Table Of ContentLecture Notes in Mathematics 2310
Herbert Lange
Rubí E. Rodríguez
Decomposition
of Jacobians
by Prym
Varieties
Lecture Notes in Mathematics
Volume 2310
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Jean-MichelMorel,CMLA,ENS,Cachan,France
BernardTeissier,IMJ-PRG,Paris,France
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Herbert Lange (cid:129) Rubí E. Rodríguez
Decomposition of Jacobians
by Prym Varieties
HerbertLange RubíE.Rodríguez
DepartmentMathematik DepartmentofMathematics
UniversityofErlangen-Nuremberg UniversidaddeLaFrontera
Erlangen,Germany Temuco,Chile
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-031-10144-1 ISBN978-3-031-10145-8 (eBook)
https://doi.org/10.1007/978-3-031-10145-8
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Preface
Abelian varieties and in particular Jacobians as well as theta functions have been
studied since the nineteenth century. Whereas Jacobian varieties are used as a
device to investigate algebraic curves, theta functions occur in several branches
of mathematics and also in other natural sciences such as physics. Their relation
becameclearonlymuchlater:thetafunctionscanbeconsideredassectionsofline
bundlesonabelianvarieties.
Here, decomposable Jacobians correspond to reducible theta functions, that is,
products of theta functions in less variables, which were intensively studied at the
end of the nineteenth century. The decomposition of Jacobians started with the
introduction of Prym varieties by Mumford in the 1960s. They contribute to the
understandingofcurvesandspecialabelianvarieties.
The present volume contains, apart from some basics, several new results and
also applications, such as new proofs of the bigonal and trigonal constructions.
It should be of interest for most algebraic geometers and several arithmetical
geometersornumbertheorists.
Erlangen,Germany HerbertLange
Temuco,Chile RubíE.Rodríguez
v
Acknowledgements
We feel very grateful to our friend Sevin Recillas for his great ideas used in the
book, which unfortunately he could not see in print. We thank Angel Carocca
andBenjaminMoragaforproofreadingpartsofthemanuscriptandHans-Joachim
Schmidforhelpinguswiththepictures.
vii
Contents
1 Introduction .................................................................. 1
2 PreliminariesandBasicResults............................................ 7
2.1 LineBundlesonAbelianVarieties ................................... 8
2.2 PolarizedAbelianVarieties ........................................... 9
2.3 EndomorphismsofAbelianVarieties ................................ 11
2.4 TheWeilFormonK(L) .............................................. 13
2.5 SymmetricIdempotents ............................................... 14
2.6 AbelianSubvarietiesofaPolarizedAbelianVariety ................ 19
2.6.1 ThePrincipallyPolarizedCase .............................. 20
2.6.2 TheCaseofanArbitraryPolarization ....................... 20
2.7 Poincaré’sReducibilityTheorem .................................... 25
2.8 ComplexandRationalRepresentationsofFiniteGroups ........... 26
2.9 TheIsotypicalandGroupAlgebraDecompositions ................ 31
2.9.1 Generalities ................................................... 31
2.9.2 InducedActionontheTangentSpace ....................... 34
2.10 ActionofaHeckeAlgebraonanAbelianVariety .................. 36
3 PrymVarieties................................................................ 39
3.1 FiniteCoversofCurves ............................................... 40
3.1.1 DefinitionsandElementaryResults ......................... 40
3.1.2 TheSignatureofaGaloisCover ............................. 41
3.1.3 TheGeometricSignatureofaGaloisCover ................. 42
3.2 PrymVarietiesofCoversofCurves .................................. 45
3.2.1 DefinitionofPrymVarieties ................................. 45
3.2.2 PolarizationsofPrymVarieties .............................. 49
3.2.3 TheDegreesoftheDecompositionIsogeny ................. 50
3.2.4 Degrees of Isogenies Arising from a
Decompositionoff :C(cid:2)→C ............................... 51
3.3 Two-DivisionPointsofPrymVarietiesofDoubleCovers .......... 55
3.4 PrymVarietiesofPairsofCovers .................................... 60
ix
x Contents
3.5 GaloisCoversofCurves .............................................. 64
3.5.1 JacobiansandPrymsofIntermediateCovers ............... 65
3.5.2 IsotypicalandGroupAlgebraDecompositionsof
IntermediateCovers .......................................... 68
3.5.3 DecompositionoftheTangentSpaceofthePrym
VarietyAssociatedtoaPairofSubgroups .................. 71
3.5.4 TheDimensionofanIsotypicalComponent ................ 74
4 CoversofDegree2and3.................................................... 79
4.1 CoversofDegree2 .................................................... 79
4.2 CoversofDegree3 .................................................... 81
4.2.1 CyclicCoversofDegree3 ................................... 81
4.2.2 Non-cyclicCoversofDegree3:TheGaloisClosure ....... 82
4.2.3 TheIrreducibleRationalRepresentationsofS ............. 85
3
4.2.4 CurveswithanS -action:DecompositionofJ(cid:2)............. 85
3
4.2.5 TheDegreeoftheIsogenyψ ................................ 89
5 CoversofDegree4........................................................... 95
5.1 CyclicCoversofDegree4 ............................................ 96
5.2 TheKleinGroupofOrder4 .......................................... 99
(cid:2)
5.2.1 DecompositionsofJ ......................................... 99
5.2.2 DegreesofSomeIsogenies ................................... 101
5.2.3 ProofofProposition5.2.5 .................................... 106
5.3 TheDihedralGroupofOrder8 ...................................... 113
5.3.1 RamificationandGenera ..................................... 113
(cid:2)
5.3.2 DecompositionsofJ ......................................... 116
5.3.3 AnIsogenyComingfromanActionofaQuotient
Group .......................................................... 120
5.3.4 AGeneralizationoftheBigonalConstruction .............. 121
5.4 TheBigonalConstruction ............................................ 128
5.4.1 DefinitionandFirstProperties ............................... 129
5.4.2 DeterminationoftheBigonalConstructioninthe
Non-GaloisCase .............................................. 132
5.4.3 TheBigonalConstructionoverC =P1 ..................... 135
5.4.4 Pantazis’Theorem ............................................ 138
5.5 TheAlternatingGroupofDegree4 .................................. 140
5.5.1 RamificationandGenera ..................................... 140
(cid:2)
5.5.2 DecompositionsofJ ......................................... 143
5.5.3 AGeneralizationoftheTrigonalConstruction .............. 146
5.6 TheTrigonalConstructionforCoverswithGroupA .............. 150
4
5.7 TheSymmetricGroupS ............................................. 152
4
5.7.1 RamificationandGenera ..................................... 152
(cid:2)
5.7.2 DecompositionofJ .......................................... 155
5.7.3 IsogeniesArisingfromActionsofSubgroupsofS ........ 159
4
5.7.4 AnIsogenyArisingfromtheActionofaQuotient
ofS ........................................................... 160
4
Contents xi
5.8 AnotherGeneralizationoftheTrigonalConstruction ............... 160
5.8.1 StatementandPreparations .................................. 160
5.8.2 TheTrigonalConstruction ................................... 167
5.8.3 The Degree of γ in the General Principally
PolarizedCase ................................................ 168
5.8.4 TheNon-PrincipallyPolarizedCase ......................... 170
6 SomeSeriesofGroupActions.............................................. 173
6.1 CyclicCovers ......................................................... 174
6.1.1 NotationandFirstResults .................................... 174
6.1.2 CyclicCoversofDegreep ................................... 176
6.1.3 CyclicCoversofDegreep2 .................................. 176
6.1.4 CyclicCoversofDegreepq ................................. 178
6.2 CoverswithDihedralGroupAction ................................. 180
6.2.1 TheIrreducibleRationalRepresentationsofD ............ 181
n
6.2.2 CurveswithD -Action:RamificationsandGenera ........ 182
p
(cid:2)
6.2.3 CurveswithD -Action:DecompositionsofJ ............. 185
p
6.2.4 TheDegreeoftheIsogenyϕ ................................. 189
6.2.5 CurveswithD2α-Action,α ≥2 .............................. 195
6.2.6 Curves with a D -Action: Ramification and
2p
Genera ......................................................... 201
(cid:2)
6.2.7 CurveswithD -Action:DecompositionofJ ............. 203
2p
6.2.8 SomeIsogeniesBetweenPrymSubvarieties ................ 205
6.3 SemidirectProductsofaGroupofOrder3byPowersofK ...... 211
4
6.3.1 TheGroupG ................................................ 211
n
6.3.2 The Diagram of Subcovers of a Curve with
G -Action ..................................................... 215
n
6.3.3 AJacobianIsogenoustotheProductaBigNumber
ofJacobians ................................................... 219
7 SomeSpecialGroupsandCompleteDecomposability................... 221
7.1 TheAlternatingGroupA ............................................ 222
5
7.1.1 TheIrreducibleRationalRepresentationsofA ............ 222
5
7.1.2 RamificationandGenera ..................................... 223
(cid:2)
7.1.3 DecompositionsofJ ......................................... 226
7.1.4 Some Isogenies Between Prym Varieties of
Subcovers ..................................................... 227
7.2 GroupswithSchurIndexLargerThanOne ......................... 228
7.2.1 TheGroupofHamiltonianQuaternions ..................... 229
7.2.2 AGroupofOrder24 ......................................... 232
7.3 CompletelyDecomposableJacobians ................................ 240
7.3.1 TheTheoremofEkedahl-Serre .............................. 240
7.3.2 Examples ...................................................... 242
7.3.3 IntermediateCovers .......................................... 246
Bibliography...................................................................... 247
Index............................................................................... 249
