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On Moments of Twisted L-Functions
ARIZONA WINTER SCHOOL 2016: ANALYTIC THEORY OF TRACE FUNCTIONS PHILIPPE MICHEL, EPF LAUSANNE 1. Functions on congruence rings As was emphasized by C.F. Gauss in the Disquisitiones Arithmeticae (and probably much earlier), the concept of congruences (a.k.a arithmetic progressions of some modulus q, a.k.a the ring Z=qZ) is central in number theory. It is therefore fundamental to understand spaces of functions over the ring Z=qZ. One of the first classes of such functions to have been studied systematically (and one of the most interesting even today) is the group of Dirichlet's characters of modulus q, (Z\=qZ)×, the group homomorphisms χ :(Z=qZ)× ! C×: Indeed these functions enter crucially in Dirichlet's proof that there are infinitely many primes in any admissible arithmetic progression of modulus q [7]. The studying of integral solutions of diophantine equations is another important source of func- tions on congruence rings: these arise as a manifestation of some general local-global principles or more directly when using the circle method. For instance, while investigating the set of representa- tions of an integer d by some integral, positive, definite, quadratic form in four variables q(x; y; z; t), ie. the integral solutions to the equation (1.1) d = q(x; y; z; t); x; y; z; t 2 Z; Kloosterman came up with the very important function modulo q, the so-called "Kloosterman sums", 1 X m (mod q) 7! Kl (m; q) := p e (mx + y); 2 q q xy=1 (mod q) where eq(·) denote the additive character of (Z=qZ; +) 2πix e (x) := exp( ): q q By evaluating the fourth moment of this function over Z=qZ, Kloosterman established the following non-trivial bound 2=3−1=2+o(1) (1.2) Kl2(m; q) q and deduced from it an asymptotic formula for the number of solutions to the equations (1.1). -
On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017
On Some Variants of the Gauss Circle Problem by David Lowry-Duda B.S. in Applied Mathematics, Georgia Institute of Technology, Atlanta, GA, 2011 B.S. in International Affairs and Modern Languages, Georgia Institute of Technology, Atlanta, GA, 2011 M.Sc. in Mathematics, Brown University, Providence, RI, 2015 A dissertation submitted in partial fulfillment of the arXiv:1704.02376v2 [math.NT] 2 May 2017 requirements for the degree of Doctor of Philosophy in the Department of Mathematics at Brown University PROVIDENCE, RHODE ISLAND May 2017 c Copyright 2017 by David Lowry-Duda Abstract of “On Some Variants of the Gauss Circle Problem” by David Lowry-Duda, Ph.D., Brown University, May 2017 The Gauss Circle Problem concerns finding asymptotics for the number of lattice point lying inside a circle in terms of the radius of the circle. The heuristic that the number of points is very nearly the area of the circle is surprisingly accurate. This seemingly simple problem has prompted new ideas in many areas of number theory and mathematics, and it is now recognized as one instance of a general phenomenon. In this work, we describe two variants of the Gauss Circle problem that exhibit similar characteristics. The first variant concerns sums of Fourier coefficients of GL(2) cusp forms. These sums behave very similarly to the error term in the Gauss Circle problem. Normalized correctly, it is conjectured that the two satisfy essentially the same asymptotics. We introduce new Dirichlet series with coefficients that are squares of partial sums of Fourier coefficients of cusp forms. We study the meromorphic properties of these Dirichlet series and use these series to give new perspectives on the mean square of the size of sums of these Fourier coefficients. -
18.785 Notes
Contents 1 Introduction 4 1.1 What is an automorphic form? . 4 1.2 A rough definition of automorphic forms on Lie groups . 5 1.3 Specializing to G = SL(2; R)....................... 5 1.4 Goals for the course . 7 1.5 Recommended Reading . 7 2 Automorphic forms from elliptic functions 8 2.1 Elliptic Functions . 8 2.2 Constructing elliptic functions . 9 2.3 Examples of Automorphic Forms: Eisenstein Series . 14 2.4 The Fourier expansion of G2k ...................... 17 2.5 The j-function and elliptic curves . 19 3 The geometry of the upper half plane 19 3.1 The topological space ΓnH ........................ 20 3.2 Discrete subgroups of SL(2; R) ..................... 22 3.3 Arithmetic subgroups of SL(2; Q).................... 23 3.4 Linear fractional transformations . 24 3.5 Example: the structure of SL(2; Z)................... 27 3.6 Fundamental domains . 28 3.7 ΓnH∗ as a topological space . 31 3.8 ΓnH∗ as a Riemann surface . 34 3.9 A few basics about compact Riemann surfaces . 35 3.10 The genus of X(Γ) . 37 4 Automorphic Forms for Fuchsian Groups 40 4.1 A general definition of classical automorphic forms . 40 4.2 Dimensions of spaces of modular forms . 42 4.3 The Riemann-Roch theorem . 43 4.4 Proof of dimension formulas . 44 4.5 Modular forms as sections of line bundles . 46 4.6 Poincar´eSeries . 48 4.7 Fourier coefficients of Poincar´eseries . 50 4.8 The Hilbert space of cusp forms . 54 4.9 Basic estimates for Kloosterman sums . 56 4.10 The size of Fourier coefficients for general cusp forms . -
Lectures on Mean Values of the Riemann Zeta Function
Lectures on Mean Values of The Riemann Zeta Function By A.Ivic Tata Institute of Fundamental Research Bombay 1991 Lectures on Mean Values of The Riemann Zeta Function By A. Ivic Published for the Tata Institute of Fundamental Research SPRINGER-VERLAG Berlin Heidelberg New York Tokyo Author A. Ivic S.Kovacevica, 40 Stan 41 Yu-11000, Beograd YUGOSLAVIA © Tata Institute of Fundamental Research, 1991 ISBN 3-350-54748-7-Springer-Verlag, Berlin. Heidelberg. New York. Tokyo ISBN 0-387-54748-7-Springer-Verlag, New York, Heidelberg. Berlin. Tokyo No part of this book may e reproduced in any form by print, microfilm or any other mans with- out written permission from the Tata Institute of Fundamental Research, Colaba, Bombay 400 005 Printed by Anamika Trading Company Navneet Bhavan, Bhavani Shankar Road Dadar, Bombay 400 028 and Published by H.Goetze, Springer-Verlag, Heidelberg, Germany PRINTED IN INDIA Preface These lectures were given at the Tata Institute of Fundamental Research in the summer of 1990. The specialized topic of mean values of the Riemann zeta-function permitted me to go into considerable depth. The central theme were the second and the fourth moment on the critical line, and recent results concerning these topic are extensively treated. In a sense this work is a continuation of my monograph [1], since except for the introductory Chapter 1, it starts where [1] ends. Most of the results in this text are unconditional, that is, they do not depend on unproved hypothesis like Riemann’s (all complex zeros of ζ(s) have real parts equal to 1 ) or Lindelof’s¨ (ζ( 1 + it) tǫ). -
Kloosterman Sums
On SL2(Z) and SL3(Z) Kloosterman sums Olga Balkanova Erasmus Mundus Master ALGANT University of Bordeaux 1 Supervisors Guillaume Ricotta and Jean-Marc Couveignes 2011 Acknowledgements I wish to thank my advisor Dr. Guillaume Ricotta for designing this project for me and making my work interesting. I am grateful for his encouragements and inspiring interest, for his knowledge and clear explanations, for our discussions and emails. It is really valuable experience to work under the guidance of advisor Guillaume. I thank my co-advisor Prof. Jean-Marc Couveignes for the help with computational part of my thesis, useful advices and fruitful ideas. I am also grateful to my russian Professor Maxim Vsemirnov for in- troducing me to the world of analytic number theory and providing interesting problems. Many thanks to my family, whose love always helps me to overcome all the difficulties. Finally, I thank my ALGANT friends Andrii, Katya, Liu, Martin, Novi, Nikola for all shared moments. Contents Contents ii List of Figures iii 1 Classical Kloosterman sums3 1.1 SL(2) modular forms . .3 1.2 Construction of SL(2) Poincar´eseries . .6 1.3 Fourier expansion of Poincar´eseries . .9 1.4 Some properties of Kloosterman sums . 13 1.5 Distribution of Kloosterman angles . 15 1.6 Numerical computation of Poincar´eseries . 21 1.6.1 Poincar´eseries and fundamental domain reduction algorithm 21 1.6.2 Absolute error estimate . 23 2 SL(3) Kloosterman sums 27 2.1 Generalized upper-half space and Iwasawa decomposition . 27 2.2 Automorphic forms and Fourier expansion . 31 2.3 SL(3) Poincar´eseries . -
A Complete Bibliography of the Journal of Number Theory (2000–2009)
A Complete Bibliography of the Journal of Number Theory (2000{2009) Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 16 July 2020 Version 1.00 Title word cross-reference (0; 1) [622]. (12 +1) (n2 + 1) [1431]. (3)2 [75]. (α, α2) [490]. (U(1); U(2)) ··· 2s [805, 806]. ( nα + γ 12) [39]. (OK =I)× [163]. 2 qr [140]. 0 [587]. 1 [220, 421, 574,f 676, 932,}− 1094, 1310, 1538, 1688]. 1=2− [1477]. 11 [592]. 12 [355]. 13 [592]. 2 [69, 88, 123, 146, 181, 203, 218, 276, 295, 365, 452, 522, 526, 645, 703, 708, 718, 729, 732, 734, 860, 980, 1111, 1150, 1203, 1220, 1296, 1321, 1326, 1361, 1469, 1494, 1517, 1540, 1581]. 24m [271]. 2t [297]. 2k [108]. 2n [1568]. 2s [1268]. 3 [75, 123, 692, 755, 1157, 1321, 1353]. 3x + 1 [948, 1111]. 4 [123, 208, 466, 536, 1369, 1666]. 40 [1528]. 4` [24]. 5 [56, 323, 1525]. 7 [1580]. 8 [585, 812, 1525]. 9 [1531]. ?(x) [694]. A [1115]. a(modp) [636, 637]. A(q) [179]. 3 3 3 3 a + b + c + d = 0 [339]. A4 [1056]. A6 [891]. An [1543]. abc [50]. α [425]. α3 + kα 1 = 0 [490]. ax(x 1)=2+by(y 1)=2 [1552]. axy + bx + cy + d − − − [512]. -
The Relative Trace Formula in Analytic Number Theory 3
THE RELATIVE TRACE FORMULA IN ANALYTIC NUMBER THEORY VALENTIN BLOMER Abstract. We discuss a variety of applications of the relative trace formula of Kuznetsov type in analytic number theory and the theory of automorphic forms. 1. The Poisson summation formula The mother of all trace formulae is the Poisson summation formula. If f : R C is a sufficiently nice function, say Schwartz class for simplicity, then → (1) f(n)= f(n) nX∈Z nX∈Z b where ∞ f(y)= f(x)e( xy) dx, e(x)= e2πix Z−∞ − is the Fourier transform.b Applying (1) formally with f(x) = x−s, one obtains the functional equation of the Riemann zeta function, and this heuristic argument can easily be made rigorous. More generally, the Poisson summation formula implies the functional equations for all Dirichlet L-functions [IK, Section 4.6], and is in fact essentially a re-statement of those. Why is the Poisson summation formula a trace formula? Given a function f as above, we consider the convolution operator L : L2(R/Z) L2(R/Z) f → given by Lf (g)(x)= f(x y)g(y) dy = k(x, y)g(y) dy ZR − ZR/Z arXiv:1912.08137v1 [math.NT] 17 Dec 2019 where (2) k(x, y)= f(x y + n). − nX∈Z The functions en(x)= e(nx) are an orthonormal basis of eigenfunctions: Lf (en)= f(n)en. Computing the trace of Lf in two ways we obtain b f(n)= k(x, x) dx = f(n). ZR Z nX∈Z / nX∈Z b 2010 Mathematics Subject Classification. -
Contemporary Mathematics 314
CONTEMPORARY MATHEMATICS 314 Topology and Geometry: Commemorating SISTAG Singapore International Symposium in Topology and Geometry (SISTAG) July 2-6, 2001 National University of Singapore, Singapore A. J. Berrick Man Chun Leung Xingwang Xu Editors http://dx.doi.org/10.1090/conm/314 Topology and Geometry: Commemorating SISTAG CoNTEMPORARY MATHEMATICS 314 Topology and Geometry: Commemorating SISTAG Singapore International Symposium in Topology and Geometry (SISTAG) July 2--6, 2001 National University of Singapore, Singapore A. J. Berrick Man Chun Leung Xingwang Xu Editors American Mathematical Society Providence, Rhode Island Editorial Borad Dennis DeTurck, managing editor Andreas Blass Andy R. Magid Michael Vogelius This is a commemorative volume of SISTAG, the Singapore International Symposium in Topology and Geometry, held from July 2-6, 2001, at the National University of Singapore. 2000 Mathematics Subject Classification. Primary 14-XX, 35-XX, 53-XX, 55-XX, 57-XX, 58-XX. Library of Congress Cataloging-in-Publication Data Singapore International Symposium in Topology and Geometry (2001 : National University of Singapore) Geometry and topology : Commemorating SISTAG : Singapore International Symposium in Topology and Geometry, (SISTAG) July 2-6, 2001, National University of Singapore, Singapore/ A. J. Berrick, Man Chun Leung, Xingwang Xu, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 314) Includes bibliographical references. ISBN 0-8218-2820-7 1. Topology-Congresses. 2. Geometry-Congresses. I. Berrick, A. J. (A. Jon) II. Leung, Man Chun, 1963- III. Xu, Xingwang, 1957- IV. Title. V. Contemporary mathematics (Amer- ican Mathematical Society) ; v. 314. QA6ll.A1 S56 2001 514-dc21 2002034243 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. -
Counting Local Systems with Principal Unipotent Local Monodromy
Annals of Mathematics 178 (2013), 921{982 http://dx.doi.org/10.4007/annals.2013.178.3.3 Counting local systems with principal unipotent local monodromy By Pierre Deligne and Yuval Z. Flicker Abstract Let X1 be a curve of genus g, projective and smooth over Fq. Let S1 ⊂ X1 be a reduced divisor consisting of N1 closed points of X1. Let (X; S) be obtained from (X1;S1) by extension of scalars to an algebraic closure F of Fq. Fix a prime l not dividing q. The pullback by the Frobenius endomorphism Fr of X induces a permutation Fr∗ of the set of isomorphism classes of rank n irreducible Ql-local systems on X − S. It maps to itself the subset of those classes for which the local monodromy at each s 2 S is unipotent, with a single Jordan block. Let T (X1;S1; n; m) be the number of fixed points of Fr∗m acting on this subset. Under the assumption that N1 ≥ 2, we show that T (X1;S1; n; m) is given by a formula reminiscent of a Lefschetz fixed point formula: the function m 7! T (X1;S1; n; m) is P m of the form niγi for suitable integers ni and \eigenvalues" γi. We use Lafforgue to reduce the computation of T (X1;S1; n; m) to counting automorphic representations of GL(n), and the assumption N1 ≥ 2 to move the counting to the multiplicative group of a division algebra, where the trace formula is easier to use. Contents Introduction 922 0. Notation 923 1. -
Sums of Kloosterman Sums in Arithmetic Progressions, and the Error Term in the Dispersion Method Sary Drappeau
Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method Sary Drappeau To cite this version: Sary Drappeau. Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method. Proceedings of the London Mathematical Society, London Mathematical Society, 2017, 114 (4), pp.684-732. 10.1112/plms.12022. hal-01302604 HAL Id: hal-01302604 https://hal.archives-ouvertes.fr/hal-01302604 Submitted on 14 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. SUMS OF KLOOSTERMAN SUMS IN ARITHMETIC PROGRESSIONS, AND THE ERROR TERM IN THE DISPERSION METHOD SARY DRAPPEAU Abstract. We prove a bound for quintilinear sums of Kloosterman sums, with con- gruence conditions on the “smooth” summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet L-functions, we prove power-saving error term in the Titchmarsh divisor problem of estimat- ing τ(p 1). Unconditionally, we isolate the possible contribution of Siegel p≤x − zeroes, showing it is always negative. -
Israel Mathematical Union — American Mathemacial Society the Second International Joint Meeting, June 2014
ABSTRACTS (BY ALPHABETICAL ORDER OF THE SPEAKERS) Israel Mathematical Union | American Mathemacial Society The Second International Joint Meeting, June 2014 List of sessions: (*) Invited Speakers (1) Geometry and Dynamics (2) Dynamics and Number Theory (3) Combinatorics (4) Financial Mathematics (5) Mirror Symmetry and Representation Theory (6) Nonlinear Analysis and Optimization (7) Qualitative and Analytic Theory of ODE's (8) Quasigroups, Loops and application (9) Algebraic Groups, Division Algebras and Galois Cohomology (10) Asymptotic Geometric Analysis (11) Recent trends in History and Philosophy of Mathematics (12) History of Mathematics (13) Random Matrix Theory (14) Field Arithmetic (15) Additive Number Theory (16) Teaching with Mathematical Habits in Mind (17) Combinatorial Games (19) Applications of Algebra to Cryptography (20) Topological Graph Theory and Map Symmetry (21) The Mathematics of Menahem M. Schiffer (22) PDEs: Modeling Theory and Numberics (23) Contributed Papers (misc. topics) 1 2 IMU-AMS MEETING 2014 All abstracts (by alphabetical order) Session 17 Lowell Abrams (George Washington University). Almost-symmetry for a family of Nim-like arrays. Abstract. For multiple-pile blocking Nim, the mex rule applies symmetri- cally, and if positions are blocked symmetrically as well, then the symmetry of P-positions is ensured. What happens, though, if the blocking is not done symmetrically? We will present some results exhibiting some \almost sym- metric" games, and discuss connections with scaling-by-2 symmetry that seems -
LETTERS to the EDITOR Responses to ”A Word From… Abigail Thompson”
LETTERS TO THE EDITOR Responses to ”A Word from… Abigail Thompson” Thank you to all those who have written letters to the editor about “A Word from… Abigail Thompson” in the Decem- ber 2019 Notices. I appreciate your sharing your thoughts on this important topic with the community. This section contains letters received through December 31, 2019, posted in the order in which they were received. We are no longer updating this page with letters in response to “A Word from… Abigail Thompson.” —Erica Flapan, Editor in Chief Re: Letter by Abigail Thompson Sincerely, Dear Editor, Blake Winter I am writing regarding the article in Vol. 66, No. 11, of Assistant Professor of Mathematics, Medaille College the Notices of the AMS, written by Abigail Thompson. As a mathematics professor, I am very concerned about en- (Received November 20, 2019) suring that the intellectual community of mathematicians Letter to the Editor is focused on rigor and rational thought. I believe that discrimination is antithetical to this ideal: to paraphrase I am writing in support of Abigail Thompson’s opinion the Greek geometer, there is no royal road to mathematics, piece (AMS Notices, 66(2019), 1778–1779). We should all because before matters of pure reason, we are all on an be grateful to her for such a thoughtful argument against equal footing. In my own pursuit of this goal, I work to mandatory “Diversity Statements” for job applicants. As mentor mathematics students from diverse and disadvan- she so eloquently stated, “The idea of using a political test taged backgrounds, including volunteering to help tutor as a screen for job applicants should send a shiver down students at other institutions.