<%BANNER%>

P-Adic Theory of Exponential Sums on the Affine Line

Permanent Link: http://ufdc.ufl.edu/UFE0041424/00001

Material Information

Title: P-Adic Theory of Exponential Sums on the Affine Line
Physical Description: 1 online resource (44 p.)
Language: english
Creator: Morofushi, Yuri
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: exponential, isocrystal, newton, padic, polygon, sum
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this work, we investigate p-adic theory of certain exponential sums on the affine line. Andrea Pulita introduced an F-isocrystal on the affine line which corresponds to a character of absolute Galois group of k((t^(-1))) of p-power order. We study the exponential sums of this F-isocrystal by examining the Newton polygon of the L-function of the exponential sums. We first compute the degree of the L-function by using the formula by Philippe Robba. Then, we replace the Frobenius of Pulita's F-isocrystal by a new Frobenius with larger radius of convergence to obtain better estimates for the Newton polygon. Finally, we find a lower bound of the Newton polygon of the L-function of the F-isocrystal.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yuri Morofushi.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Crew, Richard M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041424:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041424/00001

Material Information

Title: P-Adic Theory of Exponential Sums on the Affine Line
Physical Description: 1 online resource (44 p.)
Language: english
Creator: Morofushi, Yuri
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: exponential, isocrystal, newton, padic, polygon, sum
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this work, we investigate p-adic theory of certain exponential sums on the affine line. Andrea Pulita introduced an F-isocrystal on the affine line which corresponds to a character of absolute Galois group of k((t^(-1))) of p-power order. We study the exponential sums of this F-isocrystal by examining the Newton polygon of the L-function of the exponential sums. We first compute the degree of the L-function by using the formula by Philippe Robba. Then, we replace the Frobenius of Pulita's F-isocrystal by a new Frobenius with larger radius of convergence to obtain better estimates for the Newton polygon. Finally, we find a lower bound of the Newton polygon of the L-function of the F-isocrystal.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Yuri Morofushi.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Crew, Richard M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041424:00001


This item has the following downloads:


Full Text

PAGE 1

P-ADICTHEORYOFEXPONENTIALSUMSONTHEAFFINELINEByYURIMOROFUSHIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

PAGE 2

c2010YuriMorofushi 2

PAGE 3

Tomyparents,HitoshiMorofushiandTokoMorofushi 3

PAGE 4

ACKNOWLEDGMENTS Itisapleasuretothankthosewhomadethisthesispossible.Firstandforemost,IwouldliketoshowmygratitudetomyPh.Dadvisor,RichardCrew,forhiscontinuedsupportandguidance.Thisworkwouldnothavebeenpossiblewithouthisassistance.Iwouldalsoliketothankmydissertationcommitteemembers,KevinKeating,PierreRamond,PaulRobinson,andAlexandreTurull,fortheirsuggestionsandsupport.Finally,Ithankmymotherforalwaysbeingthereforme. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 6 CHAPTER 1INTRODUCTIONANDBASICDEFINITIONS ................... 7 1.1Denitions .................................... 7 1.1.1AlgebraicVarieties ........................... 7 1.1.2FrobeniusStructure .......................... 7 1.1.3F-Isocristals ............................... 8 1.1.4WittVectors ............................... 8 1.2History ...................................... 9 1.2.1ZetaFunctionandA.Weil'sConjectures ............... 9 1.2.2B.Dwork'sWork ............................. 10 1.2.3CohomologicalInterpretation ..................... 10 1.2.4ZetaFunctionsandtheL-Functions .................. 11 1.2.5ExponentialSumsAssociatedtoanF-Isocrystal .......... 12 1.2.6Unit-RootF-Isocrystals ......................... 13 1.2.7L-FunctionsoftheExponentialSums ................. 13 2ANDREAPULITA'SF-ISOCRYSTALS ....................... 15 2.1-Exponentials ................................. 15 2.2Pulita'sF-Isocrystals .............................. 16 2.3ExponentialSumsandF-Isocrystals ..................... 17 3DEGREESOFTHEL-FUNCTIONS ........................ 18 3.1m=1Case .................................... 18 3.2GeneralCase .................................. 24 4NEWTONPOLYGONSOFTHEL-FUNCTIONS ................. 30 4.1ReplacingPulita'sFrobenius ......................... 30 4.2NewtonPolygonoftheL-Function ...................... 33 4.3m>1Case ................................... 35 5CONCLUSIONS ................................... 41 REFERENCES ....................................... 42 BIOGRAPHICALSKETCH ................................ 44 5

PAGE 6

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyP-ADICTHEORYOFEXPONENTIALSUMSONTHEAFFINELINEByYuriMorofushiMay2010Chair:RichardCrewMajor:Mathematics Inthiswork,weinvestigatep-adictheoryofcertainexponentialsumsontheafneline.AndreaPulitaintroducedanF-isocrystalontheafnelinewhichcorrespondstoacharacterofabsoluteGaloisgroupofk((t)]TJ /F6 7.97 Tf 6.59 0 Td[(1))ofp-powerorder.WestudytheexponentialsumsofthisF-isocrystalbyexaminingtheNewtonpolygonoftheL-functionoftheexponentialsums. WerstcomputethedegreeoftheL-functionbyusingtheformulabyPhilippeRobba.Then,wereplacetheFrobeniusofPulita'sF-isocrystalbyanewFrobeniuswithlargerradiusofconvergencetoobtainbetterestimatesfortheNewtonpolygon.Finally,wendalowerboundoftheNewtonpolygonoftheL-functionoftheF-isocrystal. 6

PAGE 7

CHAPTER1INTRODUCTIONANDBASICDEFINITIONS Inthischapter,wereviewthebasicdenitions,facts,andabriefhistorywewillneedfortherestofthedissertation. 1.1Denitions 1.1.1AlgebraicVarieties Letkbeanalgebraicallyclosedeld.LetAnkbeanafnen-spaceoverk,thesetofalln-tuplesofelementsofk.YAnkisanalgebraicsetifthereexistsTk[x1,,xn]suchthatY=fP2Ankjf(P)=0forallf2Tg.AnafnevarietyisanirreduciblealgebraicsetXAnk.Yisaquasi-afnevarietyifAnk-Yisanalgebraicset. LetPnkbeaprojectiven-spaceoverk,thesetofequivalenceclassesofn+1-tuplesofelementsofk,notallzero,undertheequivalencerelationgivenby(a0,,an)(a0,,an)forall2k,6=0.YPnkisanalgebraicsetifthereexistsasetTk[x1,,xn]ofhomogeneouselementssuchthatY=fP2Ankjf(P)=0forallf2Tg.AprojectivevarietyisanirreduciblealgebraicsetinPnk.Yisaquasi-projectivevarietyifPnk-Yisanalgebraicset. Analgebraicvarietyisanyafne,quasi-afne,projective,orquasi-projectivevariety. 1.1.2FrobeniusStructure LetVbeacompletediscretevaluationring,kbetheresidueeldofV,perfectofcharacteristicp,andKbethefractioneldofV.TheRobbaringat1,RK,isdenedbyRK=ff(T):=Xi2ZaiTijai2K,9>1suchthatf(T)convergesfor1
PAGE 8

1.1.3F-Isocristals AnisocrystalonRisapair(M,r)consistingofanitefreeR-moduleMandaconnectionr:M!MN1R=K. AnF-isocrystalonRisatriple(M,r,F)where(M,r)isanisocrystalonRandFisaFrobeniusstructure,i.e.,a-linearisomorphismF:M!Mcommutingwithr. 1.1.4WittVectors Wefollow[ 15 ]forthenotationsconcerningtheringofWittvectors.LetpbeaprimeandRbearing.TheWittpolynomialsaredenedby n(X0,...,Xn):=Xpn0+pXpn)]TJ /F16 5.978 Tf 5.75 0 Td[(11++pnXn. Theorem1.1. LetX=(x0,...,xn,...)andY=(y0,...,yn,...)besequencesofindeterminates.Thenforeach2Z[X,Y],thereisauniquesequence('0,...,'n,...)ofelementsofZ[x0,...;y0,...]suchthatn('0,...,'n,...)=(n(x0,...),n(y0,...)) foralln0. Let1(X,Y)=X+Y,2(X,Y)=XY.Thenwecannduniquesequences(S0,...)and(P0,...)ofelementsofZ[x0,...;y0,...]suchthatn(S0,...)=n(x0,...)+n(y0,...)n(P0,...)=n(x0,...)n(y0,...) foralln0. Nowfor,2RN,dene+=(S0(,),...)=(P0(,),...) 8

PAGE 9

ThenRNisacommutativeringwith1undertheseoperations.CallthisringtheringofWittvectorsofR,anddenoteitbyW(R). Forall:=(0,1,...)2W(R),wecallj(0,1,...)2Rthej-thphantomcomponentof. TheringofWittvectorsofnitelengthisdenedbyWm(R):=W(R)=Vm+1W(R),whereV:W(R)!W(R),(0,1,...)7!(0,0,1,...),istheVerschieburgmorphism.DeneamorphismF:Wm(R)!Wm(R)byF(0,,m)=(p0,,pm). Example1.2. S0(,)=0+0,andP0(,)=00.SoonehasW0(R)=R. Example1.3. If,2W1(R),then+=(0+0,1+1+p0+p0)]TJ /F6 7.97 Tf 6.59 0 Td[((0+0)p p)and=(00,p01+1p0+p11). Theorem1.4. W(Fp)=ZpandWn(Fp)=Z=pnZviathemap(0,...)7!P1i=0p)]TJ /F14 5.978 Tf 5.76 0 Td[(iipi. 1.2History 1.2.1ZetaFunctionandA.Weil'sConjectures Denition1. LetXbeanon-singularn-dimensionalprojectivealgebraicvarietyoveraeldFqwithqelements.ThezetafunctionofXdenedoverFqis(X,t):=exp1Xm=0Nmtm m, whereNmisthenumberofpointsofXdenedoverFqm. In1949A.Weil[ 20 ]statedhisconjectures: (Rationality)(X,t)=P(t) Q(t)whereP(t),Q(t)2Q[t]. (FunctionalEquation)(X,1 qnt)=qn=2t(X,t)where=(Eulercharacteristic). (RiemannHypothesis)(X,t)=P1(t)P3(t)P2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1(t) P0(t)P2(t)P2n(t) 9

PAGE 10

P0(t)=1)]TJ /F4 11.955 Tf 11.95 0 Td[(t,P2n(t)=1)]TJ /F4 11.955 Tf 11.96 0 Td[(qnt,andfor1r2n)]TJ /F10 11.955 Tf 11.96 0 Td[(1,Pr(t)=rYj=1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(r,jt) wherer,jarealgebraicintegersofabsolutevalueqr=2. NotethatwhenXisacurve,n=1andtheZetafunctionbecomes(X,t)=P(t) (1)]TJ /F4 11.955 Tf 11.96 0 Td[(t)(1)]TJ /F4 11.955 Tf 11.95 0 Td[(qt), whereP(t)=Q2gi=1(1)]TJ /F3 11.955 Tf 12.38 0 Td[(it),g=genusofthecurve,andiarealgebraicintegersofabsolutevalueq1=2.Ifwelett=q)]TJ /F9 7.97 Tf 6.58 0 Td[(s,thenthestatementthattherootsofP(t)allhavenormequaltoq1=2isequivalenttotheRiemannhypothesis. 1.2.2B.Dwork'sWork In1960B.Dwork[ 4 ]provedthatthezetafunctionofavarietywasrationalbyp-adicanalysis.Heshowedthat thezetafunctionismeromorphicinCpi.e.,aquotientoftwoentirepowerseries; thezetafunctionhasanon-zeroradiusofconvergenceinthecomplexplane. Thenheconcludedthatthezetafunctionisarationalfunction. 1.2.3CohomologicalInterpretation WeilpointedoutthatgivenasuitableWeilcohomologytheoryH(X),wehave(X,t)=2dYi=0det(I)]TJ /F4 11.955 Tf 11.96 0 Td[(TFjHi(X))()]TJ /F6 7.97 Tf 6.59 0 Td[(1)i+1, whereF:X!XistheqthpowerFrobenius. WhenXisaprojectivesmoothvariety,wehaveaWeilcohomologyH(X). WhenX/Fqissmoothandproper,thereareseveralsuchcohomologies: l-adicetalecohomology(Grothendieck) Crystallinecohomology(Berthelot) 10

PAGE 11

WhenX/Fqisseparableandofnitetype,wehave(X,t)=2dYi=0det(I)]TJ /F4 11.955 Tf 11.96 0 Td[(TFjHiC(X))()]TJ /F6 7.97 Tf 6.59 0 Td[(1)i+1, whereHC(X)isacohomologywithcompactsupport,and l-adiccohomologywithsupport rigidcohomologywithsupport 1.2.4ZetaFunctionsandtheL-Functions Inordertogeneralizethezetafunctions,weneed,foreachalgebraicvarietyX,todeneacategoryofcoefcientsEonXandassociatetoEsomenumberS(E,x)ateachclosedpointx2X.ThendeneSl(X,E)=Px2X(Fql)S(E,x),andL(X,E,T):=exp(P1i=1Sl(X,E)Tl l),whereX(Fql)isthesetofpointsofXoverFql. GivenanF-isocrystal(E,r,)overX/K,wecandenetheL-functionoftheF-isocrystalasfollows. DenethetracefunctionassociatedtotheF-isocrystalbySl(X,E)=Xx2X(Fql)Tr(degxx), wherexisaFrobeniusonEx=inverseimageofEonx.HereS(E,X)=Tr(degxx).Sinceforanymatrixwehavedet(I)]TJ /F3 11.955 Tf 11.95 0 Td[(t)=exp()]TJ /F11 11.955 Tf 11.29 8.96 Td[(P1l=1Tr(l)tl l),wegetL(X,E,T)=Yx2X01 detK(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Tdegxdegxx), whereX0isthesetofclosedpointsofX. BytheLefschetzTraceFormula,wehaveSl(X,E)=P()]TJ /F10 11.955 Tf 9.3 0 Td[(1)iTrljHC(X,E).HenceL(X,E,T)=2dYi=0det(I)]TJ /F4 11.955 Tf 11.95 0 Td[(TjHC(X,E))()]TJ /F6 7.97 Tf 6.58 0 Td[(1)i+1 andtheL-functionsarerational.NotethatwhenEisrankone,thetracefunctionsareexponentialsums. 11

PAGE 12

1.2.5ExponentialSumsAssociatedtoanF-Isocrystal HerearesomeexamplesoftheexponentialsumsassociatedtoanF-isocrystals. Example1.5. In[ 6 ],B.DworkstudiedtheexponentialsumsdenedbyXx2FplTr(x)]TJ /F16 5.978 Tf 7.87 3.26 Td[(a x)p, whereaq=aincharacteristiczero,aistheimageofainFpl.ThenthecorrespondingFrobeniusisdenedby:= F,where isgivenby (PBnXn)=PBpnXnandF(x,t)=exp(t+a t) exp(tp+ap tp),=()]TJ /F4 11.955 Tf 9.3 0 Td[(p)1=p)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Example1.6. In[ 19 ]and[ 21 ],D.WanandH.JZhustudiedtheexponentialsumsgivenbySl(fFp)=Xx2FplTr(f(x)NFp)p, wheref2(ZpTQ)[x]isofdegreed.For1id,letaibethecoefcientsoffwithad=1,aibethereductionofaiatp,and^aibetheTeichmullerliftingofai.Let(x)=AH(0x),whereAH(x)istheArtin-Hasseexponentialfunctionand0isarootoflogAH(x)withordp(0)=1 p)]TJ /F6 7.97 Tf 6.58 0 Td[(1.ThenthecorrespondingFrobeniusisdenedby= G(X),where isgivenby (PBnXn)=PBpnXn,andG(X)=Qdi=1(^aiXi). Example1.7. S.SperbertreatedtheKloostermanexponentialsumsoveraniteeldofq=paelementsin[ 16 ].TheKloostermanexponentialsumsaredenedbySm(fa)=X m(x1++xn+a x1xn), wherea2Fq, m:Fqm!Cisanadditivecharacter,andthesumrangesoverallelements(x1,,xn)2(Fqm)n.ThenthecorrespondingFrobeniusxisdened,onthechainlevel,byx= F(x,t),whereF(x,t)=^F(x,t)=^F(xp,tp),^F(x,t)= 12

PAGE 13

exp((t1++tn+x=t1t2tn))and islinearanddenedonmonomialsby (t)=8><>:t=pifpjiforalli,0otherwise 1.2.6Unit-RootF-Isocrystals Denition2. Aunit-rootF-crystalonA1isatriple(M,r,)where MisalocallyfreeVfTg-module,whereVfTgisap-adiccompletionofV[T]. r:M!MN1isanintegrableconnection,i.e.,risanadditivemapsatisfyingtheLeibnitzrule:r(fs)=fr(s)+sNdf,andthecurvaturer2iszero. :M!Misa-linearisomorphism,i.e.,(fg)=(f)(g)iff2VfTgandg2M. ishorizontal,i.e.,randcommute. Aunit-rootF-isocrystalisM0NK,whereM0isaunit-rootF-crystal. LetKbethesubsetofKxedby,andRepK(1(A1))bethecategoryofcontinuousrepresentationsof1(A1)innitedimensionalK-vectorspaces. Theorem1.8. (R.Crew,N.Tsuzuki)[ 2 17 ]Thereisanequivalenceofcategories G:RepK(1(A1))'(Unit-rootF-isocrystalsonA1). Infactthistheoremistrueforanysmoothcurve. 1.2.7L-FunctionsoftheExponentialSums Foreveryl1,letSlbeexponentialsums.ThentheL-functionoftheexponentialsumisdenedbyL(T):=exp1Xl=1SlTl l. Itiswell-knownthatL-functionsarepolynomials.Sowehaveexp(1Xl=1SlTl l)=nYi=1(1)]TJ /F4 11.955 Tf 11.95 0 Td[(aiT) 13

PAGE 14

forsomen.BylogarithmicdifferentiationwehaveSl=)]TJ /F4 11.955 Tf 9.3 0 Td[(al1)-227()]TJ /F4 11.955 Tf 40.69 0 Td[(aln.SounderstandingtheexponentialsumsSlisreducedtounderstandingthereciprocalzerosoftheL-function. Inthiswork,weinvestigatecertainkindsofexponentialsumsintroducedbyA.Pulitain[ 12 ]bystudyingthecorrespondingL-functions.WecomputeanupperboundofthedegreeoftheL-functionusingtheformulabyP.Robba[ 11 13 ].Wethencomputethep-adicabsolutevaluesofthezerosoftheL-functionusingtheNewtonpolygonoftheL-function. 14

PAGE 15

CHAPTER2ANDREAPULITA'SF-ISOCRYSTALS Nowwearegoingtoreviewcertainkindsofunit-rootF-isocrystalsonA1thatwereintroducedbyA.Pulitain[ 12 ],andthecorrespondingexponentialsumsandtheL-functions. 2.1-Exponentials TheArtin-HasseexponentialisdenedbyAH(T):=exp(T+Tp p+Tp2 p2+). Itiswell-knownthatAH(T)convergeswhenordp(T)>0(see[ 8 ]). LetP(X)=pX+Xp.Let~02CpbearootofP(X)=0,andlet~i2CpbesuchthatP(~i)=~i)]TJ /F6 7.97 Tf 6.58 0 Td[(1fori1.Dene Em(T):=exp(~mT+~m)]TJ /F6 7.97 Tf 6.58 0 Td[(1Tp p++~0Tpm pm). Pulitashowedin[ 12 ]thattheradiusofconvergenceofEm(T)is0,thatis,Em(T)convergeswhenordp(T)>0. LetBbeaZp[~m]-algebra.Let=(0,...,m)2Wm(B),andleth0,...,mi2Bm+1beitsphantomvector.Fixn,m,d2Nsuchthatd=npm>0and(n,p)=1. Denition3. The-exponentialattachedto,ed(,T)21+~mTB[[T]],isdenedtobe ed(,T):=exp(~m0Tn+~m)]TJ /F6 7.97 Tf 6.59 0 Td[(11Tnp p++~0mTd pm). HerearesometheoremsbyPulitaabout-exponentials(see[ 12 ]). Theorem2.1. Forall2Wm(B),wehaveed(,T)=mYj=0Em)]TJ /F9 7.97 Tf 6.59 0 Td[(j(jTnpj). 15

PAGE 16

Theorem2.2. The-exponentialed(,T)isover-convergent(i.e.,convergentforordp(T)>)]TJ /F3 11.955 Tf 9.3 0 Td[(",with">0)ifandonlyifordp(i)>0forall0im. Theorem2.3. Setm(T)=Em(T) Em(Tp). Then,m(T)isover-convergentforallm0. 2.2Pulita'sF-Isocrystals LetKbeacompletevaluedeldcontainingQp.SetKs:=K(~s)andletksbeitsresidueeld. Letf(T)2Ws(TOKs[T])andletf(t)2Ws(tOks[t])beitsreduction.A.Pulitaprovedthefollowingtheoremsin[ 12 ]. Theorem2.4. LetL:=@T)]TJ /F3 11.955 Tf 11.95 0 Td[(@T,log(eps(f(T),1)), where@T=Td dT.Then,eps(f(T),1)isasolutionofLu=0. Theorem2.5. IffF(T)2Ws(TOKs[T])isanarbitraryliftingofF(f(t)),theneps(f(T),1) eps(fF(T),1) isover-convergent Notethatwhenm=0,thisistheDworkexponentialexp(~0(f)]TJ /F4 11.955 Tf 11.95 0 Td[(fp)).Thistheoremimpliesthefollowing. Let f(T)2Wm(TOKm[T]); M=RK; r=@T+@T,log(epm(f(T),1))where@T=Td dTand@T,log(f)=@T(f) f; f:M!Misgivenbyf(x)=epm(f(T),1) epm(fF(T),1)(x). 16

PAGE 17

Then(M,r,f)isaunit-rootF-isocrystalonA1.Notethatit'sthemostgeneralunit-rootF-isocrystalonA1correspondingtoacharacterof1(Speck((t)]TJ /F6 7.97 Tf 6.59 0 Td[(1))),absoluteGaloisgroupofk((t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)),ofp-powerorder. 2.3ExponentialSumsandF-Isocrystals Letf(X)=(f0(X),,fm(X))2Wm(Fp[X]).Letq=pl.DeneTrWm(Fq)=Wm(Fp):Wm(Fq)!Wm(Fp)by TrWm(Fq)=Wm(Fp)(0,,m)=X2Gal(Fq=Fp)((0),,(m))=l)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=0(pi0,,pim). ElementsinWm(Fp)canbeidentiedwithelementsinZp=pm+1Zpviathemap(0,,m)7!Pmi=0^p)]TJ /F14 5.978 Tf 5.75 0 Td[(iipi.Sodenel:Wm(Fq)!Cpbyl((0,,m))=Pl)]TJ /F16 5.978 Tf 5.76 0 Td[(1i=0Pmj=0(^p)]TJ /F14 5.978 Tf 5.76 0 Td[(jj)pipjpm+1, wherepm+1isaprimitivepm+1-throotofunity.Thenlisanadditivecharacter.SincePl)]TJ /F6 7.97 Tf 6.58 0 Td[(1i=0(pi0,,pim)=Pl)]TJ /F6 7.97 Tf 6.58 0 Td[(1i=0(pm+i0,,pm+im)and^plj=^j,get l((0,,m))=Pl)]TJ /F16 5.978 Tf 5.75 0 Td[(1i=0^pi0+^pi1p++^pimpmpm+1. DenetheexponentialsumsoffbySl((f0,,fm))=Xx2Fpll((f0(x),,fm(x))). Mygoalistounderstandtheseexponentialsumsforallm0. TheL-functionoftheexponentialsumsisdenedbyL((f0,,fm);T)=exp(1Xl=1Sl((f0,,fm))Tl l). ThisL-functionistheL-functionofPulita'sF-isocrystal. 17

PAGE 18

CHAPTER3DEGREESOFTHEL-FUNCTIONS Inthischapter,wecomputethedegreesoftheL-functionsusingtheformulabyP.Robba[ 11 13 ]. LetL:V!Vbealinearmap.Whenkernelandcokernelarenite,thentheindexofLonVisdenedtobe(L,V)=dimkerL)]TJ /F10 11.955 Tf 11.96 0 Td[(dimcokerL. Letkbeanalgebraicallyclosedeldofcharacteristics0.LetA=B(0,1+)andA"=B(0,(1+")+).DeneHy(A):=[">0H(A"), whereH(A")isthesetofanalyticelementsofA"withcoefcientsink. Theorem3.1. (P.Robba)[ 13 ]LettbeagenericpointonthecircumferenceC(0,r)withr>1.Letx=t+y.LetDbeadifferentialoperatorofrstorder,andubeasolutionofD=0neart.Supposethattheradiusofconvergenceofuis1=rN.Then,)]TJ /F3 11.955 Tf 9.3 0 Td[((D,Hy(A))=N. Theorem3.2. (P.Robba)[ 13 ]DeneD:=Td dT)]TJ /F4 11.955 Tf 13.15 8.78 Td[(Td dTF F; and:=F)]TJ /F6 7.97 Tf 6.59 0 Td[(1 pF. ThentheL-functiongivenbyL(f;T)=det(1)]TJ /F4 11.955 Tf 11.95 0 Td[(T)=det(1)]TJ /F4 11.955 Tf 11.96 0 Td[(tp)hasadegree)]TJ /F3 11.955 Tf 9.3 0 Td[((D,Hy(A)). LetD=@T)]TJ /F3 11.955 Tf 12.14 0 Td[(@T,log(eps(f(T),1)).ThenTheorem 3.1 andTheorem 3.2 impliesthattheL-functionofPulita'sF-isocrystalL((f0,,fm);T)hasdegree)]TJ /F3 11.955 Tf 9.3 0 Td[((D,Hy(A))=N. 3.1m=1Case LetKbeaeldandObetheringofintegersofK.Let(f,g)beanelementofW1(TO[T])suchthatdeg(f)=danddeg(g)=e.Assumethatp-dandp-e. 18

PAGE 19

Writef(T)=Pdi=1aiTiandg(T)=Pei=1biTi,andassumethatad=1andbe=1. Lemma1. Supposethatp-n.Thenexp(0(tn)]TJ /F10 11.955 Tf 12.68 0 Td[((t+y)n))convergesifandonlyifjyj<1=rn)]TJ /F6 7.97 Tf 6.58 0 Td[(1,wherer=jt+yj>1. Proof. Leth(t)=tn.ThenusingtheTaylorseriesweobtain tn)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)n=h0(t+y)()]TJ /F4 11.955 Tf 9.3 0 Td[(y)+h00(t+y) 2()]TJ /F4 11.955 Tf 9.3 0 Td[(y)2+=(t+y)n)]TJ /F6 7.97 Tf 6.58 0 Td[(1y)]TJ /F4 11.955 Tf 11.95 0 Td[(n+n(n)]TJ /F10 11.955 Tf 11.95 0 Td[(1)(y) 2(t+y)+. Sinceexp(0(tn)]TJ /F10 11.955 Tf 12.28 0 Td[((t+y)n))convergesifandonlyifjtn)]TJ /F10 11.955 Tf 12.28 0 Td[((t+y)nj<1,itconvergesifandonlyifjyj<1 rn)]TJ /F16 5.978 Tf 5.76 0 Td[(1jKj,whereK=()]TJ /F4 11.955 Tf 9.3 0 Td[(n+n(n)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(y) 2(t+y)+).Notethatjnj=1asp-n.SincejKjsupfj)]TJ /F4 11.955 Tf 23.91 0 Td[(nj,jn(n+1)(t)]TJ /F9 7.97 Tf 6.58 0 Td[(t0) 2t0j,g,whenjyj
PAGE 20

exp1(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x)+0 p(tp)]TJ /F4 11.955 Tf 11.96 0 Td[(xp)=exp1(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x)+0 p(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x)p)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1pitp)]TJ /F9 7.97 Tf 6.58 0 Td[(i()]TJ /F4 11.955 Tf 9.3 0 Td[(x)i. Notethatexp(1(t)]TJ /F4 11.955 Tf 11.96 0 Td[(x)+0 p(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x)p)=E1()]TJ /F4 11.955 Tf 9.3 0 Td[(y) convergesifandonlyifjyj<1. Nowconsiderexp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1pitp)]TJ /F9 7.97 Tf 6.58 0 Td[(i()]TJ /F4 11.955 Tf 9.3 0 Td[(x)i. Notethat p)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1)]TJ /F10 11.955 Tf 9.29 0 Td[(1 ppitp)]TJ /F9 7.97 Tf 6.59 0 Td[(i()]TJ /F4 11.955 Tf 9.29 0 Td[(x)i=p)]TJ /F16 5.978 Tf 5.76 0 Td[(1 2Xi=1()]TJ /F10 11.955 Tf 9.3 0 Td[(1)i+1)]TJ /F9 7.97 Tf 5.48 -4.38 Td[(pi p(tp)]TJ /F9 7.97 Tf 6.58 0 Td[(ixi)]TJ /F4 11.955 Tf 11.95 0 Td[(tixp)]TJ /F9 7.97 Tf 6.59 0 Td[(i)=p)]TJ /F16 5.978 Tf 5.76 0 Td[(1 2Xi=1ci(tx)i(tp)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)]TJ /F4 11.955 Tf 11.96 0 Td[(xp)]TJ /F6 7.97 Tf 6.58 0 Td[(2i), wherejcij=1. Foreach1ip)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2,exp(0ci(tx)i(tp)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)]TJ /F4 11.955 Tf 11.95 0 Td[(xp)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)convergesifandonlyifjci(tx)i(tp)]TJ /F6 7.97 Tf 6.58 0 Td[(2i)]TJ /F4 11.955 Tf 11.95 0 Td[(xp)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)j<1 ifandonlyifjtp)]TJ /F6 7.97 Tf 6.58 0 Td[(2i)]TJ /F4 11.955 Tf 11.95 0 Td[(xp)]TJ /F6 7.97 Tf 6.59 0 Td[(2ij
PAGE 21

Hence,bytheproofofLemma 1 ,exp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1pitp)]TJ /F9 7.97 Tf 6.59 0 Td[(ixiconvergesforjyj<1=rp)]TJ /F6 7.97 Tf 6.58 0 Td[(1. Lemma3. exp(1(tn)]TJ /F10 11.955 Tf 11.95 0 Td[((t+y)n)+0 p(tpn)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)pn))convergesforjyj<1=rpn)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Proof. Letx=t+y. Supposethatp=2.Then, exp(1(tn)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)n)+0 2(t2n)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)2n))=exp(1(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)+0 2(t2n)]TJ /F4 11.955 Tf 11.96 0 Td[(x2n))=exp(1(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)+0 2(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)2)exp(0(tnxn)]TJ /F4 11.955 Tf 11.95 0 Td[(x2n))=E1(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)E0(tnxn)]TJ /F4 11.955 Tf 11.96 0 Td[(x2n). (3) E1(tn)]TJ /F4 11.955 Tf 12.28 0 Td[(xn)convergesifandonlyifjtn)]TJ /F10 11.955 Tf 12.27 0 Td[((t+y)nj=jtn)]TJ /F4 11.955 Tf 12.28 0 Td[(xnj<1.BytheproofofLemma 1 ,E1(tn)]TJ /F4 11.955 Tf 11.95 0 Td[(xn)convergesifandonlyifjyj<1=rn)]TJ /F6 7.97 Tf 6.59 0 Td[(1. E0(tnxn)]TJ /F4 11.955 Tf 12.29 0 Td[(x2n)convergesifandonlyifj(t+y)njjtn)]TJ /F10 11.955 Tf 12.29 0 Td[((t+y)nj=jtnxn)]TJ /F4 11.955 Tf 12.29 0 Td[(x2nj<1.Hence,bytheproofofLemma 1 ,E0(tnxn)]TJ /F4 11.955 Tf 11.95 0 Td[(x2n)convergesifandonlyifjyj<1=r2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1. Thus,Equation 3 convergesifandonlyifjyj<1=r2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1,andthelemmaistruewhenp=2. Now,supposethatp6=2.Then, exp(1(tn)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)n)+0 p(tpn)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)pn))=exp1(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)+0 p(tn)]TJ /F4 11.955 Tf 11.95 0 Td[(xn)p)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1pitn(p)]TJ /F9 7.97 Tf 6.59 0 Td[(i)()]TJ /F4 11.955 Tf 9.3 0 Td[(x)ni=E1(tn)]TJ /F4 11.955 Tf 11.96 0 Td[(xn)exp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1pitn(p)]TJ /F9 7.97 Tf 6.59 0 Td[(i)()]TJ /F4 11.955 Tf 9.3 0 Td[(x)ni. (3) E1(tn)]TJ /F4 11.955 Tf 12.62 0 Td[(xn)convergesifandonlyifjtn)]TJ /F10 11.955 Tf 12.61 0 Td[((t+y)nj=jtn)]TJ /F4 11.955 Tf 12.62 0 Td[(xnj<1ifandonlyifjyj<1=rn)]TJ /F6 7.97 Tf 6.58 0 Td[(1. 21

PAGE 22

BytheproofofLemma 2 ,wehaveexp)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 pp)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1pitn(p)]TJ /F9 7.97 Tf 6.59 0 Td[(i)()]TJ /F4 11.955 Tf 9.3 0 Td[(x)ni=p)]TJ /F16 5.978 Tf 5.76 0 Td[(1 2Xi=1exp(0ci(tnxn)i(tn(p)]TJ /F6 7.97 Tf 6.59 0 Td[(2i))]TJ /F4 11.955 Tf 11.96 0 Td[(xn(p)]TJ /F6 7.97 Tf 6.59 0 Td[(2i))), wherejcij=1.Foreach1ip)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2,exp(0ci(tnxn)i(tn(p)]TJ /F6 7.97 Tf 6.58 0 Td[(2i))]TJ /F4 11.955 Tf 11.95 0 Td[(xn(p)]TJ /F6 7.97 Tf 6.59 0 Td[(2i)))convergesifandonlyifjci(tnxn)i(tn(p)]TJ /F6 7.97 Tf 6.59 0 Td[(2i))]TJ /F4 11.955 Tf 11.99 0 Td[(xn(p)]TJ /F6 7.97 Tf 6.58 0 Td[(2i))j<1.Hence,bythesameargumentastheoneinLemma 2 ,p)]TJ /F16 5.978 Tf 5.76 0 Td[(1 2Xi=1exp(0ci(tnxn)i(tn(p)]TJ /F6 7.97 Tf 6.58 0 Td[(2i))]TJ /F4 11.955 Tf 11.95 0 Td[(xn(p)]TJ /F6 7.97 Tf 6.58 0 Td[(2i))) convergeswhenjyj<1=rpn)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thus,Equation 3 convergeswhenjyj<1=rpn)]TJ /F6 7.97 Tf 6.59 0 Td[(1andthisprovesthelemma. Example3.3. Letp=5.Thenexp(1(t)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y))+0 5(t5)]TJ /F10 11.955 Tf 11.95 0 Td[((t+y)5))=exp1()]TJ /F4 11.955 Tf 9.29 0 Td[(y)+0 5()]TJ /F4 11.955 Tf 9.29 0 Td[(y)5)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0 54Xi=15itp)]TJ /F9 7.97 Tf 6.58 0 Td[(i()]TJ /F10 11.955 Tf 9.3 0 Td[((t+y))i.exp1()]TJ /F4 11.955 Tf 9.3 0 Td[(y)+0 5()]TJ /F4 11.955 Tf 9.3 0 Td[(y)5 convergeswhenjyj<1=r. exp)]TJ /F3 11.955 Tf 13.15 8.08 Td[(0 54Xi=15itp)]TJ /F9 7.97 Tf 6.59 0 Td[(i()]TJ /F10 11.955 Tf 9.3 0 Td[((t+y))i=exp(0(t4(t+y))]TJ /F4 11.955 Tf 11.95 0 Td[(t(t+y)4))]TJ /F10 11.955 Tf 11.95 0 Td[(20(t3(t+y)2)]TJ /F4 11.955 Tf 11.95 0 Td[(t2(t+y)3)=exp(0t(t+y)(t3)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)3)+20t2(t+y)2(t)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y))), andexp(0t(t+y)(t3)]TJ /F10 11.955 Tf 11.95 0 Td[((t+y)3))convergeswhenjt3)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)3j<1=jt(t+y)j=1=r2,i.e.,whenjyj
PAGE 23

i.e.,whenjyj<1=r4. Lemma4. ep((f(T),g(T)),1)=dYi=1exp(1aiTi+0 papiTpi)maxfdp)]TJ /F6 7.97 Tf 6.58 0 Td[(1,egYi=1exp(0~biTi), wherej~bij1. Proof. Notethatif(a0,a1)and(b0,b1)areelementsofW1(K),then(a0,a1)+(b0,b1)=(a0+b0,a1+b1+ap0+bp0)]TJ /F6 7.97 Tf 6.59 0 Td[((a0+b0)p p).So,(f(T),g(T))=(Pdi=1aiTi,0)+Pei=1(0,biTi),andIwillshowthat(f(T),0)=Pdi=1(aiTi,0)+Pdp)]TJ /F6 7.97 Tf 6.58 0 Td[(1i=p+1(0,ciTi)withjcij1. Wehave (a1T+a2T2,0)=(a1T,0)+(a2T2,0)+(0,p)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1)]TJ /F9 7.97 Tf 5.48 -4.37 Td[(pi pap)]TJ /F9 7.97 Tf 6.58 0 Td[(i1ai2Tp+i)=(a1T,0)+(a2T2,0)+2p)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=p+1(0,~aiTi), wherej~aij=(pp)]TJ /F14 5.978 Tf 5.75 0 Td[(i) pa2p)]TJ /F9 7.97 Tf 6.59 0 Td[(i1ap)]TJ /F9 7.97 Tf 6.58 0 Td[(i21. Ifwehave(a1T++ad)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.59 0 Td[(1,0)=d)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1(aiTi,0)+2(d)]TJ /F6 7.97 Tf 6.58 0 Td[(1))]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=p+1(0,~aiTi), withj~aij1,thenwehave (f(T),0)=(a1T++ad)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.58 0 Td[(1,0)+(adTd,0)+(0,p)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1)]TJ /F9 7.97 Tf 5.48 -4.38 Td[(pi p(a1T++ad)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.58 0 Td[(1)p)]TJ /F9 7.97 Tf 6.59 0 Td[(i(adTd)i)=dXi=1(aiTi,0)+2(d)]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=p+1(0,~aiTi)+dp)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=p+d+1(0,^aiTi)=dXi=1(aiTi,0)+dp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=p+1(0,ciTi), wherejcij1. 23

PAGE 24

So(f(T),g(T))=dXi=1(aiTi,0)+maxfdp)]TJ /F6 7.97 Tf 6.59 0 Td[(1,egXi=1(0,~biTi)andhenceep((f(T),g(T)),1)=dYi=1exp(1aiTi+0 papiTpi)maxfdp)]TJ /F6 7.97 Tf 6.58 0 Td[(1,egYi=p+1exp(0~biTi). Theorem3.4. Letx=t+y.Then dYi=1exp(1ai(ti)]TJ /F4 11.955 Tf 11.95 0 Td[(xi)+0 papi(tpi)]TJ /F4 11.955 Tf 11.95 0 Td[(xpi))maxfdp)]TJ /F6 7.97 Tf 6.58 0 Td[(1,egYi=p+1exp(0~bi(ti)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)) (3) convergeswhenjyj<1=rmaxfdp)]TJ /F6 7.97 Tf 6.59 0 Td[(1,e)]TJ /F6 7.97 Tf 6.59 0 Td[(1g. Equation 3 canbewrittenasep(f(t),g(t),1 ep(f(t+y),g(t+y),1, whichisasolutionofLu=0neart,whereL:=@T)]TJ /F3 11.955 Tf 11.96 0 Td[(@T,log(ep(f(T),1)). Thus,usingTheorem 3.1 andTheorem 3.2 byRobba,thedegreeoftheL-functionofPulita'sF-isocrystalwhenm=1islessthanorequaltomaxfdp)]TJ /F10 11.955 Tf 11.95 0 Td[(1,e)]TJ /F10 11.955 Tf 11.95 0 Td[(1g,whered=deg(f)ande=deg(g). 3.2GeneralCase Lemma5. LetRbearingand(a0,0,,0),,(0,,0,am)2Wm(R).Then(a0,0,,0)++(0,,0,am)=(a0,,am). Proof. IfandareinWm(R),then+=(S0,S1,,Sm),whereS0=0+0,andfork1,Sk=k+k+pk)]TJ /F6 7.97 Tf 6.58 0 Td[(1+pk)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[(Spk)]TJ /F6 7.97 Tf 6.58 0 Td[(1 p++pk0+pk0)]TJ /F4 11.955 Tf 11.96 0 Td[(Spk0 pk. 24

PAGE 25

So,ifandaresuchthati=0wheneveri6=0,andi=0wheneveri6=0,then+=(0+0,1+1,,m+m).Thus,(a0,0,,0)++(0,,0,am)=(a0,,am). Lemma6. Letf(T)beanelementofTO[T]ofdegreed2,andlet(f,0,,0)beanelementofWm(TO[T]).Thenepm((f,0,,0),1)=dYi=1Em(~a0iTi)dp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Yi=1Em)]TJ /F6 7.97 Tf 6.58 0 Td[(1(~a1iTi)dpm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Yi=1E0(~amiTi), wherej~ajij1. Proof. Sinceepm(+,T)=epm(,T)epm(,T)andepm(,T)=Qmj=0Em)]TJ /F9 7.97 Tf 6.59 0 Td[(j(jTpj),itsufcestoshowthat(f,0,,0)=dXi=1(~a0iTi,0,,0)+dp)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1(0,~a1iTi,0,,0)++dpm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,~amiTi). Notethatif(a,0,,0)and(b,0,,0)areinWm(K),then(a,0,,0)+(b,0,,0)=(S0,,Sm),whereS0=a+b,S1=ap+bp)]TJ /F9 7.97 Tf 6.59 0 Td[(Sp0 p,andSk=)]TJ /F4 11.955 Tf 10.49 8.82 Td[(Spk)]TJ /F6 7.97 Tf 6.59 0 Td[(1 p)]TJ /F4 11.955 Tf 13.15 8.82 Td[(Sp2k)]TJ /F6 7.97 Tf 6.58 0 Td[(2 p2)-221()]TJ /F4 11.955 Tf 41.71 8.08 Td[(Spk)]TJ /F16 5.978 Tf 5.76 0 Td[(11 pk)]TJ /F6 7.97 Tf 6.58 0 Td[(1+apk+bpk)]TJ /F4 11.955 Tf 11.96 0 Td[(Spk0 pk. Henceifa=g(T)andb=h(T)withdegg(T)=i
PAGE 26

wheredegSk=2pk)]TJ /F10 11.955 Tf 11.95 0 Td[(1.(0,)]TJ /F4 11.955 Tf 9.3 0 Td[(S1,)]TJ /F4 11.955 Tf 9.29 0 Td[(S2,,)]TJ /F4 11.955 Tf 9.3 0 Td[(Sm)=(0,)]TJ /F4 11.955 Tf 9.3 0 Td[(S1,0,,0)++(0,,0,)]TJ /F4 11.955 Tf 9.29 0 Td[(Sm), andbyinductionwehave (0,,0,)]TJ /F4 11.955 Tf 9.3 0 Td[(Sk,0,,0)=2pk)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,bkiTi,0,,0)+2pk+1)]TJ /F9 7.97 Tf 6.58 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,bk+1,iTi,0,,0)++2pm)]TJ /F9 7.97 Tf 6.59 0 Td[(pm)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1(0,,0,bmiTi). Thusthelemmaistruewhend=2.Supposethatthelemmaistrueforthecased)]TJ /F10 11.955 Tf 12.14 0 Td[(1.Letf(T)=a1T++ad)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.58 0 Td[(1+adTd.Then, (f(T),0,,0)=(a1T++1d)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.58 0 Td[(1,0,,0)+(adTd,0,,0)+(0,)]TJ /F4 11.955 Tf 9.3 0 Td[(S1,)]TJ /F4 11.955 Tf 9.3 0 Td[(S2,,)]TJ /F4 11.955 Tf 9.3 0 Td[(Sm), wheredegSk=dpk)]TJ /F10 11.955 Tf 12.16 0 Td[(1.UsinginductionandapoweroftheVerschieburgmorphismV,wehave (0,,0,)]TJ /F4 11.955 Tf 9.3 0 Td[(Sk,0,,0)=dpk)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,bkiTi,0,,0)+dpk+1)]TJ /F9 7.97 Tf 6.58 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,bk+1,iTi,0,,0)++dpm)]TJ /F9 7.97 Tf 6.59 0 Td[(pm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,bmiTi) 26

PAGE 27

and (a1T++ad)]TJ /F6 7.97 Tf 6.59 0 Td[(1Td)]TJ /F6 7.97 Tf 6.58 0 Td[(1,0,,0)=d)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(~a0iTi,0,,0)+(d)]TJ /F6 7.97 Tf 6.59 0 Td[(1)p)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,~a1iTi,0,,0)++(d)]TJ /F6 7.97 Tf 6.59 0 Td[(1)pm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,~amiTi). Hence(f,0,,0)=dXi=1(~a0iTi,0,,0)+dp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,~a1iTi,0,,0)++dpm)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1(0,,0,~amiTi) andthelemmaholds. Let(f0,,fm)beanelementofWm(TO[T])suchthatdeg(fi)=di.Writefk(T)=Pdki=1akiTi. Proposition3.1. Let~dk=maxfdk,dk)]TJ /F6 7.97 Tf 6.59 0 Td[(1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,dk)]TJ /F6 7.97 Tf 6.58 0 Td[(2p2)]TJ /F10 11.955 Tf 11.95 0 Td[(1,d0pk)]TJ /F10 11.955 Tf 11.96 0 Td[(1g.Thenepm((f0,,fm),1)=~d0Yi=1Em(~a0iTi)~d1Yi=1Em)]TJ /F6 7.97 Tf 6.58 0 Td[(1(~a1iTi)~dmYi=1E0(~amiTi). Proof. Since(f0,,fm)=(f0,0,,0)++(0,,0,fm),bythepreviouslemma,thepropositionfollows. Lemma7. exp(k(t)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y))+k)]TJ /F6 7.97 Tf 6.58 0 Td[(1 p(tp)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)p)++0 pk(tpk)]TJ /F10 11.955 Tf 11.96 0 Td[((t+y)pk))(3) convergesforjyj<1=rpk)]TJ /F6 7.97 Tf 6.59 0 Td[(1. Proof. Putx=t+y.Let(t,0,)and()]TJ /F4 11.955 Tf 9.3 0 Td[(x,0,)beelementsofWittvectors,andletSibesuchthat(t,0,)+()]TJ /F4 11.955 Tf 9.29 0 Td[(x,0,)=(S0,S1,).So,wehaveS0=t)]TJ /F4 11.955 Tf 12.74 0 Td[(x,S1=tp)]TJ /F9 7.97 Tf 6.58 0 Td[(xp)]TJ /F9 7.97 Tf 6.59 0 Td[(Sp0 p,andSk=)]TJ /F4 11.955 Tf 10.5 8.82 Td[(Spk)]TJ /F6 7.97 Tf 6.58 0 Td[(1 p)]TJ /F4 11.955 Tf 13.15 8.82 Td[(Sp2k)]TJ /F6 7.97 Tf 6.59 0 Td[(2 p2)-222()]TJ /F4 11.955 Tf 41.71 8.09 Td[(Spk)]TJ /F16 5.978 Tf 5.75 0 Td[(11 pk)]TJ /F6 7.97 Tf 6.59 0 Td[(1+tpk)]TJ /F4 11.955 Tf 11.95 0 Td[(xpk)]TJ /F4 11.955 Tf 11.95 0 Td[(Spk0 pk. 27

PAGE 28

Then. Ek(S0)Ek)]TJ /F6 7.97 Tf 6.59 0 Td[(1(S1)E0(Sk)=exp(kS0+k)]TJ /F6 7.97 Tf 6.59 0 Td[(1 pSp0++0 pkSpk0)expk)]TJ /F6 7.97 Tf 6.59 0 Td[(1S1+k)]TJ /F6 7.97 Tf 6.59 0 Td[(2 pSp1++0 pk)]TJ /F6 7.97 Tf 6.58 0 Td[(1Spk)]TJ /F16 5.978 Tf 5.75 0 Td[(11+exp0Sk=expkS0+k)]TJ /F6 7.97 Tf 6.59 0 Td[(1 pSp0++0 pkSpk0expk)]TJ /F6 7.97 Tf 6.59 0 Td[(1(t)]TJ /F4 11.955 Tf 11.95 0 Td[(x p)]TJ /F4 11.955 Tf 13.15 8.08 Td[(Sp0 p)+k)]TJ /F6 7.97 Tf 6.59 0 Td[(2 pSp1++0 pk)]TJ /F6 7.97 Tf 6.59 0 Td[(1Spk)]TJ /F16 5.978 Tf 5.76 0 Td[(11expk)]TJ /F6 7.97 Tf 6.59 0 Td[(2()]TJ /F4 11.955 Tf 10.49 8.09 Td[(Sp1 p)]TJ /F4 11.955 Tf 13.15 8.09 Td[(Sp20 p2+tp2)]TJ /F4 11.955 Tf 11.95 0 Td[(xp2 p2)+k)]TJ /F6 7.97 Tf 6.59 0 Td[(3 pSp2++0 pk)]TJ /F6 7.97 Tf 6.59 0 Td[(2Spk)]TJ /F16 5.978 Tf 5.76 0 Td[(22exp0()]TJ /F4 11.955 Tf 10.49 8.82 Td[(Spk)]TJ /F6 7.97 Tf 6.59 0 Td[(1 p)]TJ /F4 11.955 Tf 13.15 8.82 Td[(Sp2k)]TJ /F6 7.97 Tf 6.59 0 Td[(2 p2)-222()]TJ /F4 11.955 Tf 41.71 8.08 Td[(Spk)]TJ /F16 5.978 Tf 5.76 0 Td[(11 pk)]TJ /F6 7.97 Tf 6.59 0 Td[(1+tpk)]TJ /F4 11.955 Tf 11.96 0 Td[(xpk)]TJ /F4 11.955 Tf 11.96 0 Td[(Spk0 pk)=expk(t)]TJ /F4 11.955 Tf 11.96 0 Td[(x)+k)]TJ /F6 7.97 Tf 6.58 0 Td[(1 p(tp)]TJ /F4 11.955 Tf 11.95 0 Td[(xp)++0 pk(tpk)]TJ /F4 11.955 Tf 11.95 0 Td[(xpk). AsEk)]TJ /F9 7.97 Tf 6.59 0 Td[(i(Si)convergeswhenjSij<1,itsufcestoshowthatjSij<1whenjyj<1=rpi)]TJ /F6 7.97 Tf 6.58 0 Td[(1. WehavejS0j=j)]TJ /F4 11.955 Tf 19.78 0 Td[(yj.So,jyj<1ifandonlyifjS0j<1.Fori1,wehaveSi=Ppi)]TJ /F9 7.97 Tf 6.59 0 Td[(jj=1cjtpi)]TJ /F9 7.97 Tf 6.58 0 Td[(j()]TJ /F4 11.955 Tf 9.3 0 Td[(x)j,wherecj=cpi)]TJ /F9 7.97 Tf 6.59 0 Td[(jandjcjj1.Hence, jSijpi)]TJ /F14 5.978 Tf 5.76 0 Td[(j 2Xj=1jcjjjtpi)]TJ /F9 7.97 Tf 6.59 0 Td[(jxj)]TJ /F4 11.955 Tf 11.96 0 Td[(tjxpi)]TJ /F9 7.97 Tf 6.59 0 Td[(jj=pi)]TJ /F14 5.978 Tf 5.76 0 Td[(j 2Xj=1jcjjjtxjjj(tpi)]TJ /F6 7.97 Tf 6.59 0 Td[(2j)]TJ /F4 11.955 Tf 11.96 0 Td[(xpi)]TJ /F6 7.97 Tf 6.59 0 Td[(2jj. ByLemma 1 ,j(tpi)]TJ /F6 7.97 Tf 6.58 0 Td[(2j)]TJ /F4 11.955 Tf 12.19 0 Td[(xpi)]TJ /F6 7.97 Tf 6.59 0 Td[(2jj<1=r2jifandonlyifjyj<1=rpi)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Henceifjyj<1=rpi)]TJ /F6 7.97 Tf 6.59 0 Td[(1,thenjSijpi)]TJ /F14 5.978 Tf 5.75 0 Td[(j 2Xj=1jcjjjtxjjj(tpi)]TJ /F6 7.97 Tf 6.59 0 Td[(2j)]TJ /F4 11.955 Tf 11.96 0 Td[(xpi)]TJ /F6 7.97 Tf 6.59 0 Td[(2jj<1. 28

PAGE 29

Theorem3.5. Letx=t+y.Then ~d0Yi=1exp(m~a0i(ti)]TJ /F4 11.955 Tf 11.96 0 Td[(xi)++0 pm~apm0i(tpmi)]TJ /F4 11.955 Tf 11.95 0 Td[(xpmi))~dmYi=1exp(0ami(ti)]TJ /F4 11.955 Tf 11.95 0 Td[(xi))(3) convergeswhenjyj<1=r,where=maxfdopm)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d1pm)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 11.96 0 Td[(1,,dm)]TJ /F6 7.97 Tf 6.59 0 Td[(1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,dm)]TJ /F10 11.955 Tf 11.96 0 Td[(1g. Equation 3 isasolutionofLu=0neart,whereL:=@T)]TJ /F3 11.955 Tf 11.96 0 Td[(@T,log(epm(f(T),1)). Thus,usingTheorem 3.1 andTheorem 3.2 byRobba,thedegreeoftheL-functionofPulita'sF-isocrystalwhenm>1islessthanorequaltomaxfdopm)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d1pm)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 11.96 0 Td[(1,,dm)]TJ /F6 7.97 Tf 6.59 0 Td[(1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,dm)]TJ /F10 11.955 Tf 11.96 0 Td[(1g, wheredi=deg(fi). 29

PAGE 30

CHAPTER4NEWTONPOLYGONSOFTHEL-FUNCTIONS Inthischapter,westudytheexponentialsumsofPulita'sF-isocrystalsbystudyingtheNewtonpolygonsoftheL-functions. Denition4. TheNewtonPolygonoff(X)=a0++anXn+isthelowerconvexhullofthepoints(j,ordpaj)inR2forj0. Theorem4.1. ToeachnitesideoftheNewtonPolygonofftherecorrespondlzerosoffwherelisthelengthofthehorizontalprojectionoftheside.Ifistheslopeoftheside,thenordp=)]TJ /F3 11.955 Tf 9.3 0 Td[(.Converselyifisarootthen)]TJ /F4 11.955 Tf 9.3 0 Td[(ordpistheslopeofaside. Example4.2. Letf(X)=X+Xp p+Xp2 p2+.ThentheslopesofitsNewtonpolygonare)]TJ /F6 7.97 Tf 23.22 4.71 Td[(1 pj(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)forj0.Thusf(X)hasrootsjoforder1 pj(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1). D.WanandH.J.ZhustudiedtheexponentialsumsofPulita'sF-isocrystalswhenm=0bystudyingtheNewtonpolygonsoftheL-functions[ 19 21 ].Theirresultisthefollowing. Theorem4.3. AlowerboundoftheNewtonpolygonoftheL-functionofPulita'sF-isocrystalwhenm=0isthelowerconvexhullofpoints(n,n(n+1) 2d)for0nd)]TJ /F10 11.955 Tf 12.01 0 Td[(1,i.e.,theslopesare1 d,2 d,,d)]TJ /F6 7.97 Tf 6.59 0 Td[(1 d. IwilldiscussabouttheexponentialsumsofPulita'sF-isocrystalswhenm1bystudyingtheNewtonpolygonofthecorrespondingL-functions. 4.1ReplacingPulita'sFrobenius Let(M,r,)bearankoneF-isocrystaloverRK.LetfugbeabasisofM,andlet(u)=(T)(u).Supposethatthereareover-convergent~(T)andg(T)suchthat(T) ~(T)=g(T) (g(T)).Dene~(x)=~(T)(x).ThenthemapM!M,u7!g(T)u,isanisomorphismwhichcommuteswithFrobenius.Infact,(g(T)u)=(g(T))(u)=(g(T))(T)(u)=~(T)g(T)(u)=g(T)~(u). Sowehaveanisomorphism(M,r,)'(M,~r,~). 30

PAGE 31

NotethattheexponentialsumsofF-isocrystal(M,~r,~)isequaltotheoneof(M,r,).SincethelargertheradiusofconvergenceofFrobenius,thebettertheestimatesfortheNewtonpolygon,I'dliketoreplacethePulita'sFrobeniusbyanewFrobeniuswithbetterradiusofconvergence. Theorem4.4. LetmbearootoflogAH(X)oforder1 pm(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1).Form=0,1,thereexistover-convergentfunctionsgm(T)suchthat m(T) AH(mT)=gm(T) gm(Tp). Proof. WecanwriteAH(mT)asAH(mT)=exp(mT+(m+pm p)Tp+) exp(mTp+(m+pm p)Tp2+) SincemisarootofPXpim pi=0oforder1=pm(p)]TJ /F10 11.955 Tf 12.81 0 Td[(1),ordp(m+pm p++pkm pk)=pk+1)]TJ /F14 5.978 Tf 5.75 0 Td[(m p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[(k)]TJ /F10 11.955 Tf 11.96 0 Td[(1andexpf(m+pm p++pkm pk)Tpkgisover-convergentwhenkm+1. Letm=0.Wecanwrite0(T) AH(0T)=g0(T) g0(Tp), whereg0(T)=exp((~0)]TJ /F3 11.955 Tf 11.95 0 Td[(0)T)]TJ /F10 11.955 Tf 11.95 0 Td[((0+p0 p)Tp)-222().Sotoshowg0(T)isover-convergent,itsufcestoshowthatexp((~0)]TJ /F3 11.955 Tf 11.95 0 Td[(0)T)isover-convergent. Sinceordp(0)=ordp(~0)=1 p)]TJ /F6 7.97 Tf 6.58 0 Td[(1,write0=a~0,whereordp(a)=0.LetY=a)]TJ /F10 11.955 Tf 12.11 0 Td[(1.Then, 0=1Xi=0(a~0)pi pi=1Xi=0((1+Y)~0)pi pi=~0+~p0 p+~p20 p2+~0+~p0+~p20Y+, 31

PAGE 32

andYisarootof0+1X+2X2+=0oforder0,where0=~0+~p0 p+~p20 p2+,1=~0+~p0+~p20+.Wehaveordp(0)=p2)]TJ /F6 7.97 Tf 6.59 0 Td[(2p+2 p)]TJ /F6 7.97 Tf 6.59 0 Td[(1,ordp(1)=1 p)]TJ /F6 7.97 Tf 6.59 0 Td[(1,andordp(k)1 p)]TJ /F6 7.97 Tf 6.58 0 Td[(1fork2.So,0+1X+2X2+=0hasarootoforderp)]TJ /F10 11.955 Tf 12.31 0 Td[(1.Choose0and~0suchthatord(a)]TJ /F10 11.955 Tf 12.36 0 Td[(1)=p)]TJ /F10 11.955 Tf 12.36 0 Td[(1.Thenord(0)]TJ /F10 11.955 Tf 12.75 0 Td[(~0)=ord(~0(a)]TJ /F10 11.955 Tf 12.35 0 Td[(1))=1 p)]TJ /F6 7.97 Tf 6.59 0 Td[(1+p)]TJ /F10 11.955 Tf 12.35 0 Td[(1andexp((~0)]TJ /F3 11.955 Tf 11.95 0 Td[(0)T)convergesforordp(T)>)]TJ /F4 11.955 Tf 9.3 0 Td[(p+1. Letm=1.Write1(T) AH(1T)=g1(T) g1(Tp), whereg1(T)=exp((~1)]TJ /F3 11.955 Tf 11.95 0 Td[(1)T+(~0 p)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 13.15 6.32 Td[(p1 p)Tp)]TJ /F10 11.955 Tf 11.96 0 Td[((1+p1 p+p21 p2)Tp2)-222().Sog1(T)isover-convergentifexp((~1)]TJ /F3 11.955 Tf 11.96 0 Td[(1)T)andexp((~0 p)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 13.15 6.32 Td[(p1 p)Tp)are.Wehave0=1Xi=0(a~1)pi pi=1Xi=0((1+Y)~1)pi pi=~1+~p1 p+~p21 p2+~1+~p1+~p21Y+, where1=a~1,withordp(a)=0andY=a)]TJ /F10 11.955 Tf 9.43 0 Td[(1.SoYisarootof0+1X+2X2+=0oforder0,where0=~1+~p1 p+~p21 p2and1=~0+~p0+~p20+.Wehaveordp(0)=1+1 p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1),ordp(1)=1 p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1),andordp(i)8><>:1 p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)if2ip)]TJ /F10 11.955 Tf 11.96 0 Td[(12)]TJ /F9 7.97 Tf 6.59 0 Td[(p p)]TJ /F6 7.97 Tf 6.58 0 Td[(1ifip Hence0+1X+2X2+=0hasarootoforder1.Chooseand~1sothatord(a)]TJ /F10 11.955 Tf 12.25 0 Td[(1)=1.Thenord(1)]TJ /F10 11.955 Tf 12.65 0 Td[(~1)=ord(~1(a)]TJ /F10 11.955 Tf 12.25 0 Td[(1))=1 p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)+1andexp((~1)]TJ /F3 11.955 Tf 11.95 0 Td[(1)T)convergesforordp(T)>)]TJ /F9 7.97 Tf 10.49 5.04 Td[(p2)]TJ /F6 7.97 Tf 6.59 0 Td[(2p+1 p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1).Sinceordp(~0 p)]TJ /F3 11.955 Tf 13.24 0 Td[(1)]TJ /F13 7.97 Tf 14.44 6.32 Td[(p1 p)ordp(~1)]TJ /F3 11.955 Tf 13.24 0 Td[(1),exp((~0 p)]TJ /F3 11.955 Tf 11.96 0 Td[(1)]TJ /F13 7.97 Tf 13.15 6.32 Td[(p1 p)Tp)convergesforordp(T)>)]TJ /F9 7.97 Tf 10.5 5.04 Td[(p2)]TJ /F6 7.97 Tf 6.59 0 Td[(2p+1 p2(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1). 32

PAGE 33

Letf(T)=Pki=1(0iTi,1iTi)2W1(TOK2[T])).ThenPulita'sFrobeniuscanbewrittenas ep(f(T),1) ep(fF(T),1)=kYi=1ep((0iTi,1iTi),1) ep((p0iTpi,p1iTpi),1)=kYi=1E1(0iTi) E1(p0iTpi)E0(1iTi)) E0(p1iTpi)=kYi=11(0iTi)0(1iTi). Hencethereisanover-convergentfunctiong(T)suchthat ep(f(T),1) ep(fF(T),1)=g(T) (g(T))kYi=1AH(10iTi)AH(01iTi). Soep(f(T),1)=ep(fF(T),1)canbereplacedbyQki=1AH(10iTi)AH(01iTi). 4.2NewtonPolygonoftheL-Function NotethatbytheproofofLemma4,iff(T)=(f0(T),f1(T))2W1(Fp[T])withdandedegreesoff0(T)andf1(T),respectively,thenf(T)canbewrittenaskXi=1(0iTi,1iTi), wherek=maxfe,dp)]TJ /F10 11.955 Tf 11.95 0 Td[(1g. Lemma8. DeneG(X)=kYi=1AH(1^0iXi)AH(0^1iXi). ThenG(X)=P1n=0Gn()Xn,whereordp(Gn())n=kp(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1). Proof. WriteAH(0X)=P1m=0mXmandAH(1X)=P1m=0mXm.Then,ordp(m)m p)]TJ /F10 11.955 Tf 11.95 0 Td[(1;ordp(m)m p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1); 33

PAGE 34

andfor0mp)]TJ /F10 11.955 Tf 11.95 0 Td[(1wehavem=m0 m!andordp(m)=m p)]TJ /F10 11.955 Tf 11.95 0 Td[(1; m=m1 m!andordp(m)=m p(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1). Let~=(^01,,^0k,^11,,^1k).Then,G(X):=kYi=1AH(1^0iXi)AH(0^1iXi)=(1Xm1=0m1^m101Xm1)(1Xmk=0mk^mk0kXkmk)(1X~m1=0~m1^~m111X~m1)(1X~mk=0~mk^~mk1kXk~mk)=1Xn=0Gn(~)Xn, whereGn(~)=Xmi,mj0Pki=1i(mi+~mi)=nm1mk~m1~mk~~m;~m=(m1,,mk.~m1,,~mk)and~~m=^m101^mk0k^~m111^~mk1k.Ask(m1++mk+~m1++~mk)Pki=1i(mi+~mi)=n,getordpGn()minfm1++mk p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1)+~m1++~mk p)]TJ /F10 11.955 Tf 11.95 0 Td[(1gminfm1++mk+~m1++~mkg p(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1)n kp(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1). Nowdene p(Xv)=8><>:Xv=pifpjv0otherwise 34

PAGE 35

Let:= pG(X).AnargumentusingPoincareduality[ 5 ]showsthatisthelineardualofFrobenius.LetFbeamatrixrepresentationof.Then,L((f0,f1);T)=det(I)]TJ /F4 11.955 Tf 11.95 0 Td[(FT) det(I)]TJ /F4 11.955 Tf 11.96 0 Td[(FpT). NotethatF=fGpi)]TJ /F9 7.97 Tf 6.58 0 Td[(j(~)gi,j0.LetC0(~)=1,andforeveryn1letCn(~):=X1u11Case Letf(T)=(f0,,fm)anddeg(fi)=di.ByProposition 3.1 ,APulita'sFrobeniusep(f(T),1) ep(fF(T),1) 35

PAGE 36

canbewrittenas~d0Yi=1Em(~a0iTi) Em(~ap0iTpi)~d1Yi=1Em)]TJ /F6 7.97 Tf 6.58 0 Td[(1(~a1iTi) Em)]TJ /F6 7.97 Tf 6.59 0 Td[(1(~ap1iTpi)~dmYi=1E0(~amiTi) E0(~apmiTpi), where~dk=maxfdk,dk)]TJ /F6 7.97 Tf 6.58 0 Td[(1p)]TJ /F10 11.955 Tf 11.95 0 Td[(1,dk)]TJ /F6 7.97 Tf 6.59 0 Td[(2p2)]TJ /F10 11.955 Tf 11.95 0 Td[(1,d0pk)]TJ /F10 11.955 Tf 11.95 0 Td[(1g. Whenm2,itisnotknownwhetherthestatementofTheorem 4.4 istrue.However,wecanstillreplacethisFrobeniusby~d0Yi=1Em(~a0iTi) Em(~ap0iTpi)~dm)]TJ /F16 5.978 Tf 5.76 0 Td[(2Yi=1E2(~am)]TJ /F6 7.97 Tf 6.59 0 Td[(2,iTi) E2(~apm)]TJ /F6 7.97 Tf 6.59 0 Td[(2,iTpi)~dm)]TJ /F16 5.978 Tf 5.75 0 Td[(1Yi=1AH(1am)]TJ /F6 7.97 Tf 6.59 0 Td[(1,iTi)~dmYi=1AH(0amiTi). Considerthecasem=2.Supposethatp>3. Lemma9. 2Yi=1exp(~p2)]TJ /F9 7.97 Tf 6.58 0 Td[(i)]TJ /F10 11.955 Tf 12.36 0 Td[(~pi+12)Tpi+1 pi+1 convergeswhenord(T)>)]TJ /F9 7.97 Tf 10.49 5.03 Td[(p)]TJ /F6 7.97 Tf 6.58 0 Td[(3 p4. Proof. Wehave 2Yi=1exp(~p2)]TJ /F9 7.97 Tf 6.59 0 Td[(i)]TJ /F10 11.955 Tf 12.36 0 Td[(~pi+12)Tpi+1 pi+1=exp((p~2+~p2)p)]TJ /F10 11.955 Tf 12.35 0 Td[(~p22)Tp2 p2+((p~1+~p1)p)]TJ /F10 11.955 Tf 12.35 0 Td[(~p32)Tp3 p3=expp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=0pi(p~2)p)]TJ /F9 7.97 Tf 6.58 0 Td[(i(~p2)iTp2 p2expp)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=0pi(p~1)p)]TJ /F9 7.97 Tf 6.58 0 Td[(i(~p1)iTp3 p3expp2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=0p2i(p~2)p2)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p2)iTp3 p3=expp)]TJ /F6 7.97 Tf 6.59 0 Td[(2Xi=0pi(p~2)p)]TJ /F9 7.97 Tf 6.58 0 Td[(i(~p2)iTp2 p2expp)]TJ /F6 7.97 Tf 6.59 0 Td[(2Xi=0pi(p~1)p)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p1)iTp3 p3exp~1+p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)2Tp2+~1+p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)1Tp3 pexpp2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=0p2i(p~2)p2)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p2)iTp3 p3. 36

PAGE 37

Sinceordp)]TJ /F6 7.97 Tf 6.58 0 Td[(2Xi=0pi(p~2)p)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p2)ip)]TJ /F6 7.97 Tf 6.59 0 Td[(21+2+p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(2) p2(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1) andordp)]TJ /F6 7.97 Tf 6.58 0 Td[(2Xi=0pi(p~1)p)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p1)ip)]TJ /F6 7.97 Tf 6.59 0 Td[(32+p(p)]TJ /F10 11.955 Tf 11.95 0 Td[(2) p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1),expp)]TJ /F6 7.97 Tf 6.59 0 Td[(2Xi=0pi(p~2)p)]TJ /F9 7.97 Tf 6.58 0 Td[(i(~p2)iTp2 p2expp)]TJ /F6 7.97 Tf 6.59 0 Td[(2Xi=0pi(p~1)p)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p1)iTp3 p3 convergeswhenord(T)>)]TJ /F9 7.97 Tf 10.49 5.03 Td[(p2)]TJ /F6 7.97 Tf 6.59 0 Td[(2 p4.Alsoasordp2)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=0p2i(p~2)p2)]TJ /F9 7.97 Tf 6.58 0 Td[(i(~p2)ip)]TJ /F6 7.97 Tf 6.59 0 Td[(31+p(p2)]TJ /F10 11.955 Tf 11.95 0 Td[(1) p2(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1),expp2)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=0p2i(p~2)p2)]TJ /F9 7.97 Tf 6.59 0 Td[(i(~p2)iTp3 p3 convergeswhenord(T)>)]TJ /F9 7.97 Tf 10.49 5.04 Td[(p2)]TJ /F6 7.97 Tf 6.59 0 Td[(1 p5.Finally,wehave exp~1+p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)2Tp2+~1+p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)1Tp3 p=E(~1+p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)2Tp2)exp(~1+p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)1)]TJ /F10 11.955 Tf 12.35 0 Td[(~p+p2(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)2)Tp3 p1Yi=2exp)]TJ /F10 11.955 Tf 13.15 8.09 Td[((~1+p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)2Tp2)pi pi. E(~1+p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)2Tp2)convergeswhenord(T)>)]TJ /F6 7.97 Tf 10.49 5.48 Td[(1+p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1) p4(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1),andQ1i=2exp()]TJ /F6 7.97 Tf 10.49 6.32 Td[((~1+p(p)]TJ /F16 5.978 Tf 5.75 0 Td[(1)2Tp2)pi pi)convergeswhenord(T)>)]TJ /F9 7.97 Tf 10.49 5.03 Td[(p)]TJ /F6 7.97 Tf 6.58 0 Td[(2 p4.Since exp(~1+p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)1)]TJ /F10 11.955 Tf 12.35 0 Td[(~p+p2(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)2)Tp3 p=exp)]TJ /F9 7.97 Tf 11.96 15.65 Td[(p(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1)Xi=01+p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1)i~i1()]TJ /F4 11.955 Tf 9.3 0 Td[(p~2)p(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)+1)]TJ /F9 7.97 Tf 6.59 0 Td[(iTp3 p, itconvergeswhenord(T)>)]TJ /F9 7.97 Tf 10.49 5.03 Td[(p2)]TJ /F9 7.97 Tf 6.59 0 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1 p5. Hence2Yi=1exp(~p2)]TJ /F9 7.97 Tf 6.58 0 Td[(i)]TJ /F10 11.955 Tf 12.36 0 Td[(~pi+12)Tpi+1 pi+1 37

PAGE 38

convergeswhenord(T)>maxf)]TJ /F4 11.955 Tf 16.47 8.09 Td[(p2)]TJ /F10 11.955 Tf 11.95 0 Td[(2 p4,)]TJ /F4 11.955 Tf 10.5 8.09 Td[(p2)]TJ /F10 11.955 Tf 11.96 0 Td[(1 p5,)]TJ /F4 11.955 Tf 10.49 8.09 Td[(p2)]TJ /F4 11.955 Tf 11.95 0 Td[(p+1 p4(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1),)]TJ /F4 11.955 Tf 10.49 8.09 Td[(p2)]TJ /F4 11.955 Tf 11.95 0 Td[(p+1 p5g=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(p2)]TJ /F4 11.955 Tf 11.95 0 Td[(p+1 p5,andthelemmaholds. Proposition4.1. E2(T)=E2(Tp)convergeswhenordp(T)>)]TJ /F9 7.97 Tf 10.49 5.04 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(3 p4. Proof. Set~)]TJ /F6 7.97 Tf 6.58 0 Td[(1=0.Recallthat~1)]TJ /F9 7.97 Tf 6.59 0 Td[(i)]TJ /F4 11.955 Tf 11.95 0 Td[(p~2)]TJ /F9 7.97 Tf 6.59 0 Td[(i=~p2)]TJ /F9 7.97 Tf 6.59 0 Td[(ifor0i2.Thenwehave E2(T) E2(Tp)=exp2Xi=0~2)]TJ /F9 7.97 Tf 6.59 0 Td[(iTpi pi)]TJ /F6 7.97 Tf 18.37 14.94 Td[(2Xi=0~2)]TJ /F9 7.97 Tf 6.58 0 Td[(iTpi+1 pi=exp(~2T)exp2Xi=0(~1)]TJ /F9 7.97 Tf 6.59 0 Td[(i)]TJ /F4 11.955 Tf 11.96 0 Td[(p~2)]TJ /F9 7.97 Tf 6.58 0 Td[(i)Tpi+1 pi+1=exp~2T+2Xi=0~p2)]TJ /F9 7.97 Tf 6.59 0 Td[(iTpi+1 pi+1=E(~2T)2Yi=1exp(~p2)]TJ /F9 7.97 Tf 6.59 0 Td[(i)]TJ /F10 11.955 Tf 12.35 0 Td[(~pi+12)Tpi+1 pi+11Yi=3exp)]TJ /F10 11.955 Tf 13.15 8.08 Td[((~2T)pi+1 pi+1. E(~2T)convergesifandonlyiford(T)>)]TJ /F4 11.955 Tf 9.3 0 Td[(ord(~2)=)]TJ /F6 7.97 Tf 23.79 4.71 Td[(1 p2(p)]TJ /F6 7.97 Tf 6.59 0 Td[(1). exp()]TJ /F6 7.97 Tf 10.5 5.48 Td[((~2T)pi+1 pi+1)convergesifandonlyifpi+1ord(~2T))]TJ /F10 11.955 Tf 11.96 0 Td[((i+1)>1 p)]TJ /F6 7.97 Tf 6.58 0 Td[(1ifandonlyiford(T)>1 p)]TJ /F6 7.97 Tf 6.59 0 Td[(1+i+1 pi+1)]TJ /F10 11.955 Tf 34.37 8.09 Td[(1 p2(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1). Sincetheright-handsideisanincreasingfunctionfori1,wehave1 p)]TJ /F6 7.97 Tf 6.58 0 Td[(1+i+1 pi+1)]TJ /F10 11.955 Tf 34.37 8.08 Td[(1 p2(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1)1 p)]TJ /F6 7.97 Tf 6.59 0 Td[(1+3 p3)]TJ /F10 11.955 Tf 34.37 8.08 Td[(1 p2(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1)=)]TJ /F4 11.955 Tf 9.3 0 Td[(p+3 p4. HenceQ1i=3exp()]TJ /F6 7.97 Tf 10.5 5.48 Td[((~2T)pi+1 pi+1)convergesifandonlyiford(T)>)]TJ /F9 7.97 Tf 10.49 5.03 Td[(p)]TJ /F6 7.97 Tf 6.58 0 Td[(3 p4. Bythepreviouslemma,theradiusofconvergenceof2Yi=1exp((~p2)]TJ /F9 7.97 Tf 6.58 0 Td[(i)]TJ /F10 11.955 Tf 12.36 0 Td[(~pi+12)Tpi+1 pi+1) islessthan)]TJ /F9 7.97 Tf 10.5 5.04 Td[(p)]TJ /F6 7.97 Tf 6.59 0 Td[(3 p4.Thusthepropositionfollows. 38

PAGE 39

Lemma10. Letk=maxf~d0,~d1,~d2g.DeneG(X)=~d0Yi=1E2(^0iXi) E2(^p0iXpi)~d1Yi=1AH(1^1iXi)~d2Yi=1AH(0^2iXi). ThenG(X)=P1n=0Gn()Xn,whereordp(Gn())n(p)]TJ /F10 11.955 Tf 11.96 0 Td[(3)=kp4. Proof. WriteAH(0X)=P1m=0mXm,AH(1X)=P1m=0mXmandE2(X)=E2(Xp)=P1m=0mXm.Then,ordp(m)m p)]TJ /F10 11.955 Tf 11.96 0 Td[(1;ordp(m)m p(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1);ordp(m)m(p)]TJ /F10 11.955 Tf 11.96 0 Td[(3) p4; andfor0mp)]TJ /F10 11.955 Tf 11.95 0 Td[(1wehavem=m0 m!andordp(m)=m p)]TJ /F10 11.955 Tf 11.95 0 Td[(1;m=m1 m!andordp(m)=m p(p)]TJ /F10 11.955 Tf 11.95 0 Td[(1);m=m2 m!andordp(m)=m p2(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1). Let~=(^01,,^0~d0,^11,,^1~d1,^21,,^2~d2).Then, G(X):=~d0Yi=1E2(~a0iXi) E2(~ap0iXpi)~d1Yi=1AH(1a1iXi)~d2Yi=1AH(0a2iXi)=(1Xl1=0l1^l101Xl1)(1Xl~d0=0l~d0^l~d00~d0X~d0l~d0)(1Xm1=0m1^m111Xm1)(1Xm~d1=0m~d1^m~d11~d1X~d1m~d1)(1X~m1=0~m1^~m121X~m1)(1X~m~d2=0~m~d2^~m~d22~d2X~d2~m~d2)=1Xn=0Gn(~)Xn, (4) 39

PAGE 40

whereGn(~)=Xli,mj,~mk0l1l~d0m1m~d1~m1~m~d2~~m; wherethesumistakenoverP~d0i=1ili+P~d1i=1imi+P~d2i=1i~mi=nand~m=(l1,,l~d0,m1,,m~d1,~m1,,~m~d2);~~m=^l101^l~d00~d0^m111^m~d11~d1^~m121^~m~d22~d2. ThenwegetordpGn()minn(p)]TJ /F10 11.955 Tf 11.96 0 Td[(3)(l1+l~d0) p4+m1++m~d1 p(p)]TJ /F10 11.955 Tf 11.96 0 Td[(1)+~m1++~m~d2 p)]TJ /F10 11.955 Tf 11.96 0 Td[(1o(p)]TJ /F10 11.955 Tf 11.96 0 Td[(3)minfl1++l~d0+m1++m~d1+~m1++~m~d2g p4n(p)]TJ /F10 11.955 Tf 11.96 0 Td[(3) kp4. Nowusingthesameargumentasthem=1case,weobtain(1)]TJ /F4 11.955 Tf 11.96 0 Td[(T)L((f0,f1,f2);T)=L((f0,f1,f2);T)=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(T)P1n=0()]TJ /F10 11.955 Tf 9.3 0 Td[(1)nCn(~)Tn (1)]TJ /F4 11.955 Tf 11.95 0 Td[(pT)P1n=0()]TJ /F10 11.955 Tf 9.3 0 Td[(1)nCn(~)pnTn, whereC0(~)=1,andforeveryn1Cn(~):=X1u1
PAGE 41

CHAPTER5CONCLUSIONS Inthiswork,weinvestigatedcertainkindsofexponentialsumsintroducedbyA.PulitabystudyingthecorrespondingL-functions.Inchapter2,wereviewedthemostgeneralunit-rootF-isocrystalonA1correspondingtoacharacterof1(Speck((t)]TJ /F6 7.97 Tf 6.59 0 Td[(1))),absoluteGaloisgroupofk((t)]TJ /F6 7.97 Tf 6.59 0 Td[(1)),ofp-powerorderintroducedbyPulita.ThenthecorrespondingexponentialsumsandtheL-functionswereintroduced. Inchapter3,wecomputedalowerboundofthedegreeofPulita'sF-isocrystal.Iff(X)=(f0(X),,fm(X))2Wm(Fp[X])anddi=deg(fi),thenthedegreeoftheL-functionintroducedinchapter2hasdegreelessthanorequalto,where=maxfdopm)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d1pm)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 11.96 0 Td[(1,,dm)]TJ /F6 7.97 Tf 6.59 0 Td[(1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,dm)]TJ /F10 11.955 Tf 11.96 0 Td[(1g. Inchapter4,weestimatedthep-adicabsolutevaluesofthezerosoftheL-function.Toobtainbetterestimates,werstreplacedtheFrobeniusintroducedbyPulitabyanewFrobeniuswithlargerradiusofconvergence.UsinganewFrobenius,weestimatedthep-adicabsolutevaluesofthezerosoftheL-function.Whenm=1,alowerboundoftheNewtonpolygonoftheL-functionoftheF-isocrystalisthelowerconvexhullofpoints(n,n(n+1) 2kp)for0nmaxfd0p)]TJ /F10 11.955 Tf 11.95 0 Td[(1,d1)]TJ /F10 11.955 Tf 11.96 0 Td[(1g,wherek=maxfd0p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d1g.Whenm=2,alowerboundoftheNewtonpolygonoftheL-functionoftheF-isocrystalisthelowerconvexhullofpoints(n,n(n+1)(p)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(p)]TJ /F6 7.97 Tf 6.58 0 Td[(3) 2kp4)for0nmaxfd0p2)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d2)]TJ /F10 11.955 Tf 11.95 0 Td[(1g,wherek=maxfd0p2)]TJ /F10 11.955 Tf 11.95 0 Td[(1,d1p)]TJ /F10 11.955 Tf 11.96 0 Td[(1,d2g. 41

PAGE 42

REFERENCES [1] E.Bombieri,Onexponentialsumsinniteelds,Amer.J.Math.88(1966),71. [2] R.Crew,F-isocrystalsandp-adicrepresentations,Proc.Symp.PureMath.46(1987),111. [3] ,Canonicalextensions,irregularities,andtheSwanconductor,MathematischeAnnalen316(2000),19. [4] B.Dwork,Ontherationalityofthezetafunctionofanalgebraicvariety,Amer.J.Math.82(1960),631. [5] ,Onthezetafunctionofahypersurface,Inst.HautesE'tudesSci.Publ.Math.12(1962),5. [6] ,Besselfunctionsasp-adicfunctionsoftheargument,DukeMath.J.41(1974),711. [7] B.Dwork(ed.),Lecturesonp-adicdifferentialequations,Springer-Verlag,NewYork,1980. [8] B.Dwork,G.Gerotto,andF.J.Sullivan,AnintroductiontoG-functions,PrincetonUniversityPress,Princeton,1994. [9] N.Katz,TravauxdeDwork,SeminaireBourbaki24(1973),167. [10] ,Gausssums,kloostermansums,andmonodromygroups,PrincetonUniversityPress,NewJersey,1988. [11] S.Matsuda,Localindicesofp-adicdifferentialoperatorscorrespondingtoArtin-Schreier-Wittcoverings,DukeMathematicalJournal77(1995),607. [12] A.Pulita,Equationsdifferentiellesp-adiquesd'ordreunetapplications,PhDThesis,2006. [13] P.Robba,Indexofp-adicdifferentialoperators.III.Applicationtotwistedexponen-tialsums.p-adiccohomology,Aste'risqueNo.119-120(1984),191. [14] ,UneintroductionnaiveauxcohomologiesdeDwork,Me'm.Soc.Math.FranceNo.23(1986),61. [15] J.P.Serre,Localelds,Springer-Verlag,NewYork,1979. [16] S.Sperber,Congruencepropertiesofthehyperkloostermansum,CompositioMath.40(1980),3. [17] N.Tsuzuki,Finitelocalmonodromyofoverconvergentunit-rootF-isocrystalsonacurve,Amer.J.Math.120(1998),1165. 42

PAGE 43

[18] ,ThelocalindexandtheSwanconductor,CompositioMath.111(1998),2711. [19] D.Wan,NewtonpolygonsofzetafunctionsandL-functions,Ann.ofMath.137(1993),249. [20] A.Weil,Numbersofsolutionsofequationsinniteelds,Bull.Amer.Math.Soc.55(1949),497. [21] H.J.Zhu,p-adicvariationofL-functionsofonevariableexponentialsums.I.,Amer.J.Math.125(2003),669. 43

PAGE 44

BIOGRAPHICALSKETCH YuriMorofushiwasbornin1978inJapan.SheearnedherBachelorofArtsdegreeinmathematicsfromUniversityofArizonain2002.YurithenstartedgraduateschoolattheUniversityofFloridatocontinueherstudiesinmathematics.SheearnedaMasterofSciencedegreein2004,andcompletedherPh.D.in2010. 44