Many Variations of Mahler Measures: A Lasting Symphony (Australian Mathematical Society Lecture Series)

184 Pages·2020·1.861 MB·English

by  François Brunault, Wadim Zudilin

Many Variations of Mahler Measures
The Mahler measure is a fascinating notion and an exciting topic in contemporary
mathematics, interconnecting with subjects as diverse as number theory, analysis,
arithmeticgeometry,specialfunctionsandrandomwalks.Thisfriendlyandconcise
introductiontotheMahlermeasureisavaluableresourceforbothgraduatecourses
andself-study.Itprovidesthereaderwiththenecessarybackgroundmaterial,before
presentingrecentachievementsandremainingchallengesinthefield.
The first part introduces the univariate Mahler measure and addresses Lehmer’s
problem,andthendiscussestechniquesofreducingmultivariatemeasurestohyper-
geometric functions. The second part touches on the novelties of the subject,
especially the relation with elliptic curves, modular forms and special values of
L-functions. Finally, the appendix presents the modern definition of motivic coho-
mology and regulator maps, as well as Deligne–Beilinson cohomology. The text
includes many exercises to test comprehension and to challenge readers of all
abilities.

AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES
Editor-in-chief:ProfessorJ.Ramagge,SchoolofMathematicsandStatistics,
UniversityofSydney,NSW2006,Australia
Editors:
ProfessorG.Froyland,SchoolofMathematicsandStatistics,UniversityofNewSouthWales,NSW2052,Australia
ProfessorM.Murray,SchoolofMathematicalSciences,UniversityofAdelaide,SA5005,Australia
ProfessorC.Praeger,SchoolofMathematicsandStatistics,UniversityofWesternAustralia,Crawley,WA6009,
Australia
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress.
Foracompleteserieslistingvisitwww.cambridge.org/mathematics.
8 LowRankRepresentationsandGraphsforSporadicGroups,
C.E.PRAEGER&L.H.SOICHER
9 AlgebraicGroupsandLieGroups,G.I.LEHRER(ed.)
10 ModellingwithDifferentialandDifferenceEquations,
G.FULFORD,P.FORRESTER&A.JONES
11 GeometricAnalysisandLieTheoryinMathematicsandPhysics,
A.L.CAREY&M.K.MURRAY(eds.)
12 FoundationsofConvexGeometry,W.A.COPPEL
13 IntroductiontotheAnalysisofNormedLinearSpaces,J.R.GILES
14 Integral:AnEasyApproachafterKurzweilandHenstock,
L.P.YEE&R.VYBORNY
15 GeometricApproachestoDifferentialEquations,
P.J.VASSILIOU&I.G.LISLE(eds.)
16 IndustrialMathematics,G.R.FULFORD&P.BROADBRIDGE
17 ACourseinModernAnalysisanditsApplications,G.COHEN
18 Chaos:AMathematicalIntroduction,J.BANKS,V.DRAGAN&A.JONES
19 QuantumGroups,R.STREET
20 UnitaryReflectionGroups,G.I.LEHRER&D.E.TAYLOR
21 LecturesonRealAnalysis,F.LÁRUSSON
22 RepresentationsofLieAlgebras,A.HENDERSON
23 NeverendingFractions,J.BORWEINetal.
24 Wavelets:AStudentGuide,P.NICKOLAS
25 ClassicalGroups,DerangementsandPrimes,T.BURNESS&M.GIUDICI
26 NotesonCounting:AnIntroductiontoEnumerativeCombinatorics,P.J.CAMERON
27 OrthogonalPolynomialsandPainlevéEquations,W.VANASSCHE

AustralianMathematicalSocietyLectureSeries:28
Many Variations of Mahler Measures
A Lasting Symphony
FRANÇOIS BRUNAULT
ÉcoleNormaleSupérieuredeLyon
WADIM ZUDILIN
RadboudUniversiteitNijmegen
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Informationonthistitle:www.cambridge.org/9781108794459
DOI:10.1017/9781108885553
(cid:2)c FrançoisBrunaultandWadimZudilin2020
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Firstpublished2020
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Contents
Preface pagexiii
1 Somebasics 1
1.1 Kronecker’stheorem 1
1.2 Factorisationofcyclotomicexpressions 3
1.3 Jensen’sformula 6
1.4 FamiliesofMahlermeasures 9
Chapternotes 11
Additionalexercises 12
2 Lehmer’sproblem 14
2.1 Smyth’stheorem 14
2.2 Dobrowolski’sbound 19
2.3 Adiscriminatedproblemaboutdiscriminants 22
Chapternotes 27
Additionalexercises 28
3 Multivariatesetting 30
3.1 Mahler’sinvention 30
3.2 Newtonpolytopes 33
3.3 SpecialMahlermeasures 36
3.4 LimitsofmultivariateMahlermeasures 37
3.5 Reciprocalvsnon-reciprocalpolynomials 40
Chapternotes 44
Additionalexercises 45
4 Thedilogarithm 49
4.1 Theq-binomialtheoremandpentagonalidentity 49
4.2 TheRogersandBloch–Wignerdilogarithms 53
4.3 Maillot’sformula 56
Chapternotes 58
Additionalexercises 58
ix