica was mentioned as a point of reference for the document, as was Tolstoi’s War and Peace. Unlike the romantic writer, the poet of Roman antiquity was supposed, not only just to capture an immediate impression and conjure up its deeper meaning with astroke–... die V¨oglein schweigen im Walde; warte nur, balde .... –butalsoto prove himself a poeta doctus, able to build global knowledge from geography and natural history into his polished narration. Samuel J. Patterson is for me a rare and challenging example of mathematicus litterarius – a species of which we could use many more, provided they are as capable as he is to use the literary, the humanistic vantage point for detaching themselves from naive identifications with mathematical heroes of the past. Cf. The expanded writeup of the talk: S.J. Patterson, Gauss sums. In: The Shaping of Arithmetic after C.F. Gauss’s Disquisitiones Arithmeticae (C. Goldstein, N. Schappacher, J. Schwermer, eds.), Springer, 2007; pp. 505–528. Norbert Schappacher Es gibt wenige Personen im Leben, die dauerhaft im Gedachtnis¨ haften bleiben. Samuel J. Patterson ist eine davon. Nicht nur als Mensch, sondern mehr noch durch seine Impulse und Ideen, die entscheidende Weichenstellungen geben, und zu gl¨ucklichen und wertvollen Entwicklungen f¨uhren. Er nahm zum Wintersemester 1981 den Ruf auf die Nachfolge von Max Deuring an, die Antrittsvorlesung hielt er am 20. Januar 1982: “Gauß’sche Summen: ein Thema mit Variationen”, ein Gebiet, dass ihn stets fasziniert hat. Es ist bezeichnend, dass er in seiner Antrittsvorlesung Gauß mit den Worten zitiert1:“... n¨amlich die Bestimmung des Wurzelzeichens ist es gerade, was mich immer gequ¨alt hat. Dieser Mangel hat mir alles Ubrige,¨ was ich fand, verleidet; und seit vier Jahren wird selten eine Woche hingegangen sein, wo ich nicht einen oder den anderen vergeblichen Versuch, diesen Knoten zu l¨osen, gemacht h¨atte — besonders lebhaft wieder in der letzten Zeit. Aber alles Br¨uten, alles Suchen ist umsonst gewesen, traurig habe ich jedesmal die Feder weglegen m¨ussen. Endlich vor ein paar Tagen ist’s gelungen – aber nicht meinem m¨uhsamen Suchen, sondern bloß durch die Gnade Gottes m¨ochte ich sagen.
Load more

Recommended publications
  • In Honor of Bertram Kostant
    Progress in Mathematics Volume 123 Series Editors J. Oesterle A. Weinstein Lie Theory and Geometry In Honor of Bertram Kostant Jean-Luc Brylinski Ranee Brylinski Victor Guillemin Victor Kac Editors Springer Science+Business Media, LLC Jean-Luc Brylinski Ranee Brylinski Department of Mathematics Department of Mathematics Penn State University Penn State University University Park, PA 16802 University Park, PA 16802 Victor Guillemin Victor Kac Department of Mathematics Department of Mathematics MIT MIT Cambridge, MA 02139 Cambridge, MA 02139 Library of Congress Cataloging In-Publication Data Lie theory and geometry : in honor of Bertram Kostant / Jean-Luc Brylinski... [et al.], editors. p. cm. - (Progress in mathematics ; v. 123) Invited papers, some originated at a symposium held at MIT in May 1993. Includes bibliographical references. ISBN 978-1-4612-6685-3 ISBN 978-1-4612-0261-5 (eBook) DOI 10.1007/978-1-4612-0261-5 1. Lie groups. 2. Geometry. I. Kostant, Bertram. II. Brylinski, J.-L. (Jean-Luc) III. Series: Progress in mathematics : vol. 123. QA387.L54 1994 94-32297 512'55~dc20 CIP Printed on acid-free paper © Springer Science+Business Media New York 1994 Originally published by Birkhäuser Boston in 1994 Softcover reprint of the hardcover 1st edition 1994 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy• ing, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Springer Science+Business Media, LLC, for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A.
    [Show full text]
  • WHITTAKER MODELS and MULTIPLICITY ONE Contents 1
    WHITTAKER MODELS AND MULTIPLICITY ONE TONY FENG Contents 1. Cuspidal automorphic representations 1 2. The Fourier expansion of a cusp form 2 3. Uniqueness of Whittaker models 4 4. Multiplicity One 6 5. Proof of uniqueness of local Whittaker models 7 References 9 1. Cuspidal automorphic representations Let k be a global field and G a connected reductive group over k. Recall that the cuspidal L2-spectrum of G decomposes discretely: Theorem 1.1 ([Howe1], [Howe2]). There is a decomposition 2 ∼ Md Lcusp(G(k)nG(Ak);!) = Vi: i with the Vi being topologically irreducible, closed subrepresentations, with each isomorphism type appearing with finite multiplicity. One of the main goals of this talk is to prove that for G = GL2, each isomorphism type appears with multiplicity one. 2 2 Theorem 1.2 (Multiplicity One for L ). Every irreducible component of Lcusp(GL2(k)n GL2(Ak);!) has multiplicity one, i.e. is not isomorphic to any other irreducible component. We will approach this theorem by passing to the associated admissible automorphic rep- resentation. Recall that A(G; !) denotes the space of automorphic forms on G, meaning smooth functions φ: G(A) ! C such that: (1) φ has central character !, (2) φ is right K-finite, (3) φ is Z(U(g))-finite, (4) φ has moderate growth at the cusps. 2 We let Acusp(G; !) := A(G; !) \ Lcusp(G(F )nG(A);!). Theorem 1.3 ([Ngo] Theorem 5.4.4). Let π be an irreducible G(A)-invariant subspace of L2(G(F )nG(A);!). Then π \A(G; !) is an irreducible admissible G(A)-representation.
    [Show full text]
  • Whittaker Functions: Number Theory, Geometry and Physics 13W5154
    Whittaker Functions: Number Theory, Geometry and Physics 13w5154 Ben Brubaker (Univ. of Minnesota), Daniel Bump (Stanford University), Gautam Chinta (City College – New York) Solomon Friedberg (Boston College) Paul E. Gunnells (UMass – Amherst) 10/13/2013–10/18/2013 1 An Overview of the Workshop What are now understood as GL(2; R) Whittaker functions were initially defined by Whit- taker as solutions to the confluent hypergeometric differential equation [19]. H. Jacquet used the term “Whittaker function” in his thesis to refer to a more general class of special functions on a reductive group G over a local field F [13]. In the case of G = SL(2) and F = R, this reduces to the original definition. As we discussed during the workshop, and will explain in the following pages, Whit- taker functions exhibit what one might call an “unreasonable eclecticness” as they seem to arise in critical roles in many different contexts in mathematics and physics. At present, many of these connections remain relatively unexplored. The goal of the workshop was to report on and chart future progress toward such connections. Here are several prominent appearances of Whittaker functions, in areas where there has been recent important activity. We observe that these connections tend to point in the direction of deeper connections with quantum groups and geometry (including geometric Langlands theory). • In number theory, Whittaker functions appear as the Fourier coefficients of automor- phic forms, used to define their associated L-functions (cf. [17]). In particular, when applied to Eisenstein series, they arise in both the Langlands-Shahidi method and in Rankin-Selberg constructions.
    [Show full text]
  • GENERALIZED and DEGENERATE WHITTAKER MODELS Contents 1
    GENERALIZED AND DEGENERATE WHITTAKER MODELS RAUL GOMEZ, DMITRY GOUREVITCH, AND SIDDHARTHA SAHI Abstract. We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent or- bit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger or- bits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analo- gous results for Whittaker-Fourier coefficients of automorphic represen- tations. For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wave- front set WF(π). Previously this was only known to hold for F non- archimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky deriva- tives [BZ77, AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor. Contents 1. Introduction and main results 2 1.1. General results 2 1.2. Further results for GLn(F) 5 1.3. The structure of our proofs 6 1.4. Acknowledgements 8 2. Preliminaries 8 2.1. Notation 8 2.2. sl2-triples 8 2.3. Schwartz induction 9 2.4.
    [Show full text]
  • Automorphic Forms on GL(2)
    AUTOMORPHIC FORMS ON GL2 { JACQUET-LANGLANDS Contents 1. Introduction and motivation 2 2. Representation Theoretic Notions 3 2.1. Preliminaries on adeles, groups and representations 3 2.2. Some categories of GF -modules and HF -modules 3 2.3. Contragredient representation 8 3. Classification of local representations: F nonarchimedean 9 3.1. (π; V ) considered as a K-module 10 3.2. A new formulation of smoothness and admissibility 10 3.3. Irreducibility 11 3.4. Induced representations 13 3.5. Supercuspidal representations and the Jacquet module 14 3.6. Matrix coefficients 17 3.7. Corollaries to Theorem 3.6.1 20 3.8. Composition series of B(χ1; χ2) 22 3.9. Intertwining operators 26 3.10. Special representations 36 3.11. Unramified representations and spherical functions 39 3.12. Unitary representations 44 3.13. Whittaker and Kirillov models 45 3.14. L-factors attached to local representations 57 4. Classification of local representations: F archimedean 72 4.1. Two easier problems 72 4.2. From G-modules to (g;K)-modules 73 4.3. An approach to studying (gl2;O(2; R))-modules 77 4.4. Composition series of induced representations 78 4.5. Hecke algebra for GL2(R) 81 4.6. Existence and uniqueness of Whittaker models 82 4.7. Explicit Whittaker functions and the local functional equation 84 4.8. GL2(C) 88 5. Global theory: automorphic forms and representations 88 5.1. Restricted products 88 5.2. The space of automorphic forms 89 5.3. Spectral decomposition and multiplicity one 90 5.4. Archimedean parameters 92 5.5.
    [Show full text]
  • Genericity of Supercuspidal Representations of P-Adic Sp4
    COMPOSITIO MATHEMATICA Genericity of supercuspidal representations of p-adic Sp4 Corinne Blondel and Shaun Stevens Compositio Math. 145 (2009), 213{246. doi:10.1112/S0010437X08003849 FOUNDATION COMPOSITIO MATHEMATICA Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 01 Oct 2021 at 08:36:44, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0010437X08003849 Compositio Math. 145 (2009) 213{246 doi:10.1112/S0010437X08003849 Genericity of supercuspidal representations of p-adic Sp4 Corinne Blondel and Shaun Stevens Abstract We describe the supercuspidal representations of Sp4(F ), for F a non-archimedean local field of residual characteristic different from two, and determine which are generic. Introduction Let F be a locally compact non-archimedean local field, with ring of integers oF , maximal ideal pF , and residue field kF . Whereas every smooth irreducible supercuspidal (complex) representation of GLN (F ) is generic, i.e. has a Whittaker model, this is no longer true for classical groups over F . The existence of non-generic supercuspidal representations of classical groups is significant and has many consequences; let us cite a few of them in local representation theory (global consequences are heavy as well, see for instance [HP79]). First, of course, is the fact that the definition of the L-function attached to a representation of a classical group is available only for generic representations, thanks to the work of Shahidi; in particular, a characterisation through local factors of the local Langlands correspondence for a classical group is not fully available.
    [Show full text]
  • Intertwining Operators, L–Functions, and Representation Theory*
    Intertwining Operators, L{Functions, and Representation Theory* by Freydoon Shahidi** Purdue University *Lecture Notes of the Eleventh KAIST Mathematics Workshop 1996, Taejon, Korea; Edited by Ja Kyung Koo **Partially supported by NSF Grant DMS9622585 1 2 Table of Contents Introduction 1. Notation 2. Modular forms as forms for GL2 3. Maass forms and Ramanujan{Petersson's Conjecture 4. Automorphic cuspidal representations 5. L{functions for GL2 6. Cuspidal representations for GLn 7. Rankin{Selberg L{functions for GLn 8. Applications of properties of L(s; π π0) × 9. The method 9.1 L{groups 9.2 Parabolic subgroups 9.3 Cusp forms and Eisenstein series 9.4 Constant term and intertwining operators 9.5 L{functions in the constant term 9.6 Generic representations 9.7 Local coefficients 9.8 Main global results 9.9 Local factors and functional equations 10. Some local applications 3 Introduction These notes are based on a series of twelve lectures delivered at the Korea Ad- vanced Institute of Science and Technology (KAIST), Taejeon, during their bian- nual workshop in algebra that took place over the period of July 29{August 4, 1996. As its main goal, I have chosen to sketch a proof of the estimate 1=5 1=5 q− < αv < q v j j v for Hecke eigenvalues of a cusp form on GL2(AF ), where F is an arbitrary number field, using the method initiated by Langlands and developed by myself. The first half of the notes (Sections 1{8) is devoted to developing the problem and introducing some important L{functions (symmetric power L{functions for GL(2) and Rankin{ Selberg product L{functions for GL(m) GL(n)).
    [Show full text]
  • Rationality of Darmon Points Over Genus Fields of Nonmaximal Orders
    Sede amministrativa: Universit`adegli Studi di Padova Dipartimento di Matematica “Tullio Levi-Civita” CORSO DI DOTTORATO DI RICERCA IN SCIENZE MATEMATICHE CURRICOLO MATEMATICA CICLO XXXI RATIONALITY OF DARMON POINTS OVER GENUS FIELDS OF NONMAXIMAL ORDERS Coordinatore d’indirizzo: Prof. Martino BARDI Supervisore: Prof. Matteo LONGO Dottorando: Yan HU Abstract Stark-Heegner points, also known as Darmon points, were introduced by H. Darmon in [11], as certain local points on rational elliptic curves, conjecturally defined over abelian extensions of real quadratic fields. The rationality conjecture for these points is only known in the unramified case, namely, when these points are specializations of + global points defined over the strict Hilbert class field HF of the real quadratic field + F and twisted by (unramified) quadratic characters of Gal(HF =F ). We extend these results to the situation of ramified quadratic characters; we show that Darmon points + + + of conductor c ≥ 1 twisted by quadratic characters of Gc =Gal(Hc =F ), where Hc is the strict ring class field of F of conductor c, come from rational points on the elliptic + curve defined over Hc . iv Abstract Riassunto I punti di Stark-Heegner, noti anche come punti di Darmon, furono introdotti da H. Darmon in [11], come certi punti locali su curve ellittiche razionali, congettualmente definiti su estensioni abeliane di campi quadratici reali. La congettura sulla razionalit`a di questi punti `enota solo nel caso non ramificato, vale a dire, quando questi punti + sono specializzazioni di punti globali definiti sull'Hilbert class field stretto HF del campo quadratico reale F e twistati tramite caratteri quadratici (non ramificati) di + Gal(HF =F ).
    [Show full text]
  • SYMPLECTIC-WHITTAKER MODELS for Gln
    Pacific Journal of Mathematics SYMPLECTIC-WHITTAKER MODELS FOR Gln MICHAEL JUSTIN HEUMOS AND STEPHEN JAMES RALLIS Vol. 146, No. 2 December 1990 PACIFIC JOURNAL OF MATHEMATICS Vol. 146, No. 2, 1990 SYMPLECTIC-WHITTAKER MODELS FOR Glw MICHAEL J. HEUMOS AND STEPHEN RALLIS We consider the Klyachko models of admissible irreducible rep- resentations of the group GLn(F) where F is a non-Archimedean local field of characteristic 0. These are models which generalize the usual Whittaker model by allowing the inducing subgroup a symplec- tic component. We prove the uniqueness of the symplectic models and the disjointness for unitary representations of the different models. Moreover, for n < 4 we prove that all unitary irreducible represen- tations admit a Klyachko model. Introduction. Let F be a non-Archimedean local field of character- istic zero. This paper studies the realization of irreducible, admissible representation of G\n{F) in certain induced representations general- izing the Whittaker model. In contrast to generalizing by allowing degenerate Whittaker characters or smaller unipotent groups arising from some degenerate data (cf. [Mo-Wa]), we generalize the inducing subgroup by allowing a symplectic component. Our investigation is motivated by results of A. A. Klyachko [Kl], who exhibited a model, in the sense of I. M. Gel'fand, for G\n over a finite field. He found a set of representations (which we will re- fer to as models) which are disjoint, multiplicity free and exhaust the set of irreducible representations. The representations he considers form a family Jίn ^ , 0 < k < [§]. One extreme Jtn^9 is the Whit- taker model, a representation induced off a character on the subgroup of unipotent, upper triangular matrices.
    [Show full text]
  • Notes on the Generalized Ramanujan Conjectures
    Clay Mathematics Proceedings Volume 4, 2005 Notes on the Generalized Ramanujan Conjectures Peter Sarnak Contents 1. GLn 659 2. General G 667 3. Applications 680 4. References 681 1. GLn Ramanujan’s original conjecture is concerned with the estimation of Fourier coefficients of the weight 12 holomorphic cusp form ∆ for SL(2, Z) on the upper half plane H. The conjecture may be reformulated in terms of the size of the eigen- values of the corresponding Hecke operators or equivalently in terms of the local representations which are components of the automorphic representation associated with ∆. This spectral reformulation of the problem of estimation of Fourier coef- ficients (or more generally periods of automorphic forms) is not a general feature. For example, the Fourier coefficients of Siegel modular forms in several variables carry more information than just the eigenvalues of the Hecke operators. Another example is that of half integral weight cusp forms on H where the issue of the size of the Fourier coefficients is equivalent to special instances of the Lindelo¨f Hypoth- esis for automorphic L-functions (see [Wal], [I-S]). As such, the general problem of estimation of Fourier coefficients appears to lie deeper (or rather farther out of reach at the present time). In these notes we discuss the spectral or representation theoretic generalizations of the Ramanujan Conjectures (GRC for short). While we are still far from being able to establish the full Conjectures in general, the approximations to the conjectures that have been proven suffice for a number of the intended applications. We begin with some general comments.
    [Show full text]
  • Arxiv:1502.06483V5 [Math.RT] 16 Jan 2017 Eeisydrvtv,Wv-Rn E,Oclao Repres Oscillator Set, Wave-Front Derivative, Zelevinsky 17B08
    GENERALIZED AND DEGENERATE WHITTAKER MODELS RAUL GOMEZ, DMITRY GOUREVITCH, AND SIDDHARTHA SAHI Abstract. We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent or- bit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger or- bits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analo- gous results for Whittaker-Fourier coefficients of automorphic represen- tations. For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wave- front set WF(π). Previously this was only known to hold for F non- archimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky deriva- tives [BZ77, AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor. Contents 1. Introduction and main results 2 1.1. General results 2 1.2. Further results for GLn(F) 5 1.3. The structure of our proofs 6 1.4. Acknowledgements 8 2. Preliminaries 8 2.1. Notation 8 2.2. sl2-triples 8 arXiv:1502.06483v5 [math.RT] 16 Jan 2017 2.3.
    [Show full text]
  • Certain L^ 2-Norm and Asymptotic Bounds of Whittaker Function for GL
    Certain L2-norm and Asymptotic bounds of Whittaker functions for GL(n, R) Hongyu He Department of Mathematics Louisiana State University email: [email protected] Abstract Whittaker functions of GL(n, R) ([11][12]), are most known for its role in the Fourier-Whittaker expansion of cusp forms ( [19] [17]). Their behavior in the Siegel set, in large, is well-understood. In this paper, we insert into the literature some potentially useful properties of Whit- taker function over the group GL(n, R) and the mirobolic group Pn. We proved the square integrabilty of the Whittaker functions with re- spect to certain measures, extending a theorem of Jacquet and Shalika ([13]). For principal series representations, we gave various asymptotic bounds of smooth Whittaker functions over the whole group GL(n, R). Due to the lack of good terminology, we use whittaker functions to refer to K-finite or smooth vectors in the Whittaker model. 1 Introduction Let G = GL(n) = GL(n, R). Let U be the group of unipotent lower triangular matrices, U be the group of unipotent upper triangular ma- trices, A be the group of positive diagonal matrices, and M be the centralizer of A in G. Let K be the standard orthogonal group O(n). arXiv:2007.04914v1 [math.RT] 9 Jul 2020 Then we have the Iwasawa decomposition G = KAU. Let (π, H) be a irreducible Hilbert representation of G. Let π∞ be the Frechet space of smooth vectors in H. Let (π∗)−∞ be the topolog- ical dual space of π∞. This dual space is equipped with the natural action of π∗.
    [Show full text]