Table Of ContentLecture Notes in Mathematics 2222
History of Mathematics Subseries
Peter Roquette
The Riemann
Hypothesis in
Characteristic p in
Historical Perspective
Lecture Notes in Mathematics 2222
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryBoard:
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Athens
GáborLugosi,Barcelona
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Peter Roquette
The Riemann Hypothesis
in Characteristic p
in Historical Perspective
123
PeterRoquette
MathematicalInstitute
HeidelbergUniversity
Heidelberg,Germany
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISSN2193-1771
HistoryofMathematicsSubseries
ISBN978-3-319-99066-8 ISBN978-3-319-99067-5 (eBook)
https://doi.org/10.1007/978-3-319-99067-5
LibraryofCongressControlNumber:2018955713
MathematicsSubjectClassification(2010):01A60,11R58,14H05
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Preface
Thisbookistheresultofmanyyears’work.IamtellingthestoryoftheRiemann
hypothesis for function fields, or curves, of characteristic p starting with Artin’s
thesis in the year 1921, covering Hasse’s work in the 1930s on elliptic fields
and more, until Weil’s final proof in 1948. The main sources are letters which
wereexchangedamongtheprotagonistsduringthattimewhichIfoundinvarious
archives,mostly in the UniversityLibrary in Göttingen butalso at other places in
the world.I am tryingto show howthe ideas formed,and howthe propernotions
andproofswerefound.Thisisagoodillustration,fortunatelywelldocumented,of
howmathematicsdevelopsingeneral.
Some of the chapters have already been prepublished in the “Mitteilungen
der Mathematischen Gesellschaft in Hamburg.” But before including them into
this book, they have been thoroughly reworked, extended, and polished. I have
writtenthisbookformathematicians,butessentiallyitdoesnotrequireanyspecial
knowledge of particular mathematical fields. I have tried to explain whatever is
needed in the particular situation, even if this may seem to be superfluous to the
specialist.
Every chapter is written such that it can be read independently of the other
chapters.Sometimesthisentailsarepetitionofinformationwhichhasalreadybeen
giveninanotherchapter.Thechaptersareaccompaniedby“summaries.”Perhaps,
itmaybeexpedientfirsttolookatthesesummariesinordertogetanoverviewof
whatisgoingon.
The preparation of this book had been supported in part by the Deutsche
Forschungsgemeinschaft and the Marga und Kurt Möllgaard Stiftung. A number
of colleagues and friends have shown interest and helped me with their critical
comments; it is impossible for me to mention all of them here. Nevertheless, I
wouldliketoexpressmyspecialthankstoSigridBöge,KarinReich,Franz-Viktor
Kuhlmann,andFranzLemmermeyer.
References Mostof the lettersanddocumentscited in thisbookare containedin
theHandschriftenabteilungoftheUniversityLibraryofGöttingen(exceptifanother
sourceis mentioned),mostbutnotall of themin the Hasse files. The lettersfrom
v
vi Preface
Hasse to Davenport are contained in the archive of Trinity College, Cambridge.
The letters from Hasse to Mordellare containedin the archiveof King’s College,
Cambridge. The letters from Hasse to Fraenkel are in the archive of the Hebrew
University,Jerusalem.
IhavetranslatedthecitedtextintoEnglishforeasierreading.Transcriptionsof
thefullletters(andmore)intheiroriginallanguageareavailableatmyhomepage:
https://www.mathi.uni-heidelberg.de/~roquette/
Heidelberg,Germany PeterRoquette
Contents
1 Overture..................................................................... 1
1.1 WhyHistoryofMathematics?....................................... 1
1.2 ArtinandtheInterventionbyHilbert ............................... 2
1.3 Hasse’sProject ....................................................... 4
1.4 Weil’sContribution................................................... 6
Summary..................................................................... 7
2 SettingtheStage............................................................ 9
2.1 OurTerminology..................................................... 9
2.2 TheTheorem:RHp................................................... 11
3 TheBeginning:Artin’sThesis ............................................ 13
3.1 QuadraticFunctionFields ........................................... 14
3.1.1 TheArithmeticPart......................................... 17
3.1.2 TheAnalyticPart........................................... 20
3.2 Artin’sLetterstoHerglotz........................................... 25
3.2.1 ExtensionoftheBaseField................................ 26
3.2.2 ComplexMultiplication.................................... 27
3.2.3 BirationalTransformation.................................. 30
3.3 HilbertandtheConsequences....................................... 31
Summary..................................................................... 31
3.4 SideRemark:Gauss’LastEntry..................................... 32
Summary..................................................................... 37
4 BuildingtheFoundations.................................................. 39
4.1 F.K.Schmidt.......................................................... 39
4.2 ZetaFunctionandRiemann-RochTheorem........................ 43
4.3 F.K.Schmidt’sL-polynomial........................................ 47
4.3.1 SomeComments............................................ 48
4.4 TheFunctionalEquation............................................. 49
4.5 Consequences......................................................... 51
Summary..................................................................... 53
vii
viii Contents
5 EnterHasse.................................................................. 55
6 DiophantineCongruences ................................................. 59
6.1 Davenport............................................................. 59
6.2 TheChallenge........................................................ 61
6.3 TheDavenport-HassePaper ......................................... 67
6.3.1 Davenport’sLetterandGeneralizedFermatFields....... 67
6.3.2 GaussSums................................................. 70
6.3.3 Davenport-HasseFields.................................... 73
6.3.4 Stickelberger’sTheorem ................................... 75
6.4 ExponentialSums.................................................... 76
Summary..................................................................... 78
7 EllipticFunctionFields .................................................... 81
7.1 TheBreakthrough.................................................... 81
7.2 TheProblem.......................................................... 83
7.3 Hasse’sFirstProof:ComplexMultiplication....................... 85
7.4 TheSecondProof .................................................... 90
7.4.1 MeromorphismsandtheJacobian ......................... 92
7.4.2 TheDoubleField........................................... 96
7.4.3 NormAdditionFormula.................................... 97
7.4.4 TheFrobeniusOperator.................................... 102
7.5 SomeComments ..................................................... 104
7.5.1 Rosati’sAnti-automorphism............................... 104
Summary..................................................................... 105
8 MoreonEllipticFields..................................................... 107
8.1 TheHasseInvariantA................................................ 107
8.2 UnramifiedCyclicExtensionsofDegreep......................... 109
8.2.1 TheHasse-WittMatrix..................................... 111
8.3 GroupStructureoftheJacobian..................................... 112
8.3.1 HigherDerivations ......................................... 115
8.4 TheStructureoftheEndomorphismRing .......................... 117
8.4.1 TheSupersingularCase .................................... 118
8.4.2 SingularInvariants.......................................... 121
8.4.3 EllipticSubfields ........................................... 122
8.4.4 GoodReduction ............................................ 124
8.5 ClassFieldTheoryandComplexMultiplication ................... 126
Summary..................................................................... 128
9 TowardsHigherGenus..................................................... 131
9.1 Preliminaries.......................................................... 131
9.2 TheYears1934–1935................................................ 134
9.2.1 Deuring...................................................... 136
9.2.2 MorePoliticalProblems.................................... 139
9.3 Deuring’sLettertoHasse............................................ 141
9.3.1 Correspondences............................................ 143
Contents ix
9.4 Hasse’sLettertoWeil................................................ 146
9.5 Weil’sReplyandLefschetz’sNote.................................. 148
9.6 TheWorkshoponAlgebraicGeometry............................. 152
Summary..................................................................... 156
10 AVirtualProof.............................................................. 157
10.1 TheQuadraticForm.................................................. 157
10.1.1 TheDoubleField........................................... 157
10.1.2 TheDifferent................................................ 159
10.2 Positivity.............................................................. 166
10.2.1 ApplyingtheRiemann-HurwitzFormula ................. 166
10.2.2 TheDiscriminantEstimate................................. 168
10.3 TheRHp.............................................................. 173
10.4 SomeComments ..................................................... 175
10.4.1 FrobeniusMeromorphism.................................. 175
10.4.2 Differents ................................................... 176
10.4.3 IntegerDifferentials ........................................ 177
10.5 TheGeometricLanguage............................................ 177
Summary..................................................................... 179
11 Intermission................................................................. 181
11.1 ArtinLeaves.......................................................... 181
11.2 TheItalianConnection............................................... 183
11.2.1 Severi........................................................ 183
11.2.2 TheVoltaCongress......................................... 186
11.3 TheFrenchConnection.............................................. 190
11.3.1 OnFunctionFields ......................................... 190
11.3.2 TheBook.................................................... 195
11.3.3 ParisandStrassbourg....................................... 198
Summary..................................................................... 202
12 A.Weil ....................................................................... 203
12.1 BonneNouvelle ...................................................... 204
12.2 TheFirstNote1940.................................................. 207
12.3 TheSecondNote1941............................................... 209
Summary..................................................................... 212
13 Appendix .................................................................... 215
13.1 Bombieri.............................................................. 216
References......................................................................... 221
Index............................................................................... 231
