Topics in Number Theory: In Honor of B. Gordon and S. Chowla PDF
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Topics in Number Theory Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Volume 467 Topics in Number Theory In Honor of B. Gordon and S. Chawla edited by Scott D. Ahlgren George E. Andrews and KenOno Department of Mathematics, Pennsylvania State University, University Park, PA, U.S.A. KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13:978-1-4613-7988-1 e-ISBN-13:978-1-4613-0305-3 001: 10.1007/978-1-4613-0305-3 Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AU Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved ©1999 Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any infonnation storage and retrieval system, without written permission from the copyright owner v TABLE OF CONTENTS Preface G. Andrews .................................................................. vii Motivating the multiplicative spectrum A. Granville and K. Soundararajan ............................................. 1 Modular Mahler measures I F. Rodriguez Villegas ......................................................... 17 Diophantine problems in many variables: The role of additive number theory T. Wooley .................................................................... 49 Equations F(x,y,(il:) = 0 S. Ahlgren .................................................................... 85 A fundamental invariant in the theory of partitions K. Alladi .................................................................... 101 Congruence properties of values of L-functions and applications J. Bruinier, K. James, W. Kohnen, K. Ono, C. Skinner, and V. Vatsal ....... 115 On Tschirnhaus transformations J. Buhler and Z. Reichstein ............................" . ..................... 127 Hecke operators and the nonvanishing of L-functions J. B. Conrey and D. W. Farmer .............................................. 143 A new minor-arcs estimate for number fields M. Davidson ................................................................. 151 Modular functions, Maple and Andrews' tenth problem F. Garvan ................................................................... 163 Hecke characters and formal group characters L. Guo ....................................................................... 181 On certain Gauss periods with common decomposition field S. Gurak ..................................................................... 193 Theorems and conjectures involving rook polynomials with only real zeros J. Haglund, K. Ono, and D. Wagner .......................................... 207 vi An example of an elliptic curve with a positive density of prime quadratic twists which have rank zero K. Jaznes .................................................................... 223 Exponents of class groups of quadratic fields M. R. Murty ....................................... , ......................... 229 A local-global principle for densities B. Poonen and M. Stoll ...................................................... 241 Divisibility of the specialization map for twists of abelian varieties J. Silverman ................................................................. 245 vii Preface From July 31 through August 3,1997, the Pennsylvania State University hosted the Topics in Number Theory Conference. The conference was organized by Ken Ono and myself. By writing the preface, I am afforded the opportunity to express my gratitude to Ken for beng the inspiring and driving force behind the whole conference. Without his energy, enthusiasm and skill the entire event would never have occurred. We are extremely grateful to the sponsors of the conference: The National Sci ence Foundation, The Penn State Conference Center and the Penn State Depart ment of Mathematics. The object in this conference was to provide a variety of presentations giving a current picture of recent, significant work in number theory. There were eight plenary lectures: H. Darmon (McGill University), "Non-vanishing of L-functions and their derivatives modulo p." A. Granville (University of Georgia), "Mean values of multiplicative functions." C. Pomerance (University of Georgia), "Recent results in primality testing." C. Skinner (Princeton University), "Deformations of Galois representations." R. Stanley (Massachusetts Institute of Technology), "Some interesting hyperplane arrangements." F. Rodriguez Villegas (Princeton University), "Modular Mahler measures." T. Wooley (University of Michigan), "Diophantine problems in many variables: The role of additive number theory." D. Zeilberger (Temple University), "Reverse engineering in combinatorics and number theory." The papers in this volume provide an accurate picture of many of the topics presented at the conference including contributions from four of the plenary lectures. On Saturday evening, August 2, the conference banquet was held at the Nittany Lion Inn. The evening had a two-fold purpose. The first was to honor Basil Gordon at 65. Basil has been the Ph.D. advisor to many ofthe most active young number theorists of today. In addition, his work has been a powerful inspiration to many, many others, including me. viii Besides honoring Basil Gordon, we also paid tribute to the late Sarvadaman Chowla. Chowla was a professor at Penn State from 1963 to 1976. He was a towering presence in number theory, throughout his tenure at Penn State. He brought a continuous stream of famous mathematical visitors. In addition he inspired and directed many graduate students and young assistant professors (me included) with his exciting and eccentric "Chowla Seminar." It was only fitting that evening that Penn State honor his memory by announcing the creation of the S. Chowla Assistant Professorship and by also recognizing Kevin James (one of the contributors to this volume) as the second S. Chowla Assistant Professor. The plaque honoring S. Chowla Assistant Professorship and a framed photo of Chowla were unveiled in the presence of his daughter Paromita Chowla, who recently retired from Penn State. It is then with great pleasure that I commend to you these proceedings of the Topics in Number Theory Conference. George E. Andrews University Park, Pennsylvania July 25, 1998. MOTIVATING THE MULTIPLICATIVE SPECTRUM ANDREW GRANVILLE AND K. SOUNDARARAJAN Dedicated to the memory of s. D. Chowla ABSTRACT. In this article, we describe and motivate some of the results and notions from our ongoing project [2]. The results stated here are substantially new (unless otherwise attributed) and detailed proofs will appear in [2]. 1. DEFINITIONS AND PROPERTIES OF THE SPECTRUM Let S be a subset of the unit disc U. By F(S) we denote the class of completely multiplicative functions f such that f(P) E S for all primes p. Our main concern is: What numbers arise as mean-values of functions in F(S)? Precisely, we define L = {~ = J~oo rN(S) f(n) : f E F(S)} and r(S) rN(S), n5,N Here and henceforth, if we have a sequence of subsets IN of the unit disc U := {Izl ~ I}, then by writing limN-+oo IN = J we mean that z E J if and only if there is a sequence of points ZN E IN with ZN -+ Z as N -+ 00. We call r(S) the spectrum of the set S and our main object is to understand the spectrum. Although we can determine the spectrum explicitly only in a few cases (see Theorem 1 below for the most interesting of these cases), we are able to qualitatively describe it, and obtain a lot of its geometric structure. For example, we can always determine the boundary points of the spectrum (that is the elements ofr(S)n'll' where 'll' is the unit circle). Another property is that the spectrum is always connected. Qualitatively, the spectrum may be described in terms of Euler products and solutions to certain integral equations. We begin with a few immediate consequences of our definition: • r(S) is a closed subset of the unit disc U. • r(S) = r(S) (where S denotes the closure of S). Henceforth, we shall assume that S is always closed. • If S1 C S2 then r(S1) c r(S2)' • r({I}) = {I}. One of the main results of [2], which formed the initial motivation to study the questions discussed herein, is a precise description of the spectrum of [-1,1]. The first author is a Presidential Faculty Fellow. He is also supported, in part, by the National Science Foundation. The second author is supported by an Alfred P. Sloan dissertation fellowship S.D. Ahlgren et al. (eds.). Topics in Number Theory. 1-15. @ 1999 Kluwer Academic Publishers.
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