Download or read book Moments of Automorphic L functions and Related Problems written by Ian Petrow and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of L-functions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is -1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number -1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of full-level GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions.

Download or read book Moments of Automorphic L Functions written by Ming-Ho Ng and published by . This book was released on 2017-01-26 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "Moments of Automorphic L-functions" by Ming-ho, Ng, 吳銘豪, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: This thesis is devoted to investigation of moments of automorphic L-functions, especially on the central values or the edges of the critical strip of automorphic L-functions. There are nine chapters. Chapter 1 is an introduction and provides some background on the analytic theory of automorphic forms. Chapters 2, 3, 4, 5 and 6 are about L-functions associated to the holomorphic cusp forms, while Chapters 7, 8 and 9 are focused on the L-functions associated to the Maass forms. Chapter 2 is the study of the first moment of the symmetric-square L-functions associated to the holomorphic cusp forms. Asymptotic formulae for the twisted first moment of central values of the symmetric-square L-functions with harmonic weight, in the weight aspect are obtained. The result (in Theorem 2.1.1) extends and improves the known results in the literature. As an application, it is applied to derive an asymptotic formula for the first moment of central values of the symmetric-square L-functions without harmonic weight, under the assumption of the non-negativity of symmetric-square L-functions at the center of critical strip. Analogous new formulae without harmonic weight of the first and second moments of the Hecke L-functions are proved in Chapter 3. Unlike the case in Chapter 2, the results in Chapter 3 are unconditional. In Chapter 4, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the symmetric square L-functions, with or without harmonic weight, are investigated in the weight aspect. The similar problem in the level aspect was treated by Lau, Royer and Wu. The theme of Chapter 5, as well as Chapter 4, is to examine, in the weight aspect, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the square L-functions or the square of L-functions, with or without harmonic weight. All the above mentioned results, appeared in Chapters 4 and 5, are in the asymptotic form, that is, given by a formula consisting of a main term and an error term. Chapter 6 investigates the asymptotic behavior of the main terms of the results in Chapters 4 and 5. In particular, precise expansions of high moments are given. In Chapters 7, 8 and 9, the previous studies are carried over to Maass forms in the spectral aspect. The first two moments of central values of symmetric square L-functions associated to Maass forms are computed in Chapter 7. The first four moments of central values of L-functions associated to Maass forms are obtained in Chapter 8. Chapter 9 is to research the mixed moments of central values of symmetric square L-functions twisted by the central values of L-functions or the square of L-functions. These investigations for Maass form are not yet done in the literature. Subjects: L-functions Automorphic functions

Download or read book Six Short Chapters on Automorphic Forms and L functions written by Ze-Li Dou and published by Springer Science & Business Media. This book was released on 2012-12-15 with total page 131 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Six Short Chapters on Automorphic Forms and L-functions" treats the period conjectures of Shimura and the moment conjecture. These conjectures are of central importance in contemporary number theory, but have hitherto remained little discussed in expository form. The book is divided into six short and relatively independent chapters, each with its own theme, and presents a motivated and lively account of the main topics, providing professionals an overall view of the conjectures and providing researchers intending to specialize in the area a guide to the relevant literature. Ze-Li Dou and Qiao Zhang are both associate professors of Mathematics at Texas Christian University, USA.

Download or read book Analytic Properties of Automorphic L Functions written by Stephen Gelbart and published by Academic Press. This book was released on 2014-07-14 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analytic Properties of Automorphic L-Functions is a three-chapter text that covers considerable research works on the automorphic L-functions attached by Langlands to reductive algebraic groups. Chapter I focuses on the analysis of Jacquet-Langlands methods and the Einstein series and Langlands’ so-called “Euler products . This chapter 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). Chapter II deals with the developments and refinements of the zeta-inetgrals for GL(n). Chapter III describes the results for the L-functions L (s, ?, r), which are considered in the constant terms of Einstein series for some quasisplit reductive group. This book will be of value to undergraduate and graduate mathematics students.

Download or read book Moments of Automorphic L functions at Special Points written by Alexander Lu Beckwith and published by . This book was released on 2020 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the behavior of families of L-functions at exhibiting conductor-dropping behavior. We will derive asymptotic expansions of the short interval first and second moments of GL(2)xGL(2) L-functions at special points with power-saving error terms. As a consequence, we show that large number of cusp forms for Hecke congruence surfaces of prime level are simultaneously destroyed in two directions of the associated Teichmuller space. We also establish upper bounds for the second moment of GL(2)xGL(3) L-functions and the sixth moment of GL(2) L-functions at special points as the spectral parameter varies in a short interval.

Download or read book Automorphic Forms and Related Topics written by Samuele Anni and published by American Mathematical Soc.. This book was released on 2019-06-19 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the Building Bridges: 3rd EU/US Summer School and Workshop on Automorphic Forms and Related Topics, which was held in Sarajevo from July 11–22, 2016. The articles summarize material which was presented during the lectures and speed talks during the workshop. These articles address various aspects of the theory of automorphic forms and its relations with the theory of L-functions, the theory of elliptic curves, and representation theory. In addition to mathematical content, the workshop held a panel discussion on diversity and inclusion, which was chaired by a social scientist who has contributed to this volume as well. This volume is intended for researchers interested in expanding their own areas of focus, thus allowing them to “build bridges” to mathematical questions in other fields.

Download or read book Automorphic Forms on GL 3 TR written by D. Bump and published by Springer. This book was released on 2006-12-08 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Moments of Products of L functions Dissertation written by Caroline LaRoche Turnage-Butterbaugh and published by . This book was released on 2014 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: We first consider questions on the distribution of the primes. Using the recent advancement towards the Prime k -tuple Conjecture by Maynard and Tao, we show how to produce infinitely many strings of consecutive primes satisfying specified congruence conditions. We answer an old question of Erdos and Turan by producing strings of consecutive primes whose successive gaps form an increasing (respectively decreasing) sequence. We also show that such strings exist whose successive gaps follow a certain divisibility pattern. Finally, for any coprime integers a and D ≥ 1, we refine a theorem of D. Shiu and find strings of consecutive primes of arbitrary length in the congruence class a mod D. These results were proved jointly with William D. Banks and Tristan Freiberg. We next consider the vertical distribution of the nontrivial zeros of certain Dedekind zeta-functions. In particular, let K be a quadratic number field. Using the mixed second moments of derivatives of the Dedekind zeta-function attached to K on the critical line, we prove the existence of gaps between consecutive zeros of the Dedekind zeta-function attached to K on the critical line which are at least 2.44949... times the average spacing. Finally, assuming the Generalized Riemann Hypothesis and some standard conjectures, we prove upper bounds for moments of arbitrary products of automorphic L -functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds to estimate the variance of the coefficients of these zeta- and L -functions in short intervals. We also prove upper bounds for moments of products of central values of automorphic L -functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants. These results were proved jointly with Micah B. Milinovich.

Download or read book Automorphic Forms Representations and L Functions written by Armand Borel and published by American Mathematical Soc.. This book was released on 1979-06-30 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: Part 2 contains sections on Automorphic representations and $L$-functions, Arithmetical algebraic geometry and $L$-functions

Download or read book Automorphic Forms and L functions I written by David Ginzburg and published by American Mathematical Soc.. This book was released on 2009 with total page 315 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes articles that represent global aspects of automorphic forms. This book covers topics such as: the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions.

Download or read book Explicit Constructions of Automorphic L functions written by Stephen S. Gelbart and published by Springer. This book was released on 1987 with total page 934 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of this research monograph is to derive the analytic continuation and functional equation of the L-functions attached by R.P. Langlands to automorphic representations of reductive algebraic groups. The first part of the book (by Piatetski-Shapiro and Rallis) deals with L-functions for the simple classical groups; the second part (by Gelbart and Piatetski-Shapiro) deals with non-simple groups of the form G GL(n), with G a quasi-split reductive group of split rank n. The method of proof is to construct certain explicit zeta-integrals of Rankin-Selberg type which interpolate the relevant Langlands L-functions and can be analyzed via the theory of Eisenstein series and intertwining operators. This is the first time such an approach has been applied to such general classes of groups. The flavor of the local theory is decidedly representation theoretic, and the work should be of interest to researchers in group representation theory as well as number theory.

Download or read book Automorphic Forms and L functions II written by David Ginzburg and published by American Mathematical Soc.. This book was released on 2009 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: Includes articles that represent global aspects of automorphic forms. This book covers topics such as: the trace formula; functoriality; representations of reductive groups over local fields; the relative trace formula and periods of automorphic forms; Rankin - Selberg convolutions and L-functions; and, p-adic L-functions.

Download or read book Automorphic Forms on GL 2 written by H. Jacquet and published by Springer. This book was released on 2006-11-15 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Lectures on Automorphic L functions written by James W. Cogdell and published by American Mathematical Soc.. This book was released on 2009 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive account of the crucial role automorphic $L$-functions play in number theory and in the Langlands program, especially the Langlands functoriality conjecture. There has been a recent major development in the Langlands functoriality conjecture by the use of automorphic $L$-functions, namely, by combining converse theorems of Cogdell and Piatetski-Shapiro with the Langlands-Shahidi method. This book provides a step-by-step introduction to these developments and explains how the Langlands functoriality conjecture implies solutions to several outstanding conjectures in number theory, such as the Ramanujan conjecture, Sato-Tate conjecture, and Artin's conjecture. It would be ideal for an introductory course in the Langlands program. Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada). Table of Contents: James W.Cogdell, Lectures on $L$-functions, converse theorems, and functoriality for $GL_n$: Preface; Modular forms and their $L$-functions; Automorphic forms; Automorphic representations; Fourier expansions and multiplicity one theorems; Eulerian integral representations; Local $L$-functions: The non-Archimedean case; The unramified calculation; Local $L$-functions: The Archimedean case; Global $L$-functions; Converse theorems; Functoriality; Functoriality for the classical groups; Functoriality for the classical groups, II. Henry H.Kim, Automorphic $L$-functions: Introduction; Chevalley groups and their properties; Cuspidal representations; $L$-groups and automorphic $L$-functions; Induced representations; Eisenstein series and constant terms; $L$-functions in the constant terms; Meromorphic continuation of $L$-functions; Generic representations and their Whittaker models; Local coefficients and non-constant terms; Local Langlands correspondence; Local $L$-functions and functional equations; Normalization of intertwining operators; Holomorphy and bounded in vertical strips; Langlands functoriality conjecture; Converse theorem of Cogdell and Piatetski-Shapiro; Functoriality of the symmetric cube; Functoriality of the symmetric fourth; Bibliography. M.Ram Murty, Applications of symmetric power $L$-functions: Preface; The Sato-Tate conjecture; Maass wave forms; The Rankin-Selberg method; Oscillations of Fourier coefficients of cusp forms; Poincare series; Kloosterman sums and Selberg's conjecture; Refined estimates for Fourier coefficients of cusp forms; Twisting and averaging of $L$-series; The Kim-Sarnak theorem; Introduction to Artin $L$-functions; Zeros and poles of Artin $L$-functions; The Langlands-Tunnell theorem; Bibliography. This is a reprint of the 2004 original. (FIM/20.S)

Download or read book Advances in the Theory of Automorphic Forms and Their L functions written by Dihua Jiang and published by American Mathematical Soc.. This book was released on 2016-04-29 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the workshop on “Advances in the Theory of Automorphic Forms and Their L-functions” held in honor of James Cogdell's 60th birthday, held from October 16–25, 2013, at the Erwin Schrödinger Institute (ESI) at the University of Vienna. The workshop and the papers contributed to this volume circle around such topics as the theory of automorphic forms and their L-functions, geometry and number theory, covering some of the recent approaches and advances to these subjects. Specifically, the papers cover aspects of representation theory of p-adic groups, classification of automorphic representations through their Fourier coefficients and their liftings, L-functions for classical groups, special values of L-functions, Howe duality, subconvexity for L-functions, Kloosterman integrals, arithmetic geometry and cohomology of arithmetic groups, and other important problems on L-functions, nodal sets and geometry.

Download or read book Lectures on Automorphic L functions written by James W. Cogdell and published by American Mathematical Soc.. This book was released on with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: James W. Cogdell, Lectures on $L$-functions, converse theorems, and functoriality for $GL_n$: Preface Modular forms and their $L$-functions Automorphic forms Automorphic representations Fourier expansions and multiplicity one theorems Eulerian integral representations Local $L$-functions: The non-Archimedean case The unramified calculation Local $L$-functions: The Archimedean case Global $L$-functions Converse theorems Functoriality Functoriality for the classical groups Functoriality for the classical groups, II Henry H. Kim, Automorphic $L$-functions: Introduction Chevalley groups and their properties Cuspidal representations $L$-groups and automorphic $L$-functions Induced representations Eisenstein series and constant terms $L$-functions in the constant terms Meromorphic continuation of $L$-functions Generic representations and their Whittaker models Local coefficients and non-constant terms Local Langlands correspondence Local $L$-functions and functional equations Normalization of intertwining operators Holomorphy and bounded in vertical strips Langlands functoriality conjecture Converse theorem of Cogdell and Piatetski-Shapiro Functoriality of the symmetric cube Functoriality of the symmetric fourth Bibliography M. Ram Murty, Applications of symmetric power $L$-functions: Preface The Sato-Tate conjecture Maass wave forms The Rankin-Selberg method Oscillations of Fourier coefficients of cusp forms Poincare series Kloosterman sums and Selberg's conjecture Refined estimates for Fourier coefficients of cusp forms Twisting and averaging of $L$-series The Kim-Sarnak theorem Introduction to Artin $L$-functions Zeros and poles of Artin $L$-functions The Langlands-Tunnell theorem Bibliography

Download or read book The Second Moment Theory of Families of L Functions The Case of Twisted Hecke L Functions written by Valentin Blomer and published by American Mathematical Society. This book was released on 2023-02-13 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.