Cubic Forms and the Circle Method

175 Pages·2021·2.032 MB·English

by  Tim Browning

Progress in Mathematics
343
Tim Browning
Cubic Forms
and the 
Circle Method

Progress in Mathematics
Volume343
SeriesEditors
AntoineChambert-Loir,Paris ,UniversitéParis-Diderot,Paris,France
Jiang-HuaLu,TheUniversityofHongKong,HongKongSAR,China
MichaelRuzhansky,GhentUniversity,Belgium
QueenMaryUniversityofLondon,London,UK
YuriTschinkel,CourantInstituteofMathematicalSciences,NewYork,USA
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Tim Browning
Cubic Forms and the Circle
Method
TimBrowning
InstituteofScienceandTechnologyAustria
Klosterneuburg,Niederösterreich,Austria
ISSN0743-1643 ISSN2296-505X (electronic)
ProgressinMathematics
ISBN978-3-030-86871-0 ISBN978-3-030-86872-7 (eBook)
https://doi.org/10.1007/978-3-030-86872-7
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Ferran Sunyer i Balaguer (1912–1967) was a
self-taught Catalan mathematician who, in spite
of a serious physical disability, was very active in
research in classical mathematical analysis, an area
in which he acquired international recognition. His
heirscreatedtheFundacióFerranSunyeriBalaguer
inside the Institut d’Estudis Catalans to honour the
memoryofFerranSunyeriBalaguerandtopromote
mathematicalresearch.
Each year, the Fundació Ferran Sunyer i Balaguer
and the Institut d’Estudis Catalans award an
international research prize for a mathematical
monographof expositorynature.The prize-winning
monographs are published in this series. Details
about the prize and the Fundació Ferran Sunyer i
Balaguercanbefoundat
https://ffsb.espais.iec.cat/en/
the-ferran-sunyer-i-balaguer-prize/.
This bookhas been awarded the
Ferran Sunyer i Balaguer2021Prize.
Themembersofthescientificcommittee
ofthe2021prizewereasfollows:
AntoineChambert-Loir
UniversitéParis-Diderot(Paris7)
RuthKellerhals
UniversitédeFribourg
M.TeresaMartínez-Seara
UniversitatPolitècnicadeCatalunya
MichaelRuzhansky
GhentUniversity
QueenMaryUniversityofLondon
KristianSeip
NorwegianUniversityofScienceandTechnology
(NTNU)
FerranSunyeriBalaguerPrizewinners
since2011
2011 JayceGetzandMarkGoresky
Hilbert Modular Forms with Coefficients in
Intersection Homology and Quadratic Base
Change,PM298
2012 AngelCano, Juan Pablo Navarrete and José
Seade
ComplexKleinianGroups,PM303
2013 XavierTolsa
Analytic Capacity, the Cauchy Transform,
and Non-homogeneous Calderón–Zygmund
Theory,PM307
2014 VeroniqueFischerandMichaelRuzhansky
QuantizationonNilpotentLieGroups,Open
Access,PM314
2015 Thescientificcommitteedecidednotto
awardtheprize
2016 VladimirTuraevandAlexisVirelizier
MonoidalCategories and TopologicalField
Theory,PM322
2017 Antoine Chambert-Loir, Johannses Nicaise
andJulienSebag
MotivicIntegration,PM325
2018 MichaelRuzhanskyandDurvudkhan
Suragan
HardyInequalitiesonHomogeneous
Groups,PM327
2019 Thescientificcommitteedecidednotto
awardtheprize
2020 UrtziBuijs,YvesFélix,AnicetoMurilloand
DanielTanré
LieModelsinTopology,PM335
GiovanniCatinoandPaoloMastrolia
A Perspective on Canonical Riemannian
Metrics,PM336
To myparents
Preface
The Hardy–Littlewoodcircle method was invented a century ago to study integer
solutionstoveryspecialDiophantineequations.Ithassinceproventobeoneofthe
mostsignificantall-purposetoolsavailable,capableofhandlingremarkablygeneral
systemsofpolynomialequationsdefinedoverarbitraryglobalfields.This,inturn,
has led to the discovery that the circle method can shed light on the geometry of
moduli spaces that parameterise rational curves on algebraic varieties cut out by
polynomialequations.
In thisbook,we shalluse cubicpolynomialsasa prism throughwhichto view
themanyfacetsoftheHardy–Littlewoodcirclemethod,bothclassicalandmodern.
The aim is to take the reader from traditional topics, such as Waring’s problem
for cubes, to a discussion of the circle method over function fields of positive
characteristic, and the application of these ideas to derive basic properties of the
variety that parameterises lines contained in cubic hypersurfaces defined over C.
Alongtheway,weshallencounterthemechanicsofthecirclemethodforarbitrary
non-singularcubicformsoverQ,togetherwithanaccountofhowthecirclemethod
canbeextendedtodealwithanalogousquestionsoveralgebraicnumberfields.
This book is aimed at beginning graduate students. However, some familiarity
with the basic properties of finite fields, local fields and algebraic varieties will
be assumed, togetherwith a modestbackgroundin analytic and algebraicnumber
theory. There are already two important texts that deal with applications of the
Hardy–Littlewoodcirclemethodinnumbertheory,byDavenport[43]andVaughan
[112]. Although there is necessarily an overlap around the foundations of the
subject, we have deliberately charted a course that rapidly takes us outside the
realmsofeitherbook.
Contents Weproceedbysummarisingthewaypointsofthebook.Chapter1briefly
discusses the Hasse principle for cubic forms, in order to motivate the question
of solubility over finite fields and local fields. Basic facts about the existence and
numberofsolutionsofpolynomialequationsoverfinitefieldswillbecollectedhere.
Building on this, an analysis of the corresponding questions over local fields is
carriedoutviaHensel’slemma.Thisincludesaproofofthesolubilityofarbitrary
ix
x Preface
cubicformsoverQ ,foranyprimep (cid:2)=3,providedthatthenumberofvariablesis
p
atleast10.
Chapters2and3concerntheapplicationoftheHardy–Littlewoodcirclemethod
tocubicequationsoverQ.TheybothowemuchtothepresentationinDavenport’s
treatise[43].Chapter2dealswiththeclassicaltopicofWaring’sproblemforcubes,
whichconcernsthesolubilityinnon-negativeintegersoftheequation
x3+···+x3 =N,
1 n
foragivenpositiveintegerN.ThemainingredientsareWeyl’sinequalityandHua’s
inequality,whichconcernthemodulusoftheexponentialsum
(cid:2) (cid:3) (cid:4)
exp 2πiαx3 ,
x(cid:2)P
forgivenα ∈ R, asP → ∞. Chapter3isconcernedwith Davenport’singenious
extensionoftheseideastothesolubilityinintegersofarbitrarynon-singularcubic
formsf ∈Z[x ,...,x ],providedthattherearen(cid:2)17variables.
1 n
Chapter 4 serves to illustrate how the circle method can be extendedto handle
cubic Diophantine equations that are defined over algebraic number fields and
closelyfollowstheworkofSchindlerandSkorobogatov[99].LetK/Qbeanumber
field.AmongquadraticDiophantineequations,theequation
a x2+a x2+a x2+a x2 =0
1 1 2 2 3 3 4 4
holdsaspecialplace,forgivennon-zeroa ,...,a ∈K,sincetheequationcanbe
1 4
writtenintermsoffieldnormsfromquadraticextensionsofK.Fortheseequations,
theHasse principleisknowntohold,butlocalsolubilityis notautomaticatevery
place. Chapter 4 extends this to a Diophantine equation in 9 variables, which is
definedusingfieldnormsfromcubicextensionsofK.
Chapter5givesadetailedexplanationofhowthecirclemethodisimplemented
overthefunctionfieldF (t).Bydevelopingfunctionfieldanaloguesofthestepsin
q
Chap.2,thisisusedtostudythedensityofzerosoftheFermathypersurface
a x3+···+a x3 =0,
1 1 n n
over F (t), for given non-zero a ,...,a ∈ F , provided that n (cid:2) 9 and the
q 1 n q
characteristicofF exceeds3.Thisfollowsalineofreasoningoriginallydeveloped
q
by Kubota [81] in his 1971 thesis on the function field analogue of Waring’s
problem.
Chapter6 isbasedontheauthor’sjointworkswith Sawin[21] andVishe [24],
which shows how the circle method over F (t) can be used to make deductions
q
aboutbasicgeometricpropertiesofmodulispacesofrationalcurvesonnon-singular