Table Of ContentLECTUREOSN
p-ADILC- FUNCTIONS
BY
KENKICHIIW ASAW A
PRINCETON UNIVERSITY PRESS
AND
UNNERSITYO F TOKYO PRESS
PRINCETONN,E W JERSEY
1972
AnnalosfM athematiSctsu dies
Numbe7r4
Copyrig©h t1 972� y PrincetUonni versity Press
Allr ighrtess ervNeod .p arto ft his
bookm ay reproduicnea dn yf ormo r
be
bya nye lectroonrmi ecc hanimceaal ns
includiinnfgo rmatsitoonr aangde retrieval
systewmist hopuetr missiownr iting
in
thep ublisheexrc,e pbty a reviewer
from
mayq uote passagiens rae view.
who brief
LC Card7:8 -39058
0-691-08112-3
ISBN:
1971:1 0.14, 1102..6550,
AMS
PublisihneJ da paenx clusibvye ly
UniversoifTt oyk yoP ress;
ino thepra rtso ft hew orlbdy
PrinceUtnoinv ersPiretys s
Printeidn t heU niteSdt atoefsA merica
PREFACE
Thesaer neo teosfl ectugrievsea ntP rinceUtnoinvs eirtdyu ritnhge
falsle mesotfe1 r9 6T9h.en otes parnei snetnrto dutcopt -iaodnLi c
functions oriinKg uibnoattae-dL e[o1p0ao]slp d -ta dainca logoufce lsa ssi
calL -functoifDo inrsi chlet.
Ano utloifnt eh ceo nteinsat ssf ollowIsn.§ 1c,l assirceasluo lnt s
DirichLl-eftu'nsc tairboern ise rfelvyi ewFeodrs. o moef t hesae s,k tech
ofa p roiosfp r ovidientd h Aep pendIinx§ .2w ,ed efigneen eralBiezred
[121
noulli nfuomlbleorLwsei onpgo ldta ndd iscussosm e off untdhaem en
taplor pertioefts h esneu mberIsn§. 3 w,e intrpo-daudciec L-functions
andp rovteh eex isteanncdte h ue niqueonfes susc fhu nctions; our method
iss lhitgldyi ffefrreontmth aitn[ 10§]4c. o nsiosft sp reliminary remarks
onp -adlioacgr ithamnsdp -adriecg ulatIon§r Ssw,.e p orvea f ormuolfa
Leopof1odtrth ev aluoefsp -adic L-fautns c= t1.i Tohnesf ormuwlaas
announicn[e d1 0bu]tt,h per ohoafs n oyte bte epnu blishWeidt.h pheirs
missiwoend ,e sicbrhee rLee opolodirtg'isnp aorlo f tohffe o rmu(lsae e
[1[]7,f] o rt eam1a atperp oachI)n§. 6 w,ee xplaainno tmheetrh otdod e
finpe- adLi-cf unc.t Hieornwese f ollaoniw d ea[ 9imn]o tivabtyte hde
§7,
studoyfc yclotfoimleidcsI .n wed iscussosma ep plicaotfi otnhse
resuolbttsa iinne dtp hree cedsiencgt iionnsd,i cadteiernpeg l ations
whicehx ibsett wepe-na dLi-cf unctainodcn ysc lotfoimeilcdC so.n clud
ingr emaroknps r obmlsae ndf uutrievn estigaitnti hoinassra e arael so
mentiobnreidea fttl hyee n do f§ 7.
Thruoghotuhtne o teisti, s a ssumtehdat th ree adhearsb asikcn olw
edge algoefb raniucm btehre yo arsp resenftoeerdx ,a mlpei,n B orevich
Shafare[v2oi]rc hL a[n1g1 H]o.w ev,ee xrceipnft e wp lacwehse rcee r
tianf atcso nL -futnicoannsdc lasnsu mbearrsr� e fertrone,do d eeper
v
understoaftn hdtaihtne gom rayyb e r eqruetidof lolotwhe eel mernyt a
argumienmn otssot ft hensoet es.
Asf otrh neo toantssi,o moeft hsey mbuoslestd h routghhneoo utte s
araesf olloZw,Qs ,:R ,a ndC denotthreei onfg( ratiionntaelg)e rs,
thfei eolfrd a tailno unbmertsh,fe i eolfrd e naulm ebr,sa ntdh fei eolfd
complex numbersZ, anrdeQ swpiledlce tniotvthereeli yon.gfp
p p
adiicne tgearnstd h fei eolfpd - anduimcb erress,pv eec,lt yibpe iongf,
corusaep ,r inmuem bIenrg. e neirfaR l i,sa c ommutraitwniigvt aeh
uniRtX,d enotthemesu ltiplgircooauftap il vle invertiinb le elements
R,a ndR [[xt]hr]ei onfga lflro maplo wseerr iinea sni nedteramtixen
wihtc oeffiicniR .e nts
Is houllidk e to exprheeesrt soH .Wm .yL eothpaofnlokkdrist ny d l
permiutstt oii nngc lhuidisepm ortuannptu blriesshuel§dt5 sa, n aidln s o
toR .G erenb]e.rM g.M, a lsye,a nFd. E .G erftohcr a refruelaldyi ng the
manusacnrdi ptm avkailnugas bulgeg esftroi oiintmssp rovement.
KenkiIcwhais awa
PRINCETON, OCTOBER
1971
CONTENTS
PREFAC.....E... ............... ................................ ......................... ............... ..... v
. . . . 3
§1D.ir ichlLe-tf'usn c..t... i.....o.....n... s... ........... .................. .. .... ..........
.. .
§2.G eneraBleirnzoeuNdlu lmib .e....r.....s.. ... ................. .... ........... ........ 7
§3p.- AdLi-ucfn cti.....o.....n.. s.... ....... .... ............... .................................... 17
§4.p -AdLioacgr hmista npd- AdRiecg ul.a....t....o.....r.....s..... ..... .......... 36
§5. ...
CalculoafLt pCi1oX;n).. . ................ ....... .. . ... ......................... .......... 43
.. ..
§6A.n A lterMneatth..e.o... d. .. ............. ........ ........ ......... ...... ................... 66
.................. ...........8.8. ..........................................
§7.S omAep plica.....t ions
. ...
APPEND..I....X. ... .................. ............ ..... .. .................. .............. .....1.0.0. .......
. .
BIBLIOGR...A.....P.....H....Y. .......... .......... ......... ................. ... ............... ..... . lOS
vii
LECTURE S ON
p-ADILC- FUNCTIONS
§1.
DIRICHLLE-TF'USN CTIONS
Int hsiesc otnwi,e s harlelv iseowmo eft hwee ll-kcnloawsnsr iecal
sutIso nD icrhileLt-u'fnsc tiFoonrs .p arnomdoo frdsee tawielr se,f er
[13].
three atdoe[ 2r][ ,6 ],
1.1L.e tn bea postviiei tnegenr::=:1, . A map
x:Z --> C
Driichlet
frmo threi onfg intetgote hrceso mpZfl ieexCl ids c allae d
chaacrtteotr h meo dulus
n ifh aitsth per opertthiaiet)Xs ( ad)e pends
onluyp on retshiecd luaeos fsa mond, I iX)( abX)( Xa()b f)oa rn y
=
1.
ab, inZ ,a ndi iXi()a 0) ';'i afn odn liyfa isp ritmoen :( an,)=
Obviotuhsreleiy sa,n atuornael- tcoo-rornees pboentdnwes enuecceh
Dirichclheatr atcott hemero sdl uuns antdh e cha(ritanhc uets eurasl
sen)so eft hmeu tlipligcraotu(i/pZvn eZ )oXft hree sicdluaers isn g
Z/nZH.en caeD iricchhlaertat cott hemero dulnu issu suaildleyn ti
x
fiweidt thhc eo rrespcohnadriaoncf(gt Z e/rn .Z )
'
Let bXea D iricchhlaertat cot meoard ulmu asn lde mt bea
factonr.D eoiffn eX( a),Z , a by
f
X(a)X '(a) if( an,) 1 ,
= , =
>
= 0 if( an,) 1 .
TheXn isa D iricchhlaertat cott hemerod ulnu.sW es atyh athte
'
charaXc itsei nrd ucferdoX m. DAi richletX chtamoro adacu tleurs
primitive
n isc alled ifX isn oitn dufcreoadmn cyh aratcoat m eord u
lus witmh < n .n ist hecnla letdhc eo ncdoturo fX and is denoted
m
3
--_0' � •• '-"ULUler cnaractweer cso nsidweirl lb e
assumeads p rimitive.
Let X 1a nd X2 bes uch( primitiDvier)i chclheatr actaenrdsl etf 1
and f2 bet her espectciovned uctorThse.n t heries a uni(qpuriem itive)
Dirichclheatr actXe rw ithc onductfo rd ividi£n1g£ 2 sucht hat
fori ntegear sp rimteo f1f 2:( af,l f 2)=' 1. X isc alletdh per od�tUo f
Xl and X2: =0X XlX 2· Notet hatX (a)0= Xl( aX)2 (a)i sn otn ecessarily
an
truief ( a,f 1f 2)> 1 .T hes eto fa llp rimitDiivrei chclheatr actferosrm
abeligarno uwpi thr espetcott hea bove multipliTchaeti idoenn.t iotfy
O O
theg rouips t hep rincicphaalr actXer definbeyd X (a)== 1 fore very
a in Z;t hiiss t heu niquceh aarctewri thc onduct1o.r T hei nverosfe
X ist hec haract52er w hicihs t hec omplex-conjmuagpao tfe X :X (a=0)
X(a)a, € Z.
1.2L.e t X bea Dirichclheatr aecrta ndl et
I
s
L(sX;) = X(n)n.-
n=l
Thes erieosn t her ighcto nvergaebss olutfeolray l lc omplenxu mberss
1
withR e(s>) sot hatL (sX;) definae hso lomorpfhiucn cotf is oinn
theh alf-plwahneer eR e(s>) 1.L (sX;) isc alleDdi richLl-eftu'nsc tion
fort hec haracteXr.
O
Fort hep rincicphaalr actXeOr ,L (sX; ) isn othibnugt t hez eta
functioofRn i emann�," (s).
We shalnle xtd escrisboem ef undamenptraolp ertioefs L (sX;) .F irst,
LesX;) can beex pressaesda ni nfinpitroed uct:
II
L(Xs);= (1- X(p)p-Sl) ,- Re(s>) 1 ,
P
whertehp er odiusct ta koevne arl plr inmuem bse pr.H ence
X)
L(s;'" 0
X)
forR e(>s1) .B ya nalyctoinct inuaLt(iso;n c,a bn ee xtentdoae d
X;, X)
meromhoircfp unctointo hnee n tisr-ep lainfe ; L(s; ish olo
X XC,X ) Xo,
morpheivce rywhbeuirtfe ,= L(s; hasa u niqpuoelo efo rder
1,w itrhe sid1u,ae t s 1.F urtheromnto hrees ,- plaintse a,t isfies
= reX)
af unctieoqnuaaltia osnf oll.o Lwest denottheGe a ussian sum:
f 27Tia
reXX)( -a- )
f
2. e , ,
a=1
anlde t
ifX C-1)1 ,
=
X(
if -1= )- 1.
Then
(�t yr(S(X);s �3I)-sL
= W ()-�2 re�-+O)L( l-s;X)
X
X,
wherre d enottehser �funct52i iostn h,ie n veorfs e and
Thes ameeq ualciatnby ew ritatlesnao s
s X)
L(;sX )= re§X( f 217Le)l;-rrC ss-8)
2 i resc)o s
2
X)
Actuaalllslyu ,c ahn alyptrpioecr toifeL s( s; cabn ep rovfeodar
X)
mucwhi dcelra osfsf unicotnssi mitloaL r( s; (se[e1 1H]o)w.e vienr ,
L;(X ),
s
thcea soefo ur thper oiosfS implaes rk;e tocfsh u cah ropoifs
giviennt hliei ppendix.
