Arithmetic and Geometry

539 Pages·2015·2.87 MB·English

by  Luis Dieulefait

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,
UniversityofWarwick,CoventryCV47AL,UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
http://www.cambridge.org/mathematics
299 Kleiniangroupsandhyperbolic3-manifolds, Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMöbiusdifferentialgeometry, U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem, F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchrödingersystems, M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 Numbertheoryandalgebraicgeometry, M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordI, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordII, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Geometricmechanicsandsymmetry, J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003, C.D.WENSLEY(ed.)
308 Topology,geometryandquantumfieldtheory, U.L.TILLMANN(ed)
309 Coringsandcomodules, T.BRZEZINSKI&R.WISBAUER
310 Topicsindynamicsandergodictheory, S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects, T.W.MÜLLER(ed)
312 Foundationsofcomputationalmathematics,Minneapolis2002, F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles, S.MÜLLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs, D.CVETKOVIC,P.ROWLINSON&S.SIMIC
315 Structuredringspectra, A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience, T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations, D.HENRY
319 DoubleaffineHeckealgebras, I.CHEREDNIK
320 L-functionsandGaloisrepresentations, D.BURNS,K.BUZZARD&J.NEKOVÁR(eds)
321 Surveysinmodernmathematics, V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory, F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations, S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra, C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow, P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype, J.E.HUMPHREYS
327 Surveysincombinatorics2005, B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds, R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology, A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005, L.MPARDO,A.PINKUS,E.SÜLI&M.J.TODD(eds)
332 Handbookoftiltingtheory, L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition), A.KOCK
334 TheNavier–Stokesequations, N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability, A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules, I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory ,J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory, N.YOUNG(ed)
339 GroupsStAndrews2005I, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C.SNAITH(eds)
342 Ellipticcohomology, H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI, J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII, J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry, A.NEEMAN
346 Surveysincombinatorics2007, A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics, N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis, P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI, Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII, Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas, A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials, J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis, J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis, K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence, J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations, D.DELBOURGO
357 Algebraictheoryofdifferentialequations, M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory, M.R.BRIDSON,P.H.KROPHOLLER&I.J.LEARY(eds)
359 Modulispacesandvectorbundles, L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA&S.RAMANAN(eds)
360 Zariskigeometries, B.ZILBER
361 Words:Notesonverbalwidthingroups, D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories, R.BAUTISTA,L.SALMERÓN&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008, F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics, J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009, S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems, B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena, G.BLOWER
368 GeometryofRiemannsurfaces, F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks, M.DRAIEF&L.MASSOULIÉ
370 Theoryofp-adicdistributions, S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals, F.PRZYTYCKI&M.URBANSKI
372 Moonshine:Thefirstquartercenturyandbeyond, J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection, J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction, S.ALINHAC
375 Triangulatedcategories,T.HOLM, P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns, S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications, G.BERHUY
378 Probabilityandmathematicalgenetics, N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory, J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities, M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations, D.LEVI,P.OLVER,Z.THOMOVA&P.WINTERNITZ(eds)
382 Forcingwithrandomvariablesandproofcomplexity, J.KRAJÍCEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI, R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII, R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems, B.MARCUS,K.PETERSEN&
T.WEISSMAN(eds)
386 Independence-friendlylogic, A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI, C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII, C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere, D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials, D.E.PELINOVSKY
391 Fusionsystemsinalgebraandtopology, M.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011, R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory, J.COATESetal(eds)
394 Variationalproblemsindifferentialgeometry, R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow, A.MANN
396 Arithmeticdifferentialoperatorsoverthep-adicintegers, C.C.RALPH&S.R.SIMANCA
397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology, J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics, M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs, C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs, T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory, B.DENG,J.DU&Q.FU
402 Mathematicalaspectsoffluidmechanics, J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds)
403 Foundationsofcomputationalmathematics,Budapest2011, F.CUCKER,T.KRICK,A.PINKUS&A.SZANTO(eds)
404 Operatormethodsforboundaryvalueproblems, S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds)
405 Torsors,étalehomotopyandapplicationstorationalpoints, A.N.SKOROBOGATOV(ed)
406 Appalachiansettheory, J.CUMMINGS&E.SCHIMMERLING(eds)
407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups, J.N.BRAY,D.F.HOLT&C.M.
RONEY-DOUGAL
408 Complexityscience:theWarwickmaster’scourse, R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds)
409 Surveysincombinatorics2013, S.R.BLACKBURN,S.GERKE&M.WILDON(eds)
410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups, T.CECCHERINI-SILBERSTEIN,
F.SCARABOTTI&F.TOLLI
411 Modulispaces, L.BRAMBILA-PAZ,O.GARCÍA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds)
412 Automorphismsandequivalencerelationsintopologicaldynamics, D.B.ELLIS&R.ELLIS
413 Optimaltransportation, Y.OLLIVIER,H.PAJOT&C.VILLANI(eds)
414 AutomorphicformsandGaloisrepresentationsI, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
415 AutomorphicformsandGaloisrepresentationsII, F.DIAMOND,P.L.KASSAEI&M.KIM(eds)
416 Reversibilityindynamicsandgrouptheory, A.G.O’FARRELL&I.SHORT
417 Recentadvancesinalgebraicgeometry, C.D.HACON,M.MUSTATA&M.POPA(eds)
418 TheBloch–KatoconjecturefortheRiemannzetafunction, J.COATES,A.RAGHURAM,A.SAIKIA&
R.SUJATHA(eds)
419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations, J.C.MEYER&
D.J.NEEDHAM
420 Arithmeticandgeometry, L.DIEULEFAITetal(eds)
421 O-minimalityandDiophantinegeometry, G.O.JONES&A.J.WILKIE(eds)
422 GroupsStAndrews2013, C.M.CAMPBELLetal(eds)
423 Inequalitiesforgrapheigenvalues, Z.STANIC´
424 Surveysincombinatorics2015, A.CZUMAJetal(eds)
LondonMathematicalSocietyLectureNoteSeries:420
Arithmetic and Geometry
Editedby
LUIS DIEULEFAIT
UniversitatdeBarcelona
GERD FALTINGS
Max-Planck-InstitutfürMathematik(Bonn)
D. R. HEATH-BROWN
UniversityofOxford
YU. V. MANIN
Max-Planck-InstitutfürMathematik(Bonn)
B. Z. MOROZ
Max-Planck-InstitutfürMathematik(Bonn)
JEAN-PIERRE WINTENBERGER
UniversitédeStrasbourg
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107462540
(cid:2)c CambridgeUniversityPress2015
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2015
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Arithmeticandgeometry/editedbyLuisDieulefait,UniversitatdeBarcelona
[andfiveothers].
pagescm.–(LondonMathematicalSocietylecturenoteseries;420)
Paperspresentedatthetrimesteron“ArithmeticandGeometry”attheHausdorff
ResearchInstituteforMathematics(UniversityofBonn),January–April2013.
ISBN978-1-107-46254-0
1. Numbertheory–Congresses. 2. Algebraicnumbertheory–Congresses.
3. Geometryofnumbers–Congresses. I. Dieulefait,Luis,editor.
QA241.A695 2015
510–dc23
2015001829
ISBN978-1-107-46254-0Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
Preface pagevii
Introduction ix
Listofparticipants x
TrimesterSeminar xiii
WorkshoponSerre’sconjecture xxiii
Theresearchconference xxviii
1 Galoisgroupsoflocalfields,Liealgebrasandramification 1
VictorAbrashkin
2 AcharacterisationofordinarymodulareigenformswithCM 24
RajenderAdibhatlaandPanagiotisTsaknias
3 Selmercomplexesand p-adicHodgetheory 36
DenisBenois
4 Asurveyofapplicationsofthecirclemethodtorationalpoints 89
T.D.Browning
5 ArithmeticdifferentialequationsofPainlevéVItype 114
AlexandruBuiumandYuriI.Manin
6 Differentialcalculuswithintegers 139
AlexandruBuium
7 UncalculdegroupedeBraueretuneapplicationarithmétique 188
Jean-LouisColliot-Thélène
8 Connectedness of Hecke algebras and the Rayuela
conjecture:apathtofunctorialityandmodularity 193
LuisDieulefaitandArielPacetti
v
vi Contents
9 BigimageofGaloisrepresentationsandcongruenceideals 217
HaruzoHidaandJacquesTilouine
10 Theskew-symmetricpairingontheLubin–Tateformalmodule 255
M.A.IvanovandS.V.Vostokov
11 Equationsinmatrixgroupsandalgebrasovernumberfields
and rings: prolegomena to a lowbrow noncommutative
Diophantinegeometry 264
BorisKunyavski˘ı
12 Onthe(cid:2)-adicregulatorasaningredientofIwasawatheory 283
L.V.Kuz’min
13 OnacountingproblemforG-shtukas 318
NgoDacTuan
14 ModularformsandCalabi-Yauvarieties 351
KapilParanjapeandDinakarRamakrishnan
15 Derivative of symmetric square p-adic L-functions via
pull-backformula 373
GiovanniRosso
16 Uniformboundsforrationalpointsoncubichypersurfaces 401
PerSalberger
17 Descentontoricfibrations 422
AlexeiN.Skorobogatov
18 OnfiltrationsofvectorbundlesoverP1 436
Z
A.Smirnov
19 OnthedihedralEulercharacteristicsofSelmergroupsof
Abelianvarieties 458
JeanineVanOrder
20 CMvaluesofhigherGreen’sfunctionsandregularized
Peterssonproducts 493
MarynaViazovska
Preface
The trimester on “Arithmetic and Geometry” at the Hausdorff Research
Institute for Mathematics (University of Bonn) took place in January – April
2013. In the next few pages the reader will find a list of the participants of
thetrimester,thedescriptionsofthesessiononSerre’sconjecture,conducted
by L.V. Dieulefait and J.-P. Wintenberger, and of the session on counting
rational points on algebraic varieties, conducted by D.R. Heath-Brown, the
programmes of the workshop on Serre’s conjecture and of the final research
conference, and a list of the talks given at the HIM during the trimester. The
participantswereinvitedtosubmittheirpapersforpublicationinthisvolume.
Thepapersappearinginthevolumehavebeencarefullyrefereed.
Acknowledgements. We wish to record our gratitude to the Hausdorff
ResearchInstitute,andinparticulartoitsdirector,ProfessorDrW.Lück,for
thehospitalityandfinancialsupportwereceived.
L.V.Dieulefait,G.Faltings,D.R.Heath-Brown,Yu.I.Manin,
B.Z.Moroz,andJ.-P.Wintenberger(editors).
vii