LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES ManagingEditor:ProfessorM.Reid,MathematicsInstitute, UniversityofWarwick,CoventryCV47AL,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPress athttp://www.cambridge.org/mathematics 287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MARTÍNEZ(eds) 288 Surveysincombinatorics,2001,J.W.P.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE 290 QuantumgroupsandLietheory,A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed) 292 Aquantumgroupsprimer,S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK 294 Introductiontooperatorspacetheory,G.PISIER 295 Geometryandintegrability,L.MASON&Y.NUTKU(eds) 296 Lecturesoninvarianttheory,I.DOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES 298 Higheroperads,highercategories,T.LEINSTER(ed) 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.C.WALL(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 FusionsystemsinalgebraandtopologyM.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) LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES: 396 Arithmetic Differential Operators over the p-adic Integers CLAIRE C. RALPH CornellUniversity,USA SANTIAGO R. SIMANCA UniversitédeNantes,France cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,SãoPaulo,Delhi,Tokyo,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9781107674141 ©C.RalphandS.Simanca,2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. 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Contents 1 Introduction page 1 2 The p-adic numbers Qp 8 2.1 A pragmatic realization of Qp 11 2.2 The p-adic integers Zp and their field of fractions 13 2.3 The topology of Qp 15 2.4 Analytic and algebraic properties of Qp 17 2.5 (p−1)-roots of unity in Qp 20 3 Some classical analysis on Qp 23 3.1 The Artin–Hasse exponential function 30 3.2 The completion of the algebraic closure of Qp 33 3.3 Zeta functions 38 4 Analytic functions on Zp 48 4.1 Strassmann’s theorem 53 5 Arithmetic differential operators on Zp 55 5.1 Multiple primes I 61 6 A general view of arithmetic differential operators 65 6.1 Basic algebraic concepts 67 6.2 General δp-functions: arithmetic jet spaces 73 6.3 The analogue of a δp-linear operators for group schemes 82 6.4 Multiple primes II 86 7 Analyticity of arithmetic differential operators 103 8 Characteristic functions of discs in Zp: p-adic coordinates 107 8.1 Characteristic functions of discs of radii 1/p 107 8.2 Characteristic functions of discs of radii 1/pn 116 v vi Contents 9 Characteristic functions of discs in Zp: harmonic co- ordinates 119 9.1 A matrix associated to pm 120 9.2 Analytic functions and arithmetic differential operators 122 10 Some differences between δ-operators over Zp and Z(cid:2)ur 127 p References 135 Index 138 1 Introduction Our purpose in this monograph is to provide a concise and complete introduction to the study of arithmetic differential operators over the p-adic integers Zp. These are the analogues of the usual differential op- erators over say, the ring C[x], but where the role of the variable x is replaced by a prime p, and the roles of a function f(x) and its deriva- tive df/dx are now played by an integer a ∈ Z and its Fermat quotient δpa=(a−ap)/p. Inmakingourpresentationofthesetypeofoperators,wefindnobet- ter way than discussing the p-adic numbers in detail also, and some of the classical differential analysis on the field of p-adic numbers, empha- sizing the aspects that give rise to the philosophy behind the arithmetic differentialoperators.Thereaderisurgedtocontrasttheseideasatwill, while keeping in mind that our study is neither exhaustive nor intended to be so, and most of the time we shall content ourselves by explain- ing the differential aspect of an arithmetic operator by way of analogy, rather than appealing to the language of jet spaces. But even then, the importance of these operators will be justified by their significant ap- pearanceinnumbertheoreticconsiderations.Oneofourgoalswillbeto illustrate how different these operators are when the ground field where they are defined is rather coarse, as are the p-adic integers Zp that we use. In order to put our work in proper perspective, it is convenient to introduce some basic facts first, and recall a bit of history. Given a prime p, we may define the p-adic norm (cid:3) (cid:3)p over the field of rational numbersQ.Thecompletionoftherationalsinthemetricthatthisnorm induces is the field Qp of p-adic numbers, and this field carries a non- Archimedeanp-adicnormextendingtheoriginalp-adicnormonQ.This isthedescriptionofQpasgivenbyK.Henselcirca1897(see,forinstance, 1 2 Introduction [28]). Two decades later, A. Ostrovski [39] proved that any nontrivial normonQisequivalenttoeithertheEuclideannormortoap-adicnorm forsomeprimep.Inthisway,therearosethephilosophicalprinciplethat treats the real numbers and all of the p-adic numbers on equal footing. In the twentieth century, the p-adic numbers had a rich history. We briefly mention some major results. The idea that studying a question about the field Q can be answered by putting together the answers to the same question over the fields R and Qp for all ps was born with the Hasse–Minkowski’s theorem. This states that a quadratic form over Q has a nontrivial zero in Qn if, and only if, it has a nontrivial zero in Rn and a nontrivial zero in Qn for p each prime p. This theorem was proven by Hasse in his thesis around 1921 [27], the problem having been proposed to him by Hensel who had proventhen=2caseafewyearsearlier.Suchaprinciplefailsforcubics. Thedevelopmentabovecameafterseveralinterestingresultsthatpre- cededtheintroductionofthep-adicnumbers.Thelocal-to-globalprinci- ple embodied in the Hasse–Minkowski theorem had a precedent in Rie- mannian geometric, since as recently as 1855, Bonnet had proved that if the curvature of a compact surface was bounded below by a positive constant,thenitsdiameterwasboundedabovebyaquantitydepending only on the said constant. Strictly on the arithmetic side of things, in the seventeenth century J. Bernoulli defined the(cid:3)Bernoulli numbers Bk, the coefficients in the expansion et/(et −1) = k(cid:3)Bktk/k!, used them to compute closed-form expressions for the sums m jn, and devel- j=0 oped several identities that these numbers satisfy. A century later, the Bernoulli numbers were used by Euler to show heuristically that if ζ is (cid:3) theRiemmanzetafunction,thenζ(1−k)= ∞n=11/n1−k =−Bk/k for any integer k ≥2. In the mid nineteenth century, Riemman proved that (cid:3) ζ(s) = ∞ 1/ns is a meromorphic function on the complex plane C, n=1 giving Euler’s argument complete sense. Further, he used the Gamma (cid:4) (cid:5) functiontodefineΛ(s)=π−s2Γ 2s ζ(s)andprovedthefunctionalequa- tion Λ(s) = Λ(1−s). The intimate relationship between the Bernoulli numbers and the values of ζ(s) at negative integers led to the idea that these numbers have profound arithmetical properties, a fact discovered by Kummer in his work on Fermat’s last theorem circa 1847. The ideal class group of Q(ζN), ζN a primitive N-th root of unity, is the quotient of the fractional ideals of Q(ζN) by the set of principal ideals, and it turnsouttobeagroupoffiniteorderhN withrespecttoidealmultipli- cation. A prime p is said to be regular if p(cid:2)hp, and irregular otherwise. Kummer proved that p is regular if, and only if, p does not divide the