Scholar: A Scientific Celebration Highlighting Open Lines of Arithmetic Research: Conference in Honour of M. Ram Murty’s Mathematical Legacy on His 60th Birthd

274 Pages·2016·2.06 MB·English

by  A. C. Cojocaru

655
SCHOLAR—a Scientific Celebration
Highlighting Open Lines
of Arithmetic Research
Conference in Honour of M. Ram Murty’s
Mathematical Legacy on his 60th Birthday
October 15–17, 2013
Centre de Recherches Mathématiques,
Université de Montréal, Québec, Canada
A. C. Cojocaru
C. David
F. Pappalardi
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
CentredeRecherchesMathématiques

Montréal,Québec,Canada
SCHOLAR—a Scientific Celebration
Highlighting Open Lines
of Arithmetic Research
Conference in Honour of M. Ram Murty’s
Mathematical Legacy on his 60th Birthday
October 15–17, 2013
Centre de Recherches Mathématiques,
Université de Montréal, Québec, Canada
A. C. Cojocaru
C. David
F. Pappalardi
Editors

655
SCHOLAR—a Scientific Celebration
Highlighting Open Lines
of Arithmetic Research
Conference in Honour of M. Ram Murty’s
Mathematical Legacy on his 60th Birthday
October 15–17, 2013
Centre de Recherches Mathématiques,
Université de Montréal, Québec, Canada
A. C. Cojocaru
C. David
F. Pappalardi
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
CentredeRecherchesMathématiques
Montréal,Québec,Canada
Editorial Board of Contemporary Mathematics
Dennis DeTurck, managing editor
Michael Loss Kailash Misra Martin J. Strauss
Editorial Committee of the CRM Proceedings and Lecture Notes
Jerry L. Bona Lisa Jeffrey
Va˘sek Chvatal Ram Murty
Galia Dafni Christophe Reutenauer
Donald Dawson Nicolai Reshetikhin
H´el´ene Esnault Nicole Tomczak-Jaegermann
Pengfei Guan Luc Vinet
2010 Mathematics Subject Classification. Primary 11N05, 11N37, 11M06, 11L03, 11J81,
11G05. 11F03, 11Z05.
Library of Congress Cataloging-in-Publication Data
SCHOLAR—a scientific celebration highlighting open lines of arithmetic research : conference
in honour of M. Ram Murty’s mathematical legacy on his 60th birthday, October 15–17, 2013,
CentredeRecherchesMath´ematiques,Universit´edeMontr´eal,Qu´ebec,Canada/A.C.Cojocaru,
C.David,F.Pappalardi,editors.
pagescm. –(Contemporarymathematics;volume655)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1457-3(alk. paper)
1. Mathematics–Research–Congresses. I. Cojocaru, Alina Carmen, editor. II. David, Chantal,
1964–editor. III.Pappalardi,F.(Francesco),1965–editor. IV.Murty,MarutiRam,editor.
QA11.A1S3252015
512.7–dc23 2015024521
ContemporaryMathematicsISSN:0271-4132(print);ISSN:1098-3627(online)
DOI:http://dx.doi.org/10.1090/conm/655
Color graphic policy. Any graphics created in color will be rendered in grayscale for the
printedversionunlesscolorprintingisauthorizedbythePublisher. Ingeneral,colorgraphicswill
appearincolorintheonlineversion.
Copying and reprinting. Individual readers of this publication, and nonprofit libraries
acting for them, are permitted to make fair use of the material, such as to copy select pages for
useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin
reviews,providedthecustomaryacknowledgmentofthesourceisgiven.
Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication
is permitted only under license from the American Mathematical Society. Permissions to reuse
portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2)
service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink.
Sendreq[e-mail].
Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases,
requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s).
Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe
firstpageofeacharticlewithinproceedingsvolumes.
(cid:2)c 2015bytheAmericanMathematicalSociety. Allrightsreserved.
TheAmericanMathematicalSocietyretainsallrights
exceptthosegrantedtotheUnitedStatesGovernment.
Copyrightofindividualarticlesmayreverttothepublicdomain28years
afterpublication. ContacttheAMSforcopyrightstatusofindividualarticles.
PrintedintheUnitedStatesofAmerica.
(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines
establishedtoensurepermanenceanddurability.
VisittheAMShomepageathttp://www.ams.org/
10987654321 201918171615
M. Ram Murty

Contents
Preface ix
On the Greatest Prime Factor of Some Divisibility Sequences
Amir Akbary and Soroosh Yazdani 1
A Number Field Extension of a Question of Milnor
T. Chatterjee, S. Gun, and P. Rath 15
Mixing Rates of Random Walks with Little Backtracking
Sebastian M. Cioaba˘ and Peng Xu 27
Additive and Multiplicative Functions with Similar Global Behavior
Jean-Marie De Koninck and Nicolas Doyon 59
Multidimensional Sequences Uniformly Distributed Modulo 1
Created from Normal Numbers
Jean-Marie De Koninck and Imre Ka´tai 77
The Index of a Modulo p
Adam Tyler Felix 83
Determining Optimal Test Functions for Bounding the Average Rank
in Families of L-Functions
Jesse Freeman and Steven J. Miller 97
Familles d’´equations de Thue associ´ees `a un sous-groupe de rang 1 d’unit´es
totalement r´eelles d’un corps de nombres
Claude Levesque and Michel Waldschmidt 117
Cyclicity of Quotients of Non-CM Elliptic Curves Modulo Primes
Giulio Meleleo 135
On the Euler Kronecker Constant of a Cyclotomic Field, II
Mariam Mourtada and V. Kumar Murty 143
The Generalized Dedekind Determinant
M. Ram Murty and Kaneenika Sinha 153
A Remark on Elliptic Curves with a Given Number of Points
over Finite Fields
James Parks 165
Recovering Cusp Forms on GL(2) from Symmetric Cubes
Dinakar Ramakrishnan 181
vii
viii CONTENTS
Arithmetic Nature of Some Infinite Series and Integrals
N. Saradha and Divyum Sharma 191
Points on Varieties over Finite Fields in Small Boxes
Igor E. Shparlinski 209
Bounds for the Lang–Trotter Conjectures
David Zywina 235
Preface
1. Conference overview
“There seem to be at least three layers of understanding: the first is acquain-
tancewithsyntax,words,andsymbols;thesecondismeaning; andthethirdcanbe
called the meaning of meaning. The means of entering the deeper layers are to
cogitate,research,reflectwhatweknowatthefirstlayer.” (M.RamMurty,“The
Art of Research”, Colloquium Lecture, Queen’s University at Kingston, Canada,
2001.)
This is how M. Ram Murty, Professor and Head of the Department of Math-
ematics and Statistics at Queen’s University in Kingston, Canada, might sound
when talking to a friend, a colleague, or a student.
A refined scholar, an intellectual with an unquenchable thirst for all facets of
truth, scientific and philosophic, M. Ram Murty has had a profound impact on
the development of number theory throughout the world. To honour his mathe-
matical legacy, a conference focused on new research directions in number theory
inspired by Murty’s most significant achievements was organized by Alina Car-
men Cojocaru (University of Illinois at Chicago, USA), Chantal David (Concordia
University,Montr´eal,Canada),HershyKisilevsky(ConcordiaUniversity,Montr´eal,
Canada)andFrancescoPappalardi(Universita`degliStudiRomaTre,Rome,Italy)
at the Centre de Recherches Math´ematiques in Montr´eal. The conference was ti-
tled SCHOLAR—a Scientific Celebration Highlighting Open Lines of Arithmetic
Research and took place during October 15–17 of 2013.
The conference hosted about 100 participants representing several generations
of research mathematicians, from beginning graduate students to emeritus profes-
sors. Thespeakersincludedsomeofthemostprominentresearchersinnumberthe-
ory fromaround the world: Henri Darmon (McGill University, Montr´eal, Canada);
E´tienne Fouvry (Universit´e Paris-Sud, Orsay, France); John Friedlander (Univer-
sity of Toronto, Canada); Dorian Goldfeld (Columbia University, New York City,
USA); Henryk Iwaniec (Rutgers University, New Brunswick, USA); Ernst Kani
(Queen’sUniversity,Kingston,Canada);Wen-ChingWinnieLi(PennsylvaniaState
University, State College, USA); Kumar Murty (University of Toronto, Canada);
Yiannis Petridis (University College London, UK); Carl Pomerance (Dartmouth
College, Hanover, USA); Dinakar Ramakrishnan (California Institute of Technol-
ogy, Pasadena, USA); Michael I. Rosen (Brown University, Providence, USA); N.
Saradha (Tata Institute of Fundamental Research, Mumbai, India); Joseph Silver-
man (Brown University, Providence, USA); Cam Stewart (University of Waterloo,
Canada);DineshS.Thakur(UniversityofRochester,USA);YitangZhang(Univer-
sity of New Hampshire, Durham, USA). Their presentations illustrated research of
highestcaliberonabroadspectrumofarithmetictopics: abelianvarieties;function
ix