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(NAS Colloquium) Elliptic Curves and Modular Forms (1998)

Chapter: Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms

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Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
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Page 1
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 2
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 3
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 4
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 5
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 6
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 7
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 8
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 9
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 10
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 11
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 12
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 13
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 14
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 15
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 16
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 17
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 18
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 19
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 20
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 21
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 22
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 23
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 24
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 25
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 26
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 27
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 28
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 29
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 30
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 31
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 32
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 33
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 34
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 35
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 36
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 37
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 38
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 39
Suggested Citation:"Papers from a National Academy of Sciences Colloquium on Elliptic Curves and Modular Forms." National Academy of Sciences. 1998. (NAS Colloquium) Elliptic Curves and Modular Forms. Washington, DC: The National Academies Press. doi: 10.17226/6235.
×
Page 40

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F#~w \~. If. ir/+ #T VoL 94 p. l!T09. falter 1997 ~ . . a` go oqulum tar G ~r ~ ~ I ~ ~2 ~ p-~) ~6 ~~ P~- of ~ ~~~ ~~ ('~6 ^~ ~d r ^~ >, 0~ ~ ^~ ~~r ~d ^~ R~6f~, A~ ~- 73-7 ~ /~^ f~ -~) Or Introduction BARRY ~A/ER? AND GIL RUB1N? . ~E .~/z~/ J[~\ bet <>f ~M~l-(~ll~m~ll~s, ~Il~r~Jd Diversity. CamI~:id~c, MA (j2i38; and (l)ep~rtITl~t of ~I~e~Ila1:ics, 01~<~> Solace C:~livers:ily~ (Solumbus, O1I $32~1() The col~quTum "EU~Uc arms ad ~oduIar Irma as held fit 1be National Cadet of Sciences in Washington, DC, Larch 15-17. T996. We topics catered ~ tab coBoqutum have been ^~or~narT~ actKc latch. These topics baa pled cssentiaI role in some of the excih~ recent Cork an cI~ssic~! malady of >-adic Caloisreprescrtations.L Unctions, modular Urals, and the triadic co~grtlenc~s abed satisfy" fascia tabs Oracles by John Contra Robert Coleman Fred Diamond. ~ ~ ~dettc P~rrin-Riou. and Richard T^or) to 1be stub of the problems, including FermaLs Last Theorem. #~ wig Greg d~lLatc Comer of modular curbs and Shimura v~deUes (as continue to be contra 10 ~rtb~r d~v~lopme~1s in Number in ~ arc byword F~ldng~and Ban Rib~t~totb~analylic Theory. be 1] articles lo ~}1~ are be texts of addresses ~umber~1heoretic study of Zeta Unctions and Eisenstcin series amen during this colloquium. T>csc ardors range Mom the of ~assic~I~roups(~(in1be Cute by Nero Sbimura). !lI09

Bloc. Ago. Oaf. and. ~) Vol. 94. pp. 1111 O-T I 11T O3~cr 1997 . .. Colloquium Paper #\NE~1 a. R1B~ ED SHL7O (~HI beta I)~p~lr~:~ll, I<l>~ive:~-sit:y or (~1~:1or-~li~t. Be:~keTe<, CA )472{~)-384{} ABSlR41 Fix ~ isogeny class ~ of semista~e eDiptic curves over Q. Me elements of ~ bare ~ common conductor bicb is ~ squabble positive integer. Let ~ be ~ divisor of ~ wbicb is the product of an Men number of primes-Le~ the discriminan1 of an indennite q~tern10n algebra aver 0. lo ~ me associate ~ certain Sbim~ra curve <~, Bose Jacobian is isogeno~s 10 8~ abeam subvariety offal. abed unique ~ ~ ~ for ~hicb onc tags ~ noncons1~1 map ad: <~} ~ ~ Bose pullback ~ ~ Pica) is I~ecuv~ Me delve of ~ is an tagger ~ Shah depends onk on ~ (and the Had isogeny class Ha. ~ investigate the behavior of ~ as ~ varies. Leaf = ^~: be ~ Two norm on Fop, fibers = ~ ~ the product ~ tog rolatRoJy prime Tars ~ and and Bore ~ is Me dLcdminant of ~r indite quotation division algebra over 0. Assume that the Fourier comments office rational integers so this associated with ~ isogeny days ~ of elliptic carves Nor Q. ~0~ the curves in ~ is ~ dLG~3hed eIemen1~, tbc Prom modular curve ached to Sbimur~ (1) has constructed ~ as an optional quotient of Amp. Bush is the quotioDt of^~ ~ an Began satiety of 1h~ Jacobian. Composing the standard map <~ ~ ^ Ash the quorum ~ (# ~ at, me obtain ~ coming ~ < whose degree 8 is an integer which depends only on ~ ~c integer ~ has haven regarded with intense interest ~r the last decade. For one thing. primes dlvidlng ~ arc 'congruence 6~: /~ #1# ~ ~ 1~= is a mode ~~ between ~ and ~ weTyh~t~o gasp firm on Farm mica has integers coefficients and is orthogonal thunder the Fetersson inner product. (See. c.g~ Section 5 ~ reef. 2 far ~ precise ~1emcnLj Far snoth~p ~ ~ know that ~ ~n1\ good upper bound Far ~ ~ imps the /~L Co~ec~re (3 4~. Bore precise6, as R. Warty Plains in TOt 24, the ~6 Co~ec~re :k>~lo~s Fry the co{~ct~tlral botlnd (For a partial converse, see reef. 30 Wbile ~ is ~ lo calculate in practice (6~. it scams more difficult to manage theor~1ically. Burly (243~ teas summed That bounds are known at present. Thy note concerns rela1Tons beg ~ and analogous of in which #~ is replaced ~ the Jacobian of a Shimur, curve. To deEDe Mesa analogies ~ ~ head to gas a ch~r~ctcr~ assign of ~ in Bob ~ does not appear expJ2i1~. For [ha. note ~1 the map #: X~ ~ Rich ~ Dali ~ ~ may be Pawed as a homomorphism] ~) since jacobia~s of curves (and e~ip1ic curves in particulars ~ canonically s~TPdual. ~c~ . composer ~# ~ End ~ ~ necess~ri} muTUplic~ho~ ~ 60mC 1^ ~ ~ \~d^~1 Idea ~ Sde~ ~2~84~/~7/~1 D 1~5$~/0 P\'.!S is t:lvt~.il>~l~{e <.~:lTi~le >~1 ~1lt~Tl)://'i'W'W.~:IlS,.t#}r7. in!e~I; a momenta r~fTcction sums abut this integor is 8. St 6# be the ~Jogue of [0# in which SL(2. Z) is repl~cd bytbe group of norm-1 unite maxima order ofth~rabon~1 qual~rnion algebra of discriminant at. Let (.6~0 be the Sbi~ura curve ~sociat~d~11b ~ 0~0~ndlet7'= {~ bathe Jacobian of ~ QUO. Tho correspondence of Sbimizu and JscquerLsn~l~ds{7)rcI~l~s~to'~ci~ht)~one~form/'~r she group ~ PRO; the farm /' ~ ~D darned only up to muLip[~1ion by ~ nonz~ro constanE Associated ~ /' is an eDipticcurvc]'~bicb ~pp~arsas~noplim~lquolien16': 7'-+ at' offs Using the techniques of Tibet {8) or the ~cncr~1 theorem of F~ldngs (9~. one prov~sttst ~ and a' arc Ro- ~enous-Le~1bat~'beIongs ~ ~.\V~deOnci~OV) ~ Zastbe composite ~ ~>v, ToTnclud~th~casc7~) ~ ~J iJ] ~rn~rllas bel~,~es~l~6l(~) ..- ~ - (~( Rober~(lO)~nd Berto[~iand Darmon(se~ion50frrLIl) have pointed out that the Gross-Za~ier formula and the co{ccturc of Birch and S~inncr10~Dycr imply rotations bcl~e~n ~ and 80O~) in Q6/~ Bcrtobri and Harmon gaudy lo the possib~i) th~tlh~rc may be ~ simple, precise Dracula ~rthcr~lio 8/8~(A1# Th~relation ~hichthey~vis- ~g~invoKeslocal~c10rsfordl~ebipliccurves~ ~nd/'atthe . ... primes! Wh~c1bese actors mavwellbedin~rent aortae hwoe~iptic curved ~ ~#lignorcthissubllety momc~tarl~ andirtroducc o~lytbose ~C10~ which pertainto~.Suppose.then, Data ~primediv1ding ~sothat~ has mubiplic~tLereduclionat~. Lc1 ~ bethel numberofcomponents ~ lo fiber ~ ~ Probe Baron mod~l~fJ;Le~ ~ = order. wb~rc ~ ~ 1be minimal d\=iminanlof3. As Ads mentioned above 8 control con gFuenEcs beg / and Forms other ~n / in the space S of ~ei~hr~o farms on Fo(#O: an~o~ous#, LD(io controR congruenc~sb~lwc~n~andoth~rformsinthe~-newsubsp~ce of\.At1b~ssmc dme.Iev~Jo~ri~< Daubs such Show of Ribct(12)l~adtolhc~xp~c1~1ion that the ~ contr~lcongru- enceshe~w~n~a~d Mold ~rmsin! Thi~cldsthchourkt~ i#`rmul~t . ~ 877(~) at' 81(~)/ 1 1 0.,. AID Equiv~lently.o~e can consider ~c1~rb~tions ~ = idled where Ada disli~ciprinl0nu~bers.~ ~ Its the product . . . .. <~I~llleven~tln]bOro~f~listinc~pri~es.~DdthefUtlr~u~l]lbOrs>? .'nd!~r~rclat\~lyprimc.Tbe formula displayed above mouD1s tom heuFisticr~1~tion ~ ~ ~ [1] {.~7 re~chf~clori/atio~ \ = ~ Ad. ouch simple Maples show that Eq.1 ~ ootcorr~c1~s~t~ted we ~]lprovelhata ~ubab~ m ^ Mann cf) ~ vend ~ name cabs.

Colloquium Paper: Vibes ad ~k~b~shi 8713~1I) iS Cither 232 OI Tam.* On tho other band. since ~ and an are relative prime me find that <<I, 7, 11. IS) = 1. Pus {~1(I3) = 1008/6 = T68. Simile. ~T133~7) = 1~. The First Assertion of ^~ ~ Elf ~ is an Abel vanity over V am ~ is :1 prim. Tet I(6 <) ba the group of components of tbe bar at ~ of the grog model of ~ Dais group is a Anne Stag group Scheme aver Sync IFS i.e.. a finite ~lbelia~n group fur~is~h~d with a c~l~o~nical action , ...... of Cal<F</f</ bc association P~ I</ () is UnctoriaL Par example. as we noted amp. ~ ~ is an elliptic curve with m~tipEca~v~ reduction Eta, Men ~.~) is a cam group of order r . . a, We maps ~ aIld {' induce homonloFpllisms L:~.6,C:~.~ ~. Here~is~ve~rsio~n<>flhc:lirstasscrtiol]of an Blob ((Z! ~ ~ 40 appears web a precise value. MOORED ~ ^~ ~i . 873~) BALM = --/,~ ~< (~7),~,y,if)2, ~ it, ,. , To pFovetbetbeo~ m.~ccompaTetb~char~ctcr~roupsof s}<ebraicto~i Bach arc associated ~nctoFi~l~ to Ebb mode reduction of/'~ndthe mod grad Salon of! Recall ~attbc farmer reduction is described by 1bc well known theory of Ctrodnik and Donald (20-22~.~hiletbeTatl~rDdlsinto Me genera a^~s~diedby ~e~gnea~d Rapoport(233[AlfOough Debate Add R~poportprovide and Be br2~sidl~us~on of -thecasc O ~ 1,~h~tt me need ~i11:131~1ow from rece~lt:rosults of ~ Buzzard (3).] Cur colr~>ar:isorl~is based on 1:1~c oft- exploil~d c~cu~nco 1bat tbc Lao reductions involve the . . ar~hm~t~ftbesamede~nit~rati~nalquatemi~ns~ebr8: thatal~ebr,~hoscdiscriminantL~. Tost~lethercsu) Hick ~n~eded.~ei~troduc~somc nolatio~:k Dis~nabalian~ariety ~erVand<3aprim~ number.letSb~tb~lori p~rtof1beGberoverF<oftbe \!ro~od~J~ndwrke/(Kf)~rthecharactergroup I-Io~ll~t,(6 G1~.Thus)(~F,/)is:<fr~beli~l~roup~hi~c~lis rnTshed~1thc~mpabbleacdonsofO~l(F</F<~ndEndO ~ Atl~,stinlh~c~se~be~b~ss~m~lshlereductio~at/. tbereh~canonicalbiEncarpai~n~ #: (~ ~ x ~ ~ LIZ which ~asinlroduced by Orotb~ndieck(Thcor~m 10.4 ofre[ ^ · . . ~coDl~n ~1 ~ curve or a product of Jacobian then the monod~>p~ng~\ap~tingon ~6 f)~ ~e~ Pal defined oaths productof1~o copies ofthk gF0Up). Therel~1io~ between I(6 {)s~dlh~cb~racter~roups His as ~IJo~s(Th~orem 11.! ofreE 233:lher~ ~ ~ n~turalex~3 sequence 0~< ~ #~# ~ ~< ~ ~0 ~ which ~ is obtained Mom ~r by the standard formula {~0) = ~# ad. PROFOS1T10~ ~ 7>e~ ~ ~ r~/ ~ ~ # of >~x~/ The #~bcomln~ resume of the second author Which ware mentioned readier should paw ~1 ~(1? ~ ~13. lath = 1 Id that 8723(71) = 232. .~ #~ J^2 #. ~ ~ r~# r~~ ~ = ~ \<k0.7~e~ ~ c ?y<~-z!~/' ~<fJ".~ t 7i<~/e fly pad ~ )!~7-',~) )`~! ~ ):e~/~' ## ~ = #~ ~ ~ ~{ ~ ~ T! \(hcn ~ = t d~ pa p Don ~ pond ~ ~ [> (~ inansllalogo-us~ay.t~:lnksto~ Buz~:lrd'sa~rl~logue(3])ofthe ~ClT?nc~Rapoport:tbeorelll<23~.Tbisthe~e~ise~\plorOd~intbe York ~fJorda~landl.~<n!~26jan~d~L. Y-:tn~273. Levy bathe ~parL'of~),deAned ~rexamplcastbe group otcb~ac~rs~ e (a ~> sucb1b~t7>r = ~# forsH prim to \.[R~c~E tb~1~) ~ 1hc ~1b CoOficienI of<] 11isnotb~rd 10 ch~ctlhat ~ ~ Lomorphicl~ Z ~ndth~lin Until ~ contained ~ ~ #~),vie~ed asssub~roup of >~6 V)via~.lndeed,consid~rtbed~composLionof/~saproduc1 upto progeny of~mpIe abelianv~ictiesovcr Q. One of1be Dolors ~ ~.~hicb occurs with muLiplicky 1,and the 01ber factors are none they correspond to conforms of level dividing ~ whose tab cocf~cicnlscannotcoincldc ~ilblh~ for all primeto}( Hence ~ ~ Qistbetensorproduc1 Huh ~ ofthe cb',acter group ofthcto~ic part ofJr~tbis shows that ) Has rank 1 ~ similar computation sbo~ that n ~ ~+ has rank 1. since ~ occurs up 10 bogeny cx~cUy once in/' and zinced hula mul~plic~t~ercductionat>.be Name of ~ in a, ~) x i/', ~) -is Thus recite; {1 is zero since 73~. ~) ~ torsion free. Eta a generator ~ of ~ and set ~ = ~# at. ^~ arbitrary nonhero element f of ~ ma bo Cretan <~ where ~ ~ ~ nonzero integer. Me Ban ha ~~( 0 = ~= By the theorem of Grolhendicck (25) that was cited above, r~ ~ be interpreted as <# at, ushered ~ ~ generator of Em. Jo gild where z<..1 is the m<~no~l~romy pairings arising from lisle mod ~ r~duc~on of a. Beanie, the ma {: ~ ~/ induces ^~< s homom~~sm /: }) ~ ~ ~6 ~ ad tag dual of ~ induces simian a homomorpb~m A: }6 <) ~ am. At. Thc Go homomorphisms arc adioint with respect lo the ono]~rOnly pairings: ~-(~*~.)~) =z'~13.~)foral1~ ~ )~(~.~),)~ )(~.~). oticc,ho~over. ~1{,~$i~ multiplicsti~ by 6:= i0Q.#kO on ~C4.~).sinccitisiIlduced bylaw cn(1()morp~hism 4'mlliti Sea. %) = Ace. (~(~)j = (. (at) Overall:< ~ a.). O}ltak~in~>to-t)~<encratorof ~4,~). ... . O ll~TlG 3~< = (go: {3, A) j2~T, where me viva )#t ~) as embedded h1 ~ by as. ~ Smear ~r~umcntappbed 10 3' mode yLlds8'# = () : {64' > where 5' ~ 67~7pJ). [T~ provethisr~J~tion.on~ mu vie as~sub~rcup Off #',>)andinterprct~s ~ (< 0.~bere~ = .lheiegidm~cy oftbisiDterprotatio~ st~m~from the com ~ptl1.~ibilil, Angola [, ail. ad a.? 1 We emerge ~#hthe p~(minarv warmup # . ................ ):~-~:~.~' After k~1~ 8' or one aide oflhe oqu~bo~, He son tab a, ^ . . .. /712(~T'777 ~ :1S 1nl~pllc(lbyl~h~ ~llo~i~rl>~r~su~]1: ~\ 2~ ~ ~) ~ ~ ~ (~ ~/ a) -a I64 ~) ~ #' <I-- l#4'.>j~ ~ ~r^~<y{~) C # a' ~'~< " ~ 7~ 67 : T(4. # = ~l'' ~ ~} ~ : #~ ~ = ~ ~

ColJoqulum F<per: Rib~tand Tik~h~shi 77~# Thereto ~r~I~ sre~na~ou~ w~sb~Lprove onK the assertion Veto A. Because ofthe ~ssu~pt6n tb~l( -ad ~ an optim ~ quoden61he map (v ~ -~isin~ct\~. ~ededuceshom1b~thesudec~v~yofthe maponch~r,~er groups id: )!~! <) -a ~64, By. Consider Abe cam mutative diagram with ~xactro~s O ~ ~. T -a Hom(~64 FIZZ)-+ ~(~.q)-+ O T t Q >> If.) ~ Mom.) ~(7 an) TO inwhichtbethreC~er~caImapsarcinducedby63FOF instance. thecentralverUc~1 map ~ Hom<~.~.wherc (*: {64.~} ~- ~ ~ ~j~ani~ec~ve mapbe~ween ~eab~Eangroup~of(\i1c rank] Thaexactn~ssof the rowsis~u~rantcodbv Theorem !1.3 oire[ 25.Becaus~tbeleA~handve~ic~ map T(sudectK~, ~e cokernols ~f Hom{~. Z) and 1he richl-hand ~ mav bc id~ntib~d. ~ ~ ~rar that tbe ord~, of coker(Hom<~*. Z)) coincides ~Lb Ihe order oflhetoF~on subgroup otcoker(~.Slnce ~ ~ ~/) ~ 10r<0n ~ee by tbe deEnidon of /, ~e oblair brs1 1he ., . Tormula ~cok~r(Hom(~.Z~ = () : it4 V)) 7~>~. #~( /~) \(/ 6~X V? (~7j ~11113 extended as need~d in v~ ofrosubs of Buzzard (31) ~od Jordan ~d LivnS<263. ~lt:j~l~l~owsYrolIllbe Eichl~r-Sbinlur~r~J~1io~nl~bat~l~1leim~ge of(isannibFatedby~) - ~ - I ~r ~l,.Onededuc~strom tbPlhatibe{-primaryp~rt~fthelmage ~trKi~Llinot.tbeo ~/~) ~ r + 1 mod ~ ~r U1 ~ aod 1hk impIi~ thal tbc semisimpli~calion of]I<]isr~ducible;cL The3rem 3.2(c)of .~f i 1~. i Toprovethesecondstatemcnl,~cbegi~bynotin~tb,1 ? ~ ~ ~)divides#.^sw~poinledou1carber~there~ an~ogenyX ~ ~'ofprimeeto~Edc~ree.Lldeed.~ snd/' arclso~enouso~erQ;o~tbeotberhand.lb~h~otb~shon J[{limpliestba1~^ ration~l~o~eny) ~ ~ oide~rcc dividblebyt~ctorsthrou~htbemulEpIicalion-by^<msSon .Hencethc<-prim~ryco~ponentsof~61~)and~3',~) a:~3omorpbic.solh~ttbel~r~estpow~rsoffIn ~ ~nd 4 arethes~me.Tbus~.~ #)divid~sr~ ~ls~,~e~n . . .. ~n~lo~ous re~sorln~ sto~s 1h~t r~ ~nd ~) bavc 1bc samc valuaLons~1~.~63 ~ ~ i0 d~pendssym m~tdc~l~ o~< a~d ~,~sasserted. FTnally,~(Z! ~ ~. Af) divides both r~nd ~. ~ncc ~ divides ~ and d~pe~dssym metHc~#y o~ ~nd ~. ~= '~dthenlhe desked equa[1y. ~e(~.~.~. ~) ~ ~(~.~.6 ~) lbe Second AsserGon of ^~~ We ~ssume ~Frol~ ~]~ <>n 1~hat ~ ~is squ~r~ -Frec ~1nd tll~ll ~ is p~rime ~1- ~h~icb ~[/J is irreducil~le. ~e shotlld nlentior: in pas<rS ~ 1he i=~duc~Uity ~oth~3 bolds ~r ~e ~ ~ ~ ~if ~lld on~ly if i1 ~holds for aI~l ~ ~ .~. Illdeed. 1~lle semisimpli- ~cation of lLc ~od ~ G~lob r~prcsent~tion J1~] dcpends only . . ~ ~ . , s~mlslmpl~lcatlo~isi~d~Uble. ~LE~\ 2. ~Z~ 7S ~ ~ ~ I~ #~< 1~(!) ~ (/OPi /~()/ j/1'/~3 (~.. /~< Suppos~to ~ contr~rylh~1! d~id~s~ ~r a~ d~ Th~n1~ mo<J~ Gal<O/~reprcsel~1ation ~[~1iS/7~/f~ al primes (seclion 4.1 otr~f. 28~.If ~ = 2,lhis contrsdicts a [hcor~m ofTatc(293.IfS ~ 2.athcorcm ofth~ ~rstauthor (lhe~r~m 1.I otre[ I2)impIiesth~t~[~]is modularofIev~l linlhcsensc1hatilarkes (om th~sp~c)of~eT~bt#~cusp ~rms on SL<2. Z) Sinco {b~ spac~ ~ zero. ~ bbta~ contr~dictloninlhiscasc as ~elt ~ T~ orderto prov~thesecond as~erhon of7~ 7,~bicb coj~lcerllst~h~ (-p~tr~' of{~} ~ ~.~/j,~e~lJc<~>rlsi~l~rr~)~> decom~o<1~ns ~ = (>p~! ln tb~e d~co ~ o<)o~ the bogenydass#~ndib~inte~cr~inpar~cular,~reunde~ ~od to bei~varianC W~ vie~ 1he prim~ ~ as 6\ed,~nd recaD1he hypotbesis ~a1 J[~]isirreducibJc.~[fthisirreducibiEty hy- polbesh holJs ~r one ~ ~ ~ linen ~ holds ~r~B J.) Set ~{.~ (, 9, ~) = (~t{.~:(Z) ~ ,> j7~> . . . ~, ~, ~ ~^~<~^. (~/ //!C {~-1/Z~< (~/'/ (~/ //l(Y ('/~2J)~/ <~/ ... . . .. . #~7t~ ~ ~^ ~ ? ~ <0 = ~c ? ? / ~P} ? ~ ~') )71~! #~ () (~/Z~ (',7. !~# In vi~w of7~ 2,1be Jrstsl~lemc~t mca~stb~1 ttle {-prim~lry pEtT1 of[11~m:~ ~f(~: I(.< ~) -~<~04.~)is ~iJaL Fo~ eachp~m~ r~hkhis ~ )~ k~ ~ be ther1:h}l~cko<~peral<>l^o~) ltisa~13~ln~liar~1#actl:llat ~.! ~is P3cnst~i~ bllLc~ensetha1 ~ acls~n ~ ~ ~sl + ~ ~U such ~ l~is ~spr~v~d bytbe Jrslau1horincase ~ = T(s~ Tbe~r~n 3]2 otroF T2 ~nd r~[ 30~.and 1b~ rosuItc~o b~ #~ ~ #~) = P~ ~ \~# #^ E~ch of 1he l~o integ~rs in 1he d~play~d ~qualh~ m~Y bo ~^ulat~d as the order of th~ S-prim~ry part of tbc co~er~{ of (: ~. ~) ~ ~ ~ ~. ~is coincidence {v~s lbc ... .. , slequ~t~y. ~oobt~in1b~s~cond ~om the [IS~ ~rot~tba <~: are equal 10 tbc {-pert of the quanthy ~ (~('?(''<~/~f>~/'~77~. ~ ~ ~ ~ ^ ~ . 101ln~nTne proo101~ ~ ~e as6ume ~m no~ on U~\ ~ prbnc T~ pn~r ~ ~ ~O? ~ ~ }0 = I,~ ^ ~ ~? ~ ~ #~) FF => ~ ~ ~ . ~ ~ r= ?, 1~1S J1V1S161 ]t) iS i~cludedinlhestalem~ntof~p~ S. Asst~me.llc~1.th<~l:'is~ldivis<~roF 0,an<1 ~rite ~ = r<~. .. . . ... . a pr~nc. ~e ~ave ~,~,~,Tf) ~ ~.>,~,Tfj = e(~.~.~. w~ll~r~ 1:bc s~cond equslity:~1~I~ws frol~ 1he~ (~z>. T[1~ 1~11~r number divid~sr~ ~sreq~irod. Fin~l>.supposolbatr . ~ , ~ . ~ IVld~S ~ J. blnce if ~ not prime, ~e may ~htc if = ~,~b~re . . .. . 1S {~T prl~mc. ~etlavescc~lb~lt<(lI'>~z) =~7! ~ ~ #~. Permuhngtberol~sof the ~ur primes>, ~ ~ and ~ ~c may in~ad r(! ~ ~ ~) = ~0! ~ ~ <~\ The hn~l ~ . .. . ~ numD~r 1S a or~orolr~ ~ b a pI~asure to th~nk j. Cremo~~ 11. ~rmon ~rd O. Rob~rts . . .. . or 1elpTul co~e^~(ons a~d cor~o~d~nce. This ~1ide ~s sup- por[~d in p~ ~ ~aJonsl Scienc~ ~und~hon Grant O~S 93~06898. 1. Sbimur~, ~. (T973) ~ #~. ^~ ~ 25, i~-5~. Z~gier. D. (1~83) ^~. #~. J~. 28, 372~84. ~Fr<.~. {19873~ #~. 71,39~1. 4. Frey, C. {19871~ ~ #~. (~ 31. 117-143. i. Y~i~ L. ~ ~ur~. R. (1994) ~. #~. 166. 33i/40. 6. Crel~, j. [. 3993> ~~. C/~. 64. 1233-1230 7. j~cqt~Tel 11. ~ I.arl<~la~nds. R. P. (~1970) .7'if/~{' 737771.S' ~ [,(2~) ~ l..~ClLI.re ~()~Cs i.~1 ~1~]~31T11}1TCS (\pIiTTgOr~ 5Crl~.~1 }, <{)j' 114. 8. RibOL ~ ( 1980) ~ ~ ~ ^/ ~ ~ 291. ~I213123. ). P213n#S~ C. (TV83) ^~L #~. 73. 349-~66. 10. RO>e#, D. (1989) PL.~. tLC<S (H~ Un\~. ~AJ~. ~#'1 .. ... .... I 1. BertoItnC Y. ~ ~armon, IJ. ([g#7) 3~. #~.. in press. T 2. Rib~ L K. ~ T 9~0) ~> {~. TOO, 431-476.

I1114 Colloquium Paper: Rivet and Takab~shi Birch B. j. ~ Ktlyk W.` ads. (~197i) #~ ~.~ ~$2 I- /~) Lecture Notes 1~ ~the~alics (Springer, Berlin) Vat. 476. 14. Kuribara, ^. tl977~/ ~c. Sc/. :. #~ .~. ~ 25, 277-300. 13. Vign~ras, ~.-F. ~ j980) ~r/~7i<~r ~.~ J#~ ~ ~7e'~' 6~ No1^ ~ ~thc~i~ {Spdngep Boding VoT. 800. 16. down OFF. (I981) 6~? #~ ma. #~~ 109. 21325. 17. Gabon. OFF. (1984) ~ ma. 31, 18i-197. I8. Jordan. B. ~ Livn#, R. (1986) ^~# ma. 60, 227-236. 19. Cr~mol~a. j.E. (~1992) /~-~> /~ ~1 E~ ( (C~rlbridg~ t.J~l~v. Press. C~mbrid~e^ tj.K.). 20. Cerednlk. I. V. t 1976) ~< \> 100~ i9-88; En~Esh transL. (1976j #~ U!# St. 29. i3-76. Ddn~ld V. O. (19~) ^~< J~< ~. 1Q. 29-40; ^~bsh t~-ansI~.. (1976) ^~r7. J<1~< .4~. 10, :107-:1 15. ~, . 24. ^ ,^ #~... Bouto~ J~F. ~ C~r~ol H. (1991) J;f<~ 196-197, #-I38. Deli~e P. ~ Rapol~o:r1, ~. (:1973) [~ .~ ~ ~6 C/~ ~#z/~. I~clurc ~oles in ~atbematics (~Sprin~cr. BerlTTlj V<>T. 349. pp. ~143~3~16. ~udy, R. {1997) ^~. {~., in pr~s~ GToth~ndieck, ^. (1972) !~7 ~ #~) ~ l~cture Not~s in ~ ~ . ~ ~ ~ 3~. 26. jord~n. B ~ LKn6. R. (199i) ~ <~. ~ 8Q, 119-484. 27. Y;tn~. L. (1996) Ph.D. th~sis (Ci~ U~. of \~ York, N~w York). 2S Serre, t-~. ~1987) ~r #~. ~ 34, 179-230. 29. T~1~. j. (1994) ~. ^~. 174, 153-Ti6. ~r~ J\. . ~R:ibet. K. (~1987-1988) ~-z 7~r (~-Z~!! 07~# <~/ ~2 i/~ . ~m~ (So~ N~, Un\~l? ~- de~l~. Exp<>s! 6. Bu~ard. K. (1997j ~ {~. ~ 87, 391-612.

am. #. pp. it 1 13-1 I 117. October 1997 Colloquium Pander . . jogs C~ <~ ~ Saw godson ~ep'~m~nt ~ Pug Hemps and ~1h~m~lE~l S1~5sd~^ Un\~rs~y of C'mbd-, 16 ~41 I and, C~mb~d<~. ^2 1SB Bailed Kingdom Lc13 tab {tT1 elliptic cured dcfi~lTed fives ha. For simplicity. He shag assume tbro~hout that ~ does not admit complex 1, ~ . .) far tho p~ of -dais dints ~ 2. We+ far lbe union of the ~ (~ ~ I 2: . . ./ ^16 ~ ~. and lot ~ denote thc Colon grow of ~ aver ha. ~ a Retrim of Serge (1), C~ ~ an open subgroup of 6~2, ~/ and hence is ~-ad~ Lag group of dimension ~ Assam Mom nag on 3, so abut 6~ bus no<-1~r~on. ~ a rcEnemenl (2) of a beor~m of Lizard (3), 6~ than has ~-cobomolo~ic~I dimen- ~on equal lo ~ Eta be ~ >-primary Bean Crop, Hick ~ a discrete -module. We sag tba1/ teas a Anita 6~-Eu~[cr 0) arc knee. When ~ has bade Euler char~cteris#~. we drone as Euler characlerktic <ha<, ~ ~ ~ Hi . _ ~(~.~)~. t~(7~((~^ and 1} /= it} The present note w(1 be concerned Sib the calculation of the Euler characl~r~tic of the Seamer group )~) of ~ For By We r~caD abut 1hL Selm~r group is defined ~ the Ices of the scqu~nce 0-~^,) ~ 77~(/,,, /~<,^) ~ it] Wilily<,,, E). a) 1'i]l1tc [1~1 where ~ runs Mar all Wile places of ma; horc C~ denotes 1be union of the completions ~ ~ of Abe anile extensions of contained in 6. Of course, #~} teas ~ azures slIucture b5 ~ Module. gad me apace its Rule, characteristic to be closely related to the B1rcb and ~inncrtor-~yer formula. Speci[- caJ6, lea ~ {E) denote the Tat~-Sha~revlch group of E Far ~ and. ~r each finite prime ~ 1~1~ = [~313: (~03< Chard, as usury EM is the subgroup of points Huh nonsin~ular r~ducCo~ modulo ~ 61 Id. >) be the dasse-Weil <-sparks of ~ over A. lf ~ is ~1~ abe~Jlall Wrapup ~ wraith ~) go its >-prin]~ry ~tlb~l<>ul>. ail! >~ is :1 ~p()Sil~iVC integer ~(~`~> Ilk (longly the exact peer of ~ dividing a. Finally. we denote ~ ~ the Duck of ~ modulo a. We then dobne. far ~ Chow ~ has ~ . . number good reduction ~ , , ~ /~- <~(~)~)) UC C/~(~(E~)602 If/ ~j = I. ~1 Shears ~ runs over ~1J fi~[litc pl~tcOS of A. Co~^ 1. L~ E #r ~ ~~r ~ ^~ ~r ~/ ~S ~~ ~rL Half <(E i ~ ~ 0. muff ~ i >~ <7 >' /~ ~z''''~, am //ztz' ~ (~ ~) ~.; ~ of )~ ?~) <~ ~ 6,~E~r~ -~ 2 ~'<~6 T<F~ - <age\ ~ A\ 7# ~ ~ /~/ - ~\~/ 37. This coJeclurc is suggested ~ the Eliding co~sid~r'170~s in Wasps Cat C~ denote the unique Swenson of ~ such Bat 1be Patois group [~ of ~ over ~ ~ isomorphic to {~.O~fcourse.~3containedin (~.Let/~)betheSolm~r group ofE over ~.~hicbisdcEnedby Niacin Baby ~ ~ the ex~cis~quence of Eq.1.Siakin<1b~s~m~ hypothcs~son all] ~ AS i~ (}Ol~OCtUE~ 1 . it iS ~el~lkll<~n~b:~l >~,.~has~ [n>cl~-Eulcrchar~cteristic,~hiohissiv~r by ([an )~) ~ ~/~ Jo Cr~c~thatF~bss>-cohomologicaldim~nsion~q~alio [so tbat\(l`~.~) = ~0~(l~ ~>/~0?l~l`~ /~) Crony discrete -prim~ryE~-moduleJ Thus Co~ecturelasscr~tba~ under 1b~ hypotb~ses made oil E and~.tbe C~-Euj~rcharacter~tic offs should bepreciseJyeq~altolhe [~-Ruj~rch~ractcr isticof)~.'lhisisi~ndeed ~b;~l:o~ne would C\pOCl~ f~rOl~1~1]C #owingLeur312 argumCnL Iffy ~ a~yproEnit~ group.l~ Am) = lo .,/ ~1~ F41 /./ where ~ runsovcr ~opensub~roupsof ~ Bethels adobe of!~.?Vd)~ = H ~ #L C</Z~)~ ~ Pontoon dual of ~ discrete -primary Oberon group A. Under the bypolhe~s~fCoJec~rc ~ it ~ known1hal>(~)isabnil~\ ~ner~t~d torsion module over!(F~). ~h~r~asthe structure heoryofsuch modulc~cn~blesuslo d~6n~1becharac1~r~tic C(~)of Main AFRO ~ isobar and ~lLkn~w~ to sceth~t Ct)~j) hasa g~n~r~10r ~)such1h~t I j~) = \~, #~ Hi abort me are now 1~1crpr~tin~ ibe ~l~m~nt~ of () as -valued mc~surosor Ha. We doJlolatpr~s~n~kDow ah But be structure theory offt~j-moduIes to ha able to d~Enethe~n~lo~u~ Ct)~)of6~3x)>.~crlh~les~o~e

1i11( C?~>l~l<~>qu~ium Or: Comes and Olson b peopled to guess that Hero should be a y~ncra10r ~) of C(~(F~) such that 1 j~<~) = ~ OFF [61 Forever, tab link, which ma exist between these ch~racter- 31~ ldcaR and Triadic Cons sagest tb~1 Ct~) should map lo C(~) under the canonical ~#ecdon Mom /~) onto /(F~. bb Ia1Ier property would sob that 1be 1~0 integrals O~I1 tile ~l~f1 of Eqs. 4 and 5 are eq-u~l. Alar suil~b~l~ .. . . generators of 6~) Aid C{~), sad so explain ebb cqu~i~ of Be Euler ~tch~= , ~ . ~ In Spit of 1l]c above h~urisl~ic ~rgu-menl, it does sol seeing oa~toprove Co~octure ~ Stow And Denote the OaloR<roupofF~ovcr!~.sotba1~i~apTo~-group. We say that module ~ ov~rtbe D~as~w~ly~br~7~jist~r<on Coach element of &nnibiI~ted hysome nonhero element off. Cur mainresultis He F#k~vlng. T~E(~E~ 2. ~1 ~? rO )~r I <~/ If 7, he Czar Am <~-~ >~ aged 3~ ~ 6~F/F<~.74~ (~ ~ j~S ~) ~6,~, ..... , .~ . ()) = USA ~ = 2, . 4 k h~sIon~beenco~ecturod~s~ere[4)that ~ Ltor~on overt ~rall(>tndal~lprimes~ adhere basgoodordinary red~ion.butvcrv [tbeisknowninthisdkection ~ present. , , Movie oft~llis.it may bc ~rthl~0t~ill~tllc allowing weaker rcsuIL which ~r can prove without this assumption. By a theorem ofSerre~i),th~cobomoIo~ygroups~,!~,) 0j are Snip. HOE 3. P/~<f~r ~.~ze {~ [~:~z:~r /. \# ^! age, 7~) 2~ ~ =~ >~ = ~ ~ m Ske1~bof1befroofoflbeorem 3.Le1S Axed [niles~t ofnonarchimede~n prim~scontaiuin~> and ~lprimrs where b~sbadr~duc#~n. We Bats ~\ Robe maximalcx~nJon of ~ unlimited outside ~ and a. Fore~ch ~ ~ 0.}e1 F~ = (! ~ /~/ ~ ( ~~ \( /. (7.~4 #? (ma< Farm >S /361< E~,Coker}~(~ !~ and (mpleca~ul~donstcE red 7.L~m m~13JIhcnsbo~ Hal ~1(~},,,,E~.~)~) =~:~3,/72(~,,,.E~) = 0. Suppose next1bal~ ~ a. Tbc extension F~~ off ~ d~epIy ramiEedin1b~se~se ofreE 8 because ~ containsthe deeply remixed [eId ~. There pad denotes the group of Hi empower roots petunia \(c can Cerebra Pap the principal rcsuLsoiro[ 8 10 c~lcul~1~ Ker ~ add Cokcr ~ We deduce tbal>y ~ sudec6veb~causc/~(~~,[~) = Hand that K>r >< ~ EnLe,~1~ ord~requaI ~ completing tab proof of tab lemma. LEASE 5. ~e [ <E. 1 ) ~ O. Z7z~ (i) ) ~) 2 ~. (ii) fir P(If~/ /~) ~(E(~. ^/ {scallion 0) ~ a fundamental result of ##a-. Assertions (~) and (~) Agog immediate from 1be Snideness of )~0) and Cassette' v~risn1 of tbc Poitou-Tate sequence (cL the proofofTbcorem i2 offal 7~. LEVI 63~r(~[, n ~ O. #~C . .. . . . !~<{ We makc~ssentiaIuseof1hecvclotomic{~-o~cnsion .~ Hoof ~.Tbc ~nLencssof>~)impl~s Bat )~ ~ istor~on overhead ~<ebr~l<~. ~ Beg known met then sho~sth~tthesequence O -a )~ ~ -a ~ ~/~ 3,L~-> ~ ~7,,_~0 age L~ Ha= #~ - placesof ~dDidin~ ~ Nex1~e~sseritb~t7?j(P~,)~) = Bench ~. )~) ~ O because )~) has no nonhero Fini1:c F~-submo~lll~Jetsecref.9~. ~ence.t~ki~[~l'~nva~r~nts of1be above ex~ctsequencc.w~seeth~tthe nature map ~=~< ~ A. at ~\-~>e ~/ Where ~ runs aver ate primes of ~ dividing ~ and the induc~vc [m) awaken with respect to the restHcdon maps. Our proof ~ based ~r 1be Link OR known commnt~t~e diagram Oh Pact rows 0 --an )~77~j<~;~ ___- /~71~,/~) 1~ I) o ~ ?(~) <a/ ~ to, ~ ~ (a ~ )< . ~ ## .!e.s I ~ = ~ ~ ,{ 1.,`..... >~ /~}f$) ~ , ~ ~ Hi. 1,!~,\ There the v~rtic~l~rro~s arc restriction maps ~ ~ ~ ~ Or O/~' ~ ~ ~ (~ E<, ~ ~) ~ lo ,{` an ) . "' '' , This is a purl local calculation. For each ~ E a. ~ . ,. a piggy ~ offs above ~ andIel }~ d~noto1be Ca[ok group sudec1~.Buttb~su~ectKily of ~ aDdthc6ugect~ity of to~elherclear~ show that >~ ~ s~d~ct\~.~sr~quiro<L LENA 7 LIP. Scrrc.perso~aIcom~nicalion) ~> )~C \~. all,.) = 1. ~/1) /~#, as`;) = 0. Topr~veThcor~T3,one~mply uses diagram cb~sin~inlh~ 50v~ di~ram.combinod Rib L~nun~s4-7. #.. .. S~tcb of the Froof of lb~orem 2. We begin with ~notb~r puree Ioc~1 cslc~latio~. lair Sob ~ ~ ~ let Ha, be the Q~-m~dul~ dogged ~t 1bo being of >2. ~ &~ ~ ~ ~ = 02 /# ~1. !~/ Fig a place ~ offs above ~ eaglet }~ denote tab :l~lo~is~rotlp<>T`~^,,>~>vcr fit' Len R:>r~llf ~ (), ED -~{ 6<'::' '::' T>) ~ age},, ~ /7 I {~,< ,,,, ~ ) (at ) O~tb~o~rhand.tber~ul~ ~freE8sho~tbat7f<6F<~ (~> is~is<]m<~<phic~s, >..-~t~du~le1~X,>.~tl~re~ ,isdef~jl~dto be 2~^ =~ ~=? Gnu tbon pa day ~ ~ A = 0 #~ sad ~ ~ Lsn~ Be Hoch~h~d-Sen~speclrals~que ~ .itisthc~ea~ ~osho~ Hat <~~ ~ ~ = 0 Grady ~ ~ ~ Piqued. If ~ Tan abcha~ gr~up,w~ doED~susuat 7~(HO = (Had ~her~{~d~Dotosth~k~,nelof mul~plLalion byes

:~3ig(-st,]~g.~. p32~+ :~3~8 <~8 ~(,)~700 I; ~x \~ swat [~?~ ~ ) (~3~ ~ '7 ~: 7; ~ { ~ ~( : ~] l ~ ~ it} ~ ( ) (~.~.~e .~( .~,ss ~ 1.),~ ail.- cx51L'tog-:~S c: the sc~.~e (} >~;~) ~ {~3}s7 [sY?~.~43 ~7p: ~(- ~&?~- heir t h ~ f : 3p ~ ~ ,4 ~ ~ l. I, ~ t. 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At 3 ~· ~ §; )6S) S 37>i,3~[, A 5f}(vc ?c~ c ~3 AW) ~iAt S' y f. A r cf f ~S y ,. f ., ~ , ;, f f <, y f <, f , 3 3 A ~A ~ S -?3 3. ~ ~.c 3. ~ A ~ ~5 ~ S .( ~ ~ . S. ~ , ~ ?, ~ .~f ~ 4 (\ ~ ~ {' ~ ~ <. ~A ~ ~ ~ ~ ~` ~ . (f ~)f g 94.f 3~. 3. g AW ~ 3 .3. 3. sc a f y s 3 f 7 Y i ? W , 5 ' f ' 'C~ f ~ f , ,S g l. ;. 3 ~ h ~ ~ ~ 3 3 c ~ ? :. 3' ~ ~ 2) ~ 3 ~ <~ ~ ~ ~ ~ . ~ A, ,) `3 _. ~ .[s ~ i ~ ~ . f. ~ s ~ ~ 6 - ~ 4 A f .~6. tit . A~ ... · . ~ 7 ~ i,iS3'°'',g~-7i4,

Em #? 3~ ~ ~ ma. go, Ha. 11 lIS~lI jag. ~ 1997 Colloquium Paper ~lh(;llalique. {l~lvorsi::E do P~t:ris-St~d, at 423, ~F-9~14()5 Halsey. 1::r~llce FFsl Ial us give some crumples of Trial zeroes. Eat \/~ be ~ ~ manta [^ ~ ~ ~ ^ . . . . ssocls1cU quadratic Dirichlc1 character; the Euler Actor of ala, ~) at ~ is 1 - ~ i. Moose an idcaI ~ above ~ and ~ compatible timbering of an ~l~obraic closure ~ of ~ in an Hebraic Ensure ~ of #. More =~1s ~ ~ol~opoldt . ., . Arc ~ uncnon Age. ~) sum that Or ~ ~ O and Ecu. Ace. 1 - ~) ~ U - (~ '>~-13~ ~ 1 CREW [FCrrO~~bO~ (1) () ~=0 . ~ , , . (;(~ 0) = - <~{~)~(~. 0~ ~ ~ = ~ ~ = #? ~ = ~ #~ ~ ~ m ~ ~^ wag < redaction al a. fur Of Do. (2) h~c constructed a Medic Duncan (~(E. Hi. THEOREM [Oreenb~r~terns <3~. #. ~ . . [~.j) = 0 (;~< i) = /~t(~!t~. 1) ~/7 6~) = ~ LIZ</ ~< ~ 7~ ~' <~< E/ It has been receipts proved 1h~) is boozers: Barry ~ . , . . . jam IS ~l~e franc. lean ~ is 1Fansce~de~taL ^# ~ ~ ~ a m^ ~ ~ ~ ~ Ad I + >1-> the Euler factor at ~ of its ~ Anion. 61 ~ ~ 5~1 (~) = ~2t/I(~) (2\ The Tale mist of ~ Tam* (1 j = ?/~1(E)) = b/~4I(E)~. TO Euler factor ~1 > of ~ is T --- - ~ . , U-< ~ ~1-~ ~ ~-D ~ ~ += AS -go ~ - ~ D -a U-~ 9~1-~< ~_~> ~ When ~ has ordinary reduction. amidic ~ Anchor has beer constructed by interpolation of values of tWiSIS of Aft. ~) = 0+.~~(?~#~ 0 because 0 ~ inside the convor~nc~ domain of Be Euler p~rod-Ll.ct. Under ~ Cad tcchn2~1 ~potbes~. tab Chains theorem has been ~cd: T~E~^M [Oreenbe~uin~ (53. ~ E 4~S Act ,. . , . a. 1 (p(~, 0} = 0 . ~ ~ _ , ~ ~ ~ ~ ~n ^~ (go, ~ ~1~ ^)c/7 Am peak) at:) (p(~) = (~(E). So (~(~, a) bus a simple zero >scab #(a} ~ ~on/~T3l. In Bengal, a 1~iaI zeta sbould Spear Khan T or ~- ~n~ih]~es the ~-Euler Oscar. It means test the >-sdk ~ Cocoon should ham a zag of mull~l~ity sticky biter than tbo one of the complex ~ Cocoon. We ~i~ Ark has bean done ~ Grcenbe~ (6) (in He o~rd~illary s~1u~ti<~n). {~) ~[l~e gives a d~nTtio~n off some #~) in very general case. in p~rdcul~r, ~ ~ = #~(T(E)) with bring bloody ordinary reducdon. {~) Ha ~^ a conjecture far 1hc bchrior of the Triadic ~ Dacron at Be twig zero (mubipEcky ord~rof the zero and b~bavior often dominant co~ficienlofthoexpan~on at this z~ro).~11ech~cksthst one recov~rsth~orems~keady proved. Inthis1~1k wcJook only at the c~scoftbc~ ~ ~ Afar ~n~ wig good reduction ~t>.~ec~plaini~tbb specialcas~:~)tb~construc~onof the Oreenbe~invabant iD the ordinary c~sc.~) a construction of a slmbar b~adan1 in the supersin~uTar case; 077) the conjectural definition of the >-medic ~ Dacron; (~) ~ co~ccturaI link burgeon the <-medic ~ Dacron and a co~crtuFa1 spcdal ~stem. and (~) conse qu~nces on tab p~dic ~ Unction ad the trivial zero . .. Section I. ~ota1~nns . .... .. Fir an algebraic closure ~ of 2. 6~ = ~/~/~) ln Abe R) # ig~.3f wig deigns # 2616E)~ T1~<adic ~l~t~ = = P? a<-adicrepres~nlstion of 6~ ofdimensio~ 3. ~ ~ai~sthco~-y.lf D(f) = \~ll2( ~(~))l~-2~l.1~h~re Mists fit _~U = ~.~# aC1i0~11 4~):1~ >~1~n(il~bedItr:;lioll~xpl~illy.Le1(~(~.~ ~l. fag -2) the E) b:!tSiS SUCH tllat ~! -l = ~ 1~S 1, ~!{~> = ~ Age, ~ 2 = ~ 2(' 2 ln ~c ~ age. ~ can choose ~ 1o be T~ 7~:lhe ,.... . . . . ~ nj~r~tlonls<ren by -~~ ~ = j~-2 assume non/erg. 1~ the >~, c~ (sad ~ = 0 which ~ ~u10malic k~ ~ 3~. kis~dir~clsum task ~-r~prese~t~Co~: ~ = D/I {a) gild I r#< n = ~< tam = ~ Ha. '1. Z + ~ l + ,\-~!~> 2~1: SOlllC ~ ~ )2 talent ~0 - 1 j 118

Colloquium Paper: P~rrin~Riou #= #~ = # ^ = (-at = _ ~ - lit -3 -2 ~ find ~'l) = ~2 '1 ~ ~P (~ ~----~1 = an' jr ~1; ....) 1 ma= ~ go= ~ ~ Ha. a... . . . ~ ne :~:~lJ~rt~1:l0~n is Eve bv ~ , I ~Fi1 ~( ~ ~ t Ed l~1\,( -I = ~(~ ~ Opt 2 ' Lab ~ = a- 1 - go [far amp sumac choke of (~ a_ an both cases. ~( ~ = (~ a. lo sup~rsin~ul~r case. 2.1. Ordinary Case. On ~ there mists ~ filtration of>-adic . _ representations of ~< = a/: O C F{/C F{/C such that ~{Fi#< = ~_: i n = ~ ~ 2 ~ me - ~ So there is ~ nag sudecd~r MEL ~ ~(I). ~ choose ~-1 such that the map ~ ~ ~r ~ ~r = sands ~_1 to L Q~,^U =~? ~< ~ (~e use 1h~ notation Id, ~ of ~ocb-Kato). R~c~U t~s1 Share . . , . .~ . IS ~11 1som.~rphls!~ ~ ## ~ ~ ~ ~ ~ ~ x ~ ~ ~ . . . ~ ^ ~ Ille first one Is fast Sumner theory Theirs ((,)~ >> = #/~, gad ~^ ~on#~(~.~)~ jog is the ~J[>garit~hn1 all >~ such that flog ~ 0. So there is a ,~ ....^ 1.t 1~) a,) ?~ AXE -- VeR) ~ (~l{~(~)JJ~(r)~ HI (} a, fit lo. a} at) = -A e ~ u ^~ ~ f an' ~ em depends and on the line Ha. a. ax ~ 6# ~ ~ a 73-c~clotomic exto~sio~. domino ~) = <~) e ~> ~ ma. The ureas} norms Arc contained in {{~}(~. ~ [elements of ~1(3 ~ which are unr~mjE~d oot~d~ of <t Thanks 10 blah (7) and under 1ecb~ical~co~dibons~ 0) the universal norm am ~ dimension L (~) age. ~ = 0 ~ ~ age Ha. [0 = 1. Sail in Abe abode defi~l\~1~i0ll, (~.~) = (~) Bar ally nonzeFo among x of a. ad. .^ ^ . . ~ .... . .. \uper~ngui~r apse. the canonical map ~ ~,n~ ~ ~ I/) = r) 7~!. CZAR. ~. \~. 2~ ){ raw 1 1 119 ~ an Zoomorphism. On the other hand. by BIocb-Kato, there . . a nag go: 6~ ~ = ~^ ~ ~(~ Once blind chosen Iota on (Ion ~ ~ 0), there ~ a canonical split of the inclusive ~(~. I) ~ ~(/, I), and so we obtain an extension of the Boc~to }oat log, to (. ELI ~ 1~: Amp. ^) ~ /< ~ ~l = ma' () ~ #~ ~ ~ Amp. <) ant. at. ~ <~) e ~ 0 ha. Define #(ad = go) gibe a ~i~rs~ norm ~ 2!(~. ~ [Spain, wo can just take ~ nonhero clement in ~(~ I. Lemma = 0~(~(~)/~) ~ ~ ~nd7~llfb~con6nuous group algebra of Ad. DeOnesomo algebras: ~r \ ~ ~ ~ ~ ^~ ~ Am- #~ Hera ~) is Abe Debra of elements in #[[ax]] which arc O<jo~) far a suitable': ill GleEt~ns Abet f ~ ~ .,> C{1~ be ~ritlen / = i~? - 17~ ~itb Logo IT/ ~ ~ (7 ~ ~ 1~ologic~1 Generator of the apart of 0~> ~) ~ the told Faction hrg a. }f ~ ~ ~ conli~uous character Tom U~ pith values a;, ~ can Equate ~ on any Moment of #(go 6~.lECi tare am] (]~: /~; 6Z7q' /7 ~ ^~1~( {0, /~ ~.Y CZiZ P~~ palm ~ ~)S~) ~/#r~) ~/C~ , , 1 O(~(#, at, 0) . <~))e = ~ ---- {\ ~~-~)j (~) e ~ ~ )~(Y )~ am 4~ gaff 0~< = ~( 3~ I = ^~ egret Ada ~!~< {]} ~ )t/.~\ ~ deli Mid ~r ~ (/ do E, get ~ ~ ~ = ~ ^ ~ ~ ~ ~ ~ ~ ~ So ~(~(~))~s ~ a; = ~ 0(~)~, ~ go ^ 6~) ~03. We amp see ad}- ~ 6} as ~ element of # ~ ~ Ha. go) ad ~ a gnaw J~ ~ as: is ah cyclotom~ic char. ~I(~ at} = <~>~{~1!,~}) 3tnd~,>(}f..~.~) = <~>~(~1(,~(~)), (P).Foranyf ~ ~(~).de~ne aoo ~ ~ < Section 4.Loparithm ~ ~ = () ad <{ah. Hi= ~ Age ~ ~ ~ = ~ .. .... . _.. ^ .. / ~y~~(7~(~)).1Es~ [[module ofr~nk 3. Rota Lathe preen on77j(~.7]. Toucan constructs map (9) )r = ~ : /~} ~ ` ~(~ ~ !( Recta only some prods of ~ Ohs Era ~ depends on ~ ~r~cipr~cilY 1~ cactus tbat scams 10 be proud nova. if , · . . ~ ~ 7~(~. ~ (T1):

C}~>llog~lit~llr Paper: Per~r~in-~li~i<~>u = 0 - #~ ~ # <~) ~ Act. A. (1 _ ~) 1(l _~ 1~ 1~8~/ ~ ~) mod Fb°~. Section S. Specia1 Systems and ~ad1c ~ functions there should exist a specie elemcrt (/ (~(~,,~ I) ~ such that 1~11c ~-ad~ic ~ :Fu~rlction sh(:~)u~l(1 defined ~ the Tumult r >~ = <~ ^ ~ . #, .# I, / · . rag. Defirle <~) ~ ~> {~) ~ 7(, l). CON j East RE: . 1./ ~\/ /\.~.~'.(~) ~=7l-/p-> ~' ~1~ =1 '1~1 iS /~ ~(')~71 ~7 Dl7(V)< In tbe ordinary case 1t means tbat . ~1 (; (~. 01 ~ ~ ~ ~ ~ . ~ . . , ~ 3# ~ ' ~-^ ~ . ~~ ~ a(~() = {~(~)(~J _ ~ 2)( 1 - <)~ !! {1;~, i'12 ,,,j, e ~ ~ ~ ~ e = ~ ^ ~ ~ Q In thE {din~ry case. ~; sb6ulS be tbo ~-adic ~ncdon alre~ldy k~no^Il, 1be last ~/mtlla T~s then the ~rlula corliec- turcd ~ Crec~be=. SecHon 7. ~en ~o~ Speculations C[~0h~) should co~ ko~ a motivic cl~meDI: s~ ~ ~ould exist ?t the clemen1 should again h-c ~ood reducdo~ out~dc of . Eor ~ ~ ~ l~t D/~) = \~; tbero is a map ~l(~ ~ ~ ~13 . . . . . ano IOr ~ = p. '~.~< ~ ~e ~ ~-1{6 \) ~(~")~' 10""i ~e h~e P (~ #-31-~1-~ ) ~ 7~ ~ 2 \ /7 \ ~7 n<< ~1 ~< or 1/ ~\ / ~ = ~ 1 - ~( I _ \. ,, \ Secdon 6. Some The~rems ~\ . ~ ~ /~ ~^ ..~ ~ ~ .,, .., ~ . Wc assume tb~ existe~cc of #~ and 1b~ ~ct 1ba1 th~ ~-adic ~nction can b~ calcul~led by 1h~ ~rmula F~c = ^~. U. TI]~EORE~: 777~! #~? [.,t ~ ~/T) ~/ //~2 /2177~/ C4 . ., . ;''1 r 1 ~ ~) ~} ~ F/~n>) ~ H (~ 7) ~< ~ / # = U - ~ ^ ~ ~IT1 Pa,1~iCU~1~T,, by USin? F1~C[1 S t1100rCn] (71. 1~! iS T1~/~FO if ~lnd <~lllv if <1l~65r~) i~ 1lollze-ro " . $... . . . ,` ,, ~ . . . Assu~m/(~(~/(p)~ b Let63~(~ 1~ 2)~(~) T~11~E<~E~ ~ #~? L ~ /~73 <! {/t) ~/ 1 ~/C/Z ~ S) ^< <~# ~ O ~~ 3 8~)e 1 =~ (j - ~ ~^ _. \ · .... ~ _3 ~^ ~.~ /, ~ .,. T} 1~PO~R E\1~: ~ \~ #~;~\ )~Z-~! (Z? J?~! \ (r<I<bt 7~) = I f 1 ~ ~ ~ . ~ ~ ~ \ A can(hd~lc of sucb a~ ele~ne~t has beer conslructed by Fl&ch. Or 1h~ olbor h~nd, 1h~rc exists a nstural ~-vect~r sp~Je ~ such 111[tt ~) =~ P c~n be describ~d in lcrms of 1b~ ~#ron-Sevcri group of 1he r~d~tip~ E x E a1 ~ (8~. We ~ould [ke t~ compare {~) ~r d~ro#1 / ~d ~ive ~ link ~kb th~ <~ad~ ~ ~nchon ~rk in preparation). Cr ~ ~ >. s~e cal^I~1ions of Flach (73. .! .. . ~. ~\ / )~(\ O) _~! _~ ~ L 6#e#. B. ~ G/3eGLC~. R. (1978) ~( ~. 50. 91-102. 2. ~ZU,. B., T{~t10^ J. ~ ]2i12JbJ11T1] j. (~1986) D!~. #~. 84, I-48. G:~e~. ~. ~ S1#CnS. C. (1993) ~. #~. 111. 403~? CO~tC~ j. & SC>mT]L K. (1987) ~ ~e~. /~. <~. 375. I~I56. ~.. . ~ .. , .. . ... D. wr~noe~. <. ~ ~ulne J., 1~ p~p~r~non. 6. Grc~nb~, ~. (i994) ~ #~. 163.149-18j. 7. Fjsch. #. (1992) Dr>~< #~> 109. 307-327. 8. L~cer. ^. ~ Sai10. s f 1396) ~rsion Z~ro~s o ~1h~ ~.. . ~ , , SOTPP~JuC1 Of a ~O<U1,, E[iP1L CO,R, PSCPJ~t. #. PO[d~-R10u. B. (1~94) ~6 #~. 11S, \1-149. 10. FC,Dn-R10U. B. (1993) /~e 229. !I. PeF~n-RiOG. B. (1~95) In #~ ~ ~7 ~ #~. CJ. ^~C(# S. D. (Bi~h(USOP B2SC>. PP. 4~-410. B;St{TE-Si,TCi\. KDi2Z. G()r2:~1J2:iI1. ~F. ~ PbI]ibO{1, C. (~1996) . ~ #.

Am #~ ~ am ~ Vol. 94. Up ~[1I21~11124 October 1997 ,... .. .. .. . <~oqulum Paper ~^ ~. ant modular Patois representations and their Seller groups Acadia t~don/~lass Camber ~r~ui~/main co~erturc) {ROZO ~IDA~t~ Pcoue TI~1NE$, ED ERIC URBAN! <!~pa~l~:>ly~, <:!! >~:l~cm~1ic~, !#Il:i~rsily of (half, t.os A~:geles. CA S(~){~)~5-1~; Hillel ~Irlsl:ilLl-~ (balm lJ<~:rsSlS de P'~is-~<~r<l, Age Jo~ln-B>~pl:ts1e ~ ~ Y~ Fag ABSTRACT In the las1 15 yearn maw class number formulas and main Lures brave been proven. He~, discuss such formulas on the Seamer groups ofibe 1b~e~ dimensional fit representation ad(~) of ~ thou dimensional modular Gniois representation ~.~(es1art~Rb tbe~dirGal~srep~senta1ion goofs modular e'UpHc~urve and present ~rmulae`~press1nginterm~of~1,ad(~<l)) the in~rsechon number of the elliptic auras ~ and tab complementary obelisk varies, inside 1be Jacobian of the mauler aura. Hem ~ appall baa one ~n deduce ~ ~uIa far the aver of the Selmer group Sel(~d(~) Mom 1be proof of Shies of the Sbimu~#l ani>amn co~ectu~. Rear that, He generalize the formula fin an I~sa~n bearer setups of one and tab variables. Here the Orst variabie~ ~ is 1b; Height `arintle off universal~ordin~ry Hecke alpebr~,~nd 1be second v~rinble is the cyclotomic variable a. In the ones variable case, ~ Iet ~ denote the p~ordinar, Goods repre- senf~11on fib values [~ ~ z(2,[[~]) HhI~g Ad, and the char~ct~ri~ti~po~erseriesofth~ Selmerproup Sel(ad(~))ls given by ~ ~ad1c Tycoon interpolating [~1, adage)) far ~eiphlt + 2specIn~zatio~ (~ of +.I~ibet~-vurlablecase. gestated m~1nconjectu~ onibeLbaracteristicpo~erseries in 27[~6 i]] ofSel(ad(O ~ r-1~> Mere ~istbe universe! cyclo1omIc cbarsc1er wi1b values 2~[I`ll~ F1~11y, ~ den scribe our ^centresultslo~nrd tab pro~fofthe co~1ure gad ~ possiblestr~1eg~otprov~np1be m~1nconjectu~ using -add Siegel modular Army Lee Ok atthe conic on ElEpti Carves and \iodul~ Formsatlhe N~honsl#~d~my~fSciences As presented bv H.H. Shy purpose oflh~t~lk Casio describe ~rmul~s~<n~ the ch~r~cter~ti idea fifths Selmer group ofthe Halo; represent~tionsasinth~blleintarmsoftb~ir~ <~lu~s.\Vc~x Prime ~ 5.~kho~h~ecanlrea11he~eneralcas~.~Ho~i~< ramiEca~on ~ [cited m~nyprim~s~nd ~ to kccpthr paper sad Me ~ssumethalth~r~miEca~on ~ concentrated on IP at. ~ Selmer groups [~1 ~ belch ~1~\ group ofthe maxim,Iexl~r Q>)/0 unrami]~dou1<dc1~,=~.LctObe~<alu~hon hnpEniteflat over2# Residue bald F.{V~stsrt~bh ~t~o-dimensio~aI continu~usr~pres~nt~ti~ ~ : ~ -a ~64) fir ~ complete (~oclh~ria~)l~caIC~al~ebra~ ~itbresidue Said F = //~. '' - ~ - . · .!+ J nc par Inn ha On Hi. . ~1 Us ~ Me ~ gab ~ . We ~1 ~ ,~t on ~ = /> Via ~ and ~ End( ~ ~ ~oniu~Jon: {~#~ 3~.#l~3t~ factor adage: ~ ~> CL~) acting on trace zero sub~pac~ ~ 1997 ~ ^O ~J! AC2^mS ~ #~CDCOS ~-~/97311121-~/0 CNA-S IS 21V2Si!~1:1!C (~111:~11C [}{ TI1t~TS://WW~11{t\,<~:g~ D()) in End( if. Thus ~ ~ {v = ad(^ ~ 1. Let ~ = ~ god a. We sssume 1llc glowing three co[ldit~ion:s: /D ~ ~ #) ~ ~/~'~ ~ . D~ .. .... . ~-1. -I ~1#Z~/ Hi: (Re~) ~ mt>(ln~^ ~ ~ mod Ad. C>ndidon PI ~ ~quL~len1 10 tbc absoIutei~educibUl~ of Doves. W~wr)cP/~) C Efor He ~eigensubspace,\ ~ Breach ~ ~ubmodulc ~ of P(~d<~\ ~1 \8 = ~ #\ J~ fir the Pon~y<#n dug /* = FL>+<ti. ~ /23)ofJ.\VopntP~ = {{ ~ P/ad(~)) ~ End(P] I >~y ~ a,. Ton He Dad tab Selm~r~roupfbrad(~).~s~s~ecialc~se of G~cnber~? dc~ni~on (reC l;sec also reE 2~: Scl(~d<~) = kerned-, 6~d<~*) ~ Hi {a, Oad()jj'>/~!~) rtb~ine~iasub~r~uplof~.Thisis~cenerahzadon oaths assgroup: IF ex~pIe.taki~psquadr)k ch=~cter~of U. Sel(~) = k~r67l(~,k<~j-- ~l{! Pa, the Spay ofibe>-cl~ss~roup of the quadr~6caxlon~ion ~^ uessisth~lScI(~d(40)b [ni1~ndthati~ orderisthcp-p~ of((l,ad<~)upto ~ ~nscende~tal DCtoF.Tbe hnitoncssis first shown hy Flinch (3) and than by WDcs <43. We dLcuss later some good cases ~ber~this~ucss works Bela We ~enerahz~ . . . ... . . .. lee goose opinion lo a tensor product adage ~ ~ ash a characters: ~ _- ax ~racomplelenoethe,~n Zebras. repIacT~ ~ by ~ 3~8 and Pa by P~(ad(~) ~ ~) = ~ ~ a: Sea = kerUYl<~.P<ad(~) -~71(7 3~d(~*/L<{ad(~). . . . ^ Alga 1S a discrete module overt Add. 2. ElUptic Curves over ~ ~ . . . For sl~pEclty. ~ suppose chat ~ is the Oalois represe~t~lion an 21~/~, a,) far ~ modular Wiper curve ~;6 inside the Jacobian ~ ~ (~) of the modular carve <~ Thns ~ has ... .. . . m ~pl~tIv~reUucUonal~ and hss~oodr~ducdonout~de~ Takin~lhedualofib~incIusion~ C ~ ~chavc~quoDentm -ax- e . .-o . ~:~-~ln~n7=~+~k~=kerp+tand~n3> a nile~roupofsquareord~r.For~ \~:o~di(>re~tia!~ one P , . . ~ 1~el-~[1 nl<~>ue13/z,5y arcstlltof>4:lzur(~)c<~rolI:l~y 4.1.~e nag 3rr ~< ~ = 2~2~36( ,>) b~ ~ pdD1d~e Mann e !2(1~))~rdr ~ Z.Choosi~< ~ basest of 3~i~enspa~e gf77~(C),Z)~d~rcompk~ co~u~a12n.~t degas ~ bv I<< ~ aver 1lorm~li~n~ r~ as described belo~.The Ludwig) Formula w~spr~v~ I5 y~rsa~oi~ arc 6(s~ ago rag 7j: ....... . r1~ whom reprl~lr~qu>1~shouldbesddrcssed. ~ ~ 1~)

1 TOT Colloquia F~eF: Dada ~ ~{ ~ f 1 ~d(~3 T ~ ANT) a= >~ n ~ I =/ ~23)0~ ala . ~ ~ ~ Ontersection number armpit), -~=~-n~=~r age l. We doEne the canonical period #) of ~ by C-1~2~11+O . In ref. 6 10 get formula 1~1, ~c used 1b~ period determinant ~<[1~ it\! `; - ~ ~ ! - ~ r - . _ I\ Act ~ 02~ / r a abase {~ 1. c~} of Ha (~(C). Z) in place of Q+~_ Asia reef. 6, formula 6.20bj. Writing ~ = ~ ~/2, me see Icy ~ = I<^ am, and thus a+ ~ R and ~-lO_ ~ R. R~pI~cin~ their negative T~f~ecc~ssary me may assume that I} ~ > 0 am VAT a_ > 0. -Andes this normaIizati~n, formula INI is correct Gem ~ deE~idon. 2~ = An, find sac can deduce formula lN1 from ref. 6, Theorem 6J, by just remarking chat <~/~ ~ ~ n ~ finder the notation of the theorem quoted A.ctu~ll~v a. ala simi.1.a.r 10 mule ~1~1 is proven in reef. . . 6 Tar tbc G~lois ropr~s~n1~1ion ~tt~cb~d lo any bolomorphic primp farm of wei~bl ~2. We ~uIa is generalized later lo cohomolo~1 cusp arms on ~2) aver imaginary qua . · ~ ~ · ~ ~ ~ ~ m ~ ~ ~k ^~# ~o ·~ · a. O. 61~ be the sub~l~cbra of Ada g~ncr~t~d by Scab op~r,10rs 7~. Tan ~ induces the peon a: ~ ~ Z C End(~) and anolLcr pro V: ~ ~ End^/ Men we dabne No ante mo~l^: C~ = ^~) ~ #I/') and 61 = 0~' ~ ~.~ {~) ~ kernel /k~r<A)~. prawn in rcf 7 (equation 3.8b) that ~ nag ~ (60~2 as ~ moduT~s. amp that Spec(CO) ~ the scheme ~coro~c intersection of Specks and Spoc(~/'))i~ Spec{/0 a. l bus we get (IN2) .~ >-p~1 of ~ = Coal (inl~rsection number formula in Specks. Recasts, Taylor and Wiles (4. S) have abort that Scowl = INTEL and Wars t4) has shown C1 ~ ~ Scltad()O)~. This formula is ~ Kiev 10 Woos' proof ~ Formats lost 1beorem~ , The Dc1 tbBt S~l(~d(~p)) teas ~ natural map into C1~ was :5rs1 d~covcrod bv ~azur tbrotl~h big d~rm~1ion theory of Galois , ~ , repr~solllatiolls (103. The above formula is conjectured in ref. 1I after pawing tho sure of the map hesidos other a.. . . Cleat realm. amp. under tab ~r10us assumptions on ~ tbat me made. , . .. .. me Envoy get a formula far the order of S~\ad(^j): [~1 ~df~# >apart ~ ~ = I6~ltad(~))I (order Tumult of Selmer group). Tt~cusp arm ~ ~ \~(PO~)cs~ belibed 10 -add ~ may of>-ordinary com man corms ~ ~ ~ hi Ace: ~ ~ !~+~1~),~-~( ~ 0) Earths I~ichm~ll~rch~r~cter ~<cE #~ #~ q^~ #. ~ ~ C-O ref. 12. Lbapter~. theorem 7.3.7j. For tbi~ me need 10 fax an embedding #: ~ ~ ~ Then '#~ordinarLy~ of ~ impl as that comedy ~ ~ in <-^p~s~n skim 1~;~ = 1. \01c that, ~ the muI6pllcat~e reduction ~olhe~is, ~; #) = ~ I Tbis family Melds ~ Oalois representation a: ~ 6~2 (A) far a finite Pat O[[~]-alg~br~ A (~[ 12, tycoon 73). A = speci~izabon of ~ via 1 + ~ ~ ~ far ~ = 1 + a, ~ is the Aglow repFesOn1'tion of ah cusp Army. Then the Pontryagin du~lScl$(ad(~)ofSel(~d(~)issbo~n by Wiles and T<jor to be ~10r~0n 0[[7]]~modulo of agile typ~,and as ch~rac- t~rEt~ pow~rseri~ isg~en by ~ccharacte~sUc po~ers~d~s oftbe A-~diccongruencc module 60~. Before gamy the dean of Can, me now that me have 1~k~ncobom ~o~icsl2rmulahonofO~oisrepF=cnt~t~ns.In this paper. ~ dbaracter~e Gslois representations bv the . . ~ charscteris1~ic polynomial of ~comet~ri~c Frobeni~i From at primes ~ ~ a. Far example. ~ ~ characterized - )~(Frob~# = 1 - ~(~;<~\- ~-~+> k n~malLaUon ~ dual to Lee one Men ~ rag ~ but it aH dot far our purpose because ad<~<j = add. deEnc CO#, we need to in1~ducc tbc ~c !~ of -ordinsrv Medic cusp farms. For abut me consider the , .. subspac~ \~+~Lo6# -AL Oj of S<+~EP~) -A Dada of cusp ~rms~w~h Ace: ~ ~ ~ far all a. !c consider tab Sampan \~+~(Fo63 ark; Hi Of )+~(rO~ a-!; 0~ in )[[~11 via <-expsnsio~. We write ~ #{Fogy/ A-; ~) far Be subsp~ce of i<+~<Fo6# -: ~) sp~nnedbyU1~-ordin~ryeig~n~rms. <~>e<~#l~> S ~R<11 such that the sp~cia~z~tion >~ via 1 + ~ ~ ~< ~ tho . . . .. Non of an Cement in ~ <<Foggy. A-: ~) far '[ ~ ~ ~ Men S~ ~ age of anise rank Mar ~ On which Heckc operators p~) naturals act arch T2. suction 7.3~. H~re~er me = ~ ~ ~ O. Let as- be the )-subalgebra of Ends) generated by !(~) trait a, and dense a >-al~ebFa homomorphism a: ~ ~+ ~ by J1D = /(~)~. five sbo have Nether i' of ~ inlo End~(ker(~)) Darn by mubipEc~ti~ by) S 7f on k~F(~). lben . ,. me Retake (a {a}..\ = /~ (a) ~ Buzz (at ) arid ~ L) = i170) ~ 63^ (hi) ~ kern{) /kcr(A)2. Tbc~itRea~tosceth~tC#> ~ !/~(T)) Bran ~ement~7] ~ A. Wcc~ndeducefrom ~er~ullof Wi~sand TaylorinrcE 4<thaor~m 3.3ja~d ref.9thal (~) = char>~6I >) and C1< ~ Self. flora the charactcrklic ideal chat far ~ torsion~moduIe of [Di10 tam ~ over a Norma nocthcrian ring ~ ~ <\en by the product of prime donors ~ in ~ Ash exponent given l~nylh~/~6,ofibcloc~liz,Iion~f~ a1( Notolh~1~ssho~nin r~f.7(thcorem 0.1) ~rac~no~icaIporiod [OS)associatedfo (CN2) (- I)= up to>-adic units. gals ~rmuIa is not completely s~tis~c10ry, because tbc >~adlc (-~nc~on {) is determined only up 10 Unix in a. For \-adic Arms of CM Lee, go can choose a suitable #~ Triadic <sanction in place of ~ (11, I3-Ii). In eenerat ~c can c~ make ~ conjecture on the existence of a.. . . canonical ~-~dlc <-function Z >(ad()j) with procise interpolar bon property ( 16) Bach generates chsr>(Sel6 Madam = (~( f)) fiber extending scat the >-medic integer raid all of the ~-~dR completion {1 of Ha.

Colloquium Papas: Ida ~ ~{ ~ me look al the universal cb~r~cter a: ~ ~ oft~llX de~rminy tbc id~ntEv chsracteF ~f ~. As ak~adv s~ij. our ~rmulaJon ~ cohom~logic~t and bence ~Fro~)~ q~/q)-1 , . _ , .. . .. lor ~eom~t~rlc t~l~Ot:~[liUS Frob~. Wl itillg Q. ~f~r 1~b~ cyclo10mic ^-extension of Q and F = Gal<~/~), tbe taulolo~icaJ . ·.--. , . . , ~ ~ cnar=1er: 1 ~ °L, induces the abov~ ~ ~r ~ = ~ ~ I ~r a <~ner~tor 7 of (. Tb~n ~e consider Sel~(ad(~) ~ ~-1T, ~hicb a modul~ ^cr 0[[( \]] of [nite type (Ij. ~Iassic>LY. tbe Selmer gro~ i~\in~ 1b~ c~olomic v~r1~ble ~ is deE4ed in terms of cobomology yroups ov:< 1bc cyclolomL ~ tower Q>. As sbow~ ~ Oreenbcrg (r~E i. proposkion 3.2; seS a~o ~e[ 2, s~cl~io~ 3.1), <~>ur SeTme~r ~roup Se~l~<ad(~) ~ ~> ~-1 <)ver ~ ~is 1sol~orp~hic to ~he cl~lssics~l on~ ovor O~. Recenliy. ~c 1]ave pr~en a control 1heorem ~r Sel(~d<~) ~ ~-1) ~ivf~ thr iny thcor~m. ~ ., , ~. ~ORE~ ~ ~e ~~2 ScR(~(~) ~ ~ ~ ~ ~ ~ 0[[7 i~ll-~3l7~ q<./Z':~ 0~. #~1~,, ~)e <~< >~ ~SeD(~) ~ ~-) ~ #~ !~( !) ~ 01~l7 . !11 ~ZzZ~ }~! 0~1~7) )~/~/7` {~7 ) 7'Z 0~[~] ~')~ <~27) ~1 h73 <~7zi/~-6z<~) (~17) ~ <~-T>~-~>d~ic <-~nctio~ [~6 !) i~n ~-~7~ S~ ~uch tha1 ~r cvc~ ~ ~)h -< ~ ~ ~ O. < - T)~- 1 ~~ - 1) = ~` ~j ~ ~r a ~c10r ~ lik~ ~ Eulor ~or and ~ ~mpl~ consl~n1 ~. ~ . ~ . . . ~ l]~lS 7,-: :ULlC1:10~ 776 St61Ll~I1 [T;1S ambi~u~i1y bY ull~il:s in '\. ~lttou~l] <~6 i~ is ~iquely del~rm~cd. 17 ret f6. ~ ^~cnc~ o~a . , ~ ~ C~nOOlCal<-)C (~0C60DS 0~d(~) ~ ~-I) iD 0~6 i]] [~r ad(~) ~ ~-lt~ilh p~ci~e irte~o]aiTon property ~ co~ec- t~t~r~d. I~ll ~l:~rl~c~ll;>r, ~ sll~u~Id ~ll~lVO ~n ~qu~ll~ty: ~ ~ ~ ~ ~ = ~ ~(ad(~)) A~, 1be denomina1~r and tbc num~,alor are nol ^ kno\~ lo exis1 in g~neral 1~ spite of th~ kn~ existence of ;b~ ratio /~( S). Bec~use of 1hi~ ~C ~leed 10 use (~ ~s repl~c~m~ot of (~<ad(~> TF1EORE~ 2. (R. (ire~b~r~ a~]d J. 'l~`~il<)tllliO). l~3 (~L!) = !~6!) ~ )~# ~Z I<0` 0) = ~60~) <0} ~ /~ ~! /~ C. . , W~ k~ow th~t ~/~7 (0) ~ O ~ ttc 1heor~m ~f St. Etienne (18} due lo ~ur pcopl~ at St. E1iennc in Fr~ncc. Tbu~ lf one c~n rove 1~h~ ~livis~ibility ~ ~ in (~[ 1~ 7 . i1 ] . 11le i~! [~>i~g corliect-ur~ \~1 1.~. ~ ~ ~ ~ ~ = ~ ~ ~ ~ ~ ~ J? \]]. ]~:' /~) (!~<'~Z/~Y' /.S (~'>)~ /~) >~71'3/Z, ~Z\\Z!~773/ , .` , . . .^ ,.. //z~ /~!~/ (~/Z~7'/~} ('<~('/~.~! (~/7 <)e /~:~/ >~' :~ <~/ ~'7~!/' ~ ~/~6 7~S, ~.Y <~<'77.~{'~ //Z f7Z<? /~`/I/~! <;/ (. ~! ~/ /~ {~ ~^ ~l\~/P ~), ~. () {~! #(Z/~ {~./f~/Z ~ .S/~f\. ^~/ <<f<~7/. /~! /,\ SZ //Z<~]7 ~/ (7Z?~) >-ordin~ry 0[F7 !~-~dic ~rms on ~43. de~el~p~d mai(< by TTIouin~ ~nd Urban (I#. 20~. A cohomolo~ical H~ck) ei~en~rm < ~ O#~4~/~ is c,~ed near} ~-~rdinarY if its eig~n~alu~s ~r 1~O st~ndard Heck~ opcr~rs at~ ~-~dic ~i~ under th~ bx~d ~mb~ddin~ ~ Intc V~. Here 1h~ ~ord c~h~mol~ical mc~ tha1 1he ~st~ of Hec~e ~i<~lu~s ~I app~ars i~ th~ mTddle collomolo~y ~ ~ith coc~cicnts in a polyDomial rOprCsentati~D l, of a Sic~el modul~r <~rie1> ~r GL{4~/~. lo o[b~r ~ard~. ~ beloll~s to a discret~ s~ries rcpre- s~t~Uon #~se H~r~h-Cb~ndra p~r~m~ter is 1h~ sum ~f 1h~ ~< ^~. ~. !~7 #) ~ (7~ ~lli23 llig~h~t ~i~bt of {. ~ln~l tJ~c ~l~alf sun] <>f p<~:)S~il~iVC roots. For c~lc~b cohomolo~ic~1 ei~en:~rm ~ /~\sauer ~has attached a >-adic . , ~ . . mo~ular ualols represontalion ~ inlo 63~4} ~TO~ ch~ractcr~ i~t~[c polylloDl~i~ls <~E ~F`~>b~:ni~i outs~ide ~ YiV3D h~ tlle Hec~ke polynomiaI (see reL 21~. }1~c is the ordin~rbY c~nicclurc :~r 1he GaloL r~=sen1~ion. . . ORD^^R#Y CON]ECT~RB. J~ ~/ ~ ~<> ~- {~. 7~/Z //Z~ /~2 ?/ /~P <~< ~\f~f0~ ~Z~ ~/ ~ <>r )~ ~ ~! >~- ~ 6~f<. #~\ r/~ ~e ~ ~ ~~ #~-adic reprcs~ntatiors atl~chod to ~ . ^ . /, a~t ~ ~1S o~ne <~1: ltS me~mbcrs W~hen ~ is cFystal~lirle, ~c h~lve t~o character~tic pQlynomi~ls al<. One ~ tbat of th~ crv~taF . . ...., . . ~ .... . 1~ne troD~l:~'us ~ c~is(~). alla tile otll~< ~ ~(~), is lilat of tbc Frob~rius ~1 ~ of a non~-adic memb~r ~f lbe comp~1ible . . .. . . .. >rm. i~c >-~ty co~cc~e ~1~ 1n th~ c~e if o~ c~n prove !~riS~ = .~.~. ~bicb is ~ sland~rd co~ecture ~l~ld ~ kno~ lo be ~ue al Icast ~r co~stsn1 sh~s ~ba1 i~ so speak. 1he wei~h1 0 cas~j. lt ls enough to pr~ 1he ordinarbY conicct~ ~r crvstabi~e <y ~r t~he ~llowi~n~ re~so~. W~ cs:) ~lu~ W-eiss~tuC~r'\ GaI<~is repr~sen1~tio~s ~ means of T<lor? pscudor~presenl~tions ~nd ~ct~ch to eacb 6[[( \]]-adic eT~n cusp ~rm ~ a Galois reprcsc~t~tIo~ p~: ~ ~ 53~) ~r lhe (e}d of fractions F~ ~f ~ :Finit~ c~tcllsio~n ~ <~:F G[[7~ \1]. Tllus at densc~ly ~popu~l~ted points on Sp~c(~), ~ sp~ci~#zes into (Sissau~r s ~alois repres~ntations. ~ribermorc. ~ has deDse} ~pulat~d ~e ~ ~ · ~ _ r ~ cla~z~tlons o~ ~c<~) ~hicb ar~ cryst~lhne ~t~. (us if ona p~ ~ ~d~0 ~~ ~ ~~ iza(~s, tho im~e und~r ~ of e~cb decomposkion ~2up ~t a BoreT sub~ ln 6~(4), and hence ~c o]~dtv . ... . . . corllectura: :6)r :~l1 spec~ializ~tl{:~)~s ~l~lows. W~ no~ como b~ck to 1bc ~Fate~y ~r a proof of 1he Msin Co!ject~r~. W~ l~k at thc Kli~g~n-style 0[[6 !]]-adic Eis~n- S1~1~ sorl~s ~ i~duced from 1he A-adic ~rm /. Th~ Oalois r~presontalion ~ allached 10 ~ has v~l~s i~ tbe s1~ndard . . . .. ~1~iX~1I~l 1)a[~it}~:)lIC Sll>~rOU~l), 1~h~l iS, ~it iS of ~l]e ~ll<~wing i~r~n~T: /~ ~ ~ >~ 0 ~ 1^ ~ ~ det(~) C ~<fE7- ~l1> The c~nsta~1 lerm of ~ ,1 the nonstaDdard p~rshohc s~~up ~ iS {t]:lllOSt equaT 1:~> {} t~il~ ~/j(~! .~. ~llc~c ~e ~m~r1 by nonsta~d~rd tbc par~bol~ sub~roup ~i~n ~ / :1: 0 ·~) ~ \ ~ .~, ,~, ~ t , {~= ~ o .~. ?!~(~4~)~. t \o o o ~/ j Tbus 1he Elsc~sI~in id~16 ~ivi~y co~ru~c~ b~t~ec~ ~ and ~other C[[( S]]-adic cusp ~Fm ~ shouId bo ~cner~1~d i). In pa~iculsr. u~d~r tb~-ordinari~ co~ecl~r~ . . , . ~ . ~ . ~ ~roan n~s ~ 1~n l~r sucn ~cnst~In prime~ ~ di~idin~ ~(~ i), if ~ ~ ~ m~d ~ ~r ~ cusp ~rm <~. r~ ll~s v~luc~ i~ ^34\ ~nd ~ irreducibl~. lt w~s a no~tri~ial tssk to prove this 5~ca~s~ th~ repres~1atio~ is re~idus~y reducibl~. We ~ls~ notc th~I. to p~e thi~ ~e ~<ain ~ced 1h~ rcsul1 of WEes (4) provi~y th~ co~clur~ in ref. j#. ^~ ~ct tbat ~ h~s values in ~34) css~nli~ in 1b~ proof b~causc ~ guaranI~s th~t 1h~ ~ioTnt acEo~ ~f ~ on the ~nip~ter1 r~dical of 1he ~ndard m~im~) p~r~b~bc sub<roup ~ ~cluaHy isomo~hic 10 ~d(~ ~ r~. Tb~ icnsion of ~ot(~) ~- 1 mod ~ ~ ~ mod ~ indu~d ~om =~ b~ made n~Ft b~cause of 1h~ i~du~ibly of ~. Th~ . . . . . . nG~trlYlaI #1~nsl~ ~i~s rise 10 8 nor trivi~l c~^cle in Sel(~d(~) ~ ~-!) u~d~r tb~ OrdinsrRy Co~ectur~. ~> is ~ ~(4} <~rsio~ of a~ s~ment of W(es in (22> ~pplied lo ! (2) Sincc it is true ~r ~s~h b~iph1 o~e prim; ~ 3\idin~ (~. . , . ~onclu~e 1 ~: lUc k3~nst~in ideal ~ of ~ divid~s I. assuming 1he Ordin~rily CoJectur~. {o ~slablisb 1h~ divis)

11124 Colloquium P~per:~Hida~/. legality ~|E/~,illo11]cr wo~rds,to es1~t~1islltbe con~ruCI modem ~r~lI>wc Pedro haveprec~ein~rnad~n OD ~ (~ot~illstitsexislc~c~.for cx~tmpl~, TlsFouricrco~fEcie~nts.its \Vbilta~ker modcl,and so on. ^[hou~h each author had abeam Worked out some 0~1b~ir share of the York prcsonted hare before IBM vialed 1be Webb Research Inedible of ~bema1> and Ya~ematk~ Phylum (~RI. hand, lodia) in January ad February. 1996. ~ coordination in bangs aL tabs earth into ~ ~erer~T fl-an]e~o]-~k was drink wilily they -e jsT1~1]~ All~bab:Id. Me arc gr~sttefu1 10 Prof Dipendr~ Prosed at TORI ~, gi~ViI)) us 1be Opportunity of cooking together sad to the audience at TORI ~: pa-ticntly list~u~in~ 1~> Blur l~clurcs Clan tbc subject wb<>s~ ~>r~ul~ti~n Ems not >ct dcEniIe. H.H. ~n~d~cs the Uppers from 1bc Carnal Seance Ru~(o~ Army the p~par~ion ~ We pawn L O,yenbery,R.~1~94) ~.~.~ T~.(~# 55,P':1} 1 0~-~? ~ . ^~ 4~. Hera, H. (1996j ~ ~ )~ ~ #~^ ^^ 792 V<, 1 AS fracture Tl01es series (Cambridge [nlv. Press. Cambridge. l) ~ x Voj. 23i. pp. 8~-132. ~ Mach. hi. (19~) k~~6 Am. 1^ a. 4. Wde~ ^. (1993) ala. ma. I42. 443#i 1. i. augur, B. (1978) Em ma. 44, 129-I62. ~ ~ ~ ~ ~ ~ ~ AL a. ~.~O 7. Betide I1.(1988~./ >~.110.323-382. .. ~ ~ . a.. . 14. . . - . 16. :19. ~0 ~ . ~ _ , . ~ , ~ . ~ . ~ ~ ~ ~ _ . U#~ E. (1993) ^~ In. 99. 283-324. T^l<>r. R. ~ Wiles. ^. {~199i) I. ~. 142, 333-372. Razor. B. (1987) 6~2 ~ Afar Q. ~lh~ma~c~l Sciences Reseat Iris p~l^1~ns (Spewer. ~ Yoga. VEAL 16. glazer. B. ~ T]ouine, a. {19~) ^~< (~\ 71, 6i-103. [lids, }1. (1993) Egg I ~f[-~< If E/~r~' {London gab. Soc. Student tea, C~rid~e Univ. Pig ~ ~\ I. ~ ~ . ,. T(ouine, ~ (i 988) ^~ ma. 65. 265-320. TLouino.} (1989)~~ Tang./ 59,629-673. ~1~ ~ 10~< Hida. H. (1996) ~ ~ .~( ~ Ce>~Dz~ >~.~# #mar [-~ far ~. Vampires de Society ~alb~m~tiquc do From (French math. Sock Perk), Vat. 67. Ada H. )9~.~. 1~93-1~. Bask. K., Dim, GCr~m~i~. F. & Pbil~r~ G. {1696j an. . , ~.. .... ^~< ma. 124, 1~9. Louise, j. ~ Urban. E. (T995) ~ ~ /~) ~E ~ ~ 321, RIO. T(ouinc J.\ Lrban.E.(j997)!~/ #~2~? ^ ~ S'~ Czar E~e~z~z~7 ~' 6~< ~77~7 /1~.\ preprint. 21. ~e:iss,qr. R. ( j 996) ~ .~? C~ </ size ~l67z<~7z~77777/ /,~777ztz, <~r C~e~fG3p<~. preprint. 22. Wiles, a. (-1990) adz. ~. 131, 493-540.

Vol. 94. pp. tl:I23-'l't~128. Ocl~>b~r T#97 Colloquium Paper G ~~ x,~` p^~2 ~' ~ ~~ ~ <~c bare ~ gear ^~> >> ~ ~ arc ^~r d ^~ R~> A~ ~~4 /~! 799^ ~/ '6~ (~/ ~ ~ fit -~, Ad. The structure of Seamer aroma ROTH ORE#BERC ~D~p~r-t~[T~nt: Of ~ll~e~ll~t~ics. Box 33433(), [,II~:~e:~sil:y <~:~:f ~>l~lliI~l:(~Il. S~arllc. ~ 98195-43i ABS^#T Me purpose of tbI~ article is to describe cer1~in ~su11s and co~ectu~s concerning lbe structure of Galore cobomology groups and Seimer groups, es~dal~ far abeiian varieties. These results ~^ analoques of ~ classic~1 theorem of Asia. Me ~rmulafe ~ very ~ene~1 version of the Weak GopolUt Co~eclure. One consequence of ibis cohere ~ the non^istence of proper ~submodules of Oni1e index in ~ certain Galois cobomolog~ group. Wader certain ~potheses~ one can prove Abe ~onexIs1ence of proper i~ubm~uks of OnDe lade i~ Sclmer groups. An ample subs At some ~po{heses are needed. 'T~'1~c restyles 1]1~1 1 milk describe 11ere are motiv~tO<1 by ~ eO-[n~ 1bc0r~m of Basalt. 61 \ be a finite extension of A. St () be the cy~olom~ (-tension of ~ Hero ~ is additive group of>^adic inSo~cr~ Ho let A = [[If]] be 1be completed group algebra Of ~ Ear as. Hub ~ ~om~rpbic (n~ncanonicaDy) 10 Ah formal poor furies Dan [Ian Lo1 denote the maxima abeLsn pram ~~ns10n of ~ unramb .. . . . .. .. prom 0xtUrsion of ~ urr~miD~d at PI primes of ma. 61 ~ = al{~'~/~.) and ~ = ~sl<~/~. Len ark. 1, Lass proves the bowing important result .~ .~ . . }TEORB~ ( ~asa~a): ~ yawn ~ <~) 2~<,>~) = ax, ~ a,: )~ /~ 7777~7' at/ ~'< (/'/~! at'. . . 66y#~ ~ 03 ~ (~! ~ ~ #< \-~6 We remark ago that if Ha/\ is an arbitrary 7~-ax1cnsioD. #) and tag are Duo (due to Iwas~. Statement Q/) should co{~clura~y be true. ~ is often rumored to as 1b~ Weak Lcopoldt Co#~ctur~ Or Ha/\ and teas tab ~ll~i~< int~r pretaBon. Eat ~ denote the unique subtend of ~ such that \1/\ ~ ~k of de<=o~. Let < denote the composkum of ., ·---, . ,. , . ~ . . . . .. ~ ~. ~ tam ~ ~ ~ ~ ~ ~ age+ #~ ~/# = ~ . charm 8~ ~ O. Lcopoldl) Co~octuro states that ~ ~ O. The Weak Copoldl ^~ctur~ slates that 8~ ~ bounded as a, which is equ\,J~nt to the ~sserd~ that r~nk~)j = a. >ho if s1~tomer1 (~) bolds then so dogs statement (ah. (See ~ , , ~ . T)FOT)OS!1~:>~ 4 {~[ ret. Z.) . .. . Returning to the c~lotom~ 6-e~onsion &~/~. me can restate lw~\ theorem in farms of the Fontry~in duals l l~om('\. /), H0~:ll(~/, 3~/~',). ~11[C~11 {1rC St]b~1~)t]PS Of //1( Age, ~/~) = HO1T1453~1~! / #/ #/~) darned by imposing coriain Ioc,1 conditions. They are example of what have coma 10 bc caged ~Selm~r ~ 1~7 ~ 6~ Nylons Madly ~ #~c~ <~4/~/# T 1 lSi#SI/# PEAS is a>~<til~lt~lr (':~lTl~:{lS 611 1lt!~[~:~/\vww.~ll~ls.~:>rg. roups~'D~sawasresullsiben become:~)Hom(Y /) and Hom(( C#/~)are~o~nis\~an~raled -module) Campy C#/ZO ha A~co~k a. (a) Hom(( ~/Z>) !-cotorsion. (~) Homed. ~/~) has no proper )- submodul~sof]~iteindcx. \cwcons~ranib~i~nv~dely~ de~nedover~ good. crd~aFyreducdonsaltheprlmes~ ~tin~over<. W~d~note bye 1he~-pr~n~ysub~up fifths das~calSelm~r group Arm o~er\~.O~erX#.th3Sclmergroup~dc[~cdss ^ . . 1:~)! [O~S. lee) = kern. 3[~]) ~ ~ 7~(~)). Shoe amp) = (. [])/). Here J[~j den ,,,.. notes 1[e (-power 10rsiC0 points on (I ~ rugs over ~] ~ - primes of a. ~ ovcrthe primal of63 land over ~ and far denotestheima~eofth~loc~lK(mmerhomomorphism Ark ov~l-the ~-~tdiccoll]~plctioll \~of ~?~ We d~fi[lc71.(A~) ~ P#n !~) (ash . Ten gob = (~n SeL(\z~ canbe demand by say ~ =~r~l< /l3~)~ ~736y i, = k~r(771(~/~..4~])-- ~ 71.(~)j 1.' bercE1ssEnitesetofp~mesof~containin~ ~lprimesof fibers b~sbadreduclionas~ollas~U primesdhidi~ or~.lnth~e~rlyI970s,~rm~d~tbc ~llo~in~co~ec~re. abort \~/\ ~ assumedfobctb~cyclotomic~-extensT~. 0~.;E(~J~1J~E (~Z~ll): i~,.1~> it I-. Uncap ~/kentheassumptionthat~ has~ood.ordina~y r~du~ions1 ~1) dRiding~.Forcach )~.~l~<denot~th~ he[~hlofth~:~rmal~Foupassoci~10dfolh~ Baron meddler ~. .. .. ovcrthcintc~ersin Any finite e(~sio~ of &) There a~bi~v~ssem~tabieFeduction.L~t~ = dim64~.Tben Y~z~r~ co]li~ct:ul-o shoed be true if ~/~ is the cycl<~-to~mic ?~- cn(~na~d<~ = ~ ~raUp~m~ Sofa ~over~.Us~ results otroF 3.onc can show that SeL(~\ has postage !-corsnkif<~ ~raile~slone#~nd Dr~ny?~-cxtensi~n in ~hichtisramlE~d.Onlheolherb~nd.~shouldr~maIk Hither m ~ exist nonc~lotomic?~-~xtensions~f~ Obese SO ~iJstobe!-cotor~on even J~ bas~od~ordinary r~ducti~n~tt stat I|. ~[701 exal~ple,1~bis C{111 OCCU~I if~.<~isl~h~ ~nti<<lotom~ ~-exiensionof~ima~in~ry quadratic gelded See r~[ 4 Discussion of this bsue . . .. . lo ~lDd~scrlbevarlousconsequencosif~assumeth~t \~/\ iS1be c~lotomic ?~nsion,/ has~od,crdblarv re~jucl~io~laTl~pr~im~s Or ~ oval- a, gild Sells )-c~ lor~ .. . . ^~7< The !-cor~nk Of ~1~/<,~[~]>c~n be del~rmi~ed.ForZ = 0,!.snd 2.lhe A~modul~ 27~/~. >#Il~r~<nil~lY~en~rated~ndthcircoranks~r;rOjO[Od ... ., , `., . . ~ _ . . . ~ tear clc^7olnc~ c~t~rlsllc

J 1 I26 ColloquTom Paper: Greenberg 2 .. . 7( 1~ /~) = ~m~ ~ () From tab one gets the Lear bound cork, .~[~^J)) ~ [a: dim) latch cEltlalil~y if and only- if 172~/ ma, ~ [am]) is A-col~rs[~ (since ok/. ~ [ha] j is obvi- o~sly !-co1~rsio~. The calculation of 1b~ above global [ul~r- Poinc~} cb~racl~rLtic ~ a consequence of results of Plaits and Tote far anus Patois muddies over number Outdo Used their results mar local gelds one Cal prove lho ~i~ Act: Corank~< ~ #~/ = (~:~]dim~j. 1.'~\ Me d~ridon of the Selmer group and tbc ~ssu~Jo~ that Sch(<p is i-cotor~o~ then imps that cork/ >[a ]~) ~ [a: ~.ldim~\ , .- . ^ go ~ ~ ~c ma ~13/~) ~ ~ ok) <`e<1 . . . ~ . .. . ~ suck. ~ ~ Afar ~ compar10g the )-cor~ that We cok~rnel of tab map ~1 be A-coto~lo~. ~ su<cc1\i? ~ a consequence of ~u~i~ ah beb~ior of the cor~sp~dir cok~rnels war the Am. Any uses ah Kahn Al abash ark . ... . >c ~ ,. ) ~ ,. C<~ S: 1~1 ~<dditio:n to tulle above ;Ssst~mptio~s, {tSStll~C that at 1~1 one of the tallying hold: {0 /~ has DO p~t<>rsio~. (~) For shale ~ ~ a. 3[~=lf HIS ciliate. {~) Ida solar Ha. ~> ~ ~ - ~ Tbe~ Seeks has no proper ) submodul~s of [flits index. Me proof of thy co~s~ncc ~ discussed in a much more genera contra in rcE i. TO aft, 7~ denotes 1be lncrha subgroup of 6~. If ~ ~ ~r elliptic curve, then {~) ~ equ\~le~t to ~ having ~dd1Jve reduction at some ~ ~ a. 1~ (>Z~ ~) ~ the ~ificali<>I] index 1:h~is ;lsslimptiol~ cry llolds :if~ :> [a: A] ~1. ~ssu~l~l>1~i<>ll {~) ~1~> 1lL>~lds !~{ ~ is s-tiftic~ie~l By lyric, at least r~f~ed~and\. 1 Art ~ add several remark a50u1 tees con<cq~cnc~s. Consequence I should be true more genial>, Wilbur We Fringed! ~ssu#ho~s made Abut. For any Aeolian variety Ad ~/a~ ~ ~ \~ A/{ R ~ c-< twain Bus that ~!>/<,~[~]) has Asorank equal 1o ha: ~]dim~\ ~ is ~qu\~lo~t to the Sermon that a/. J[~j) is )-cot~rsic~. I wee Late later ~ much more general conjecture which Ail ~ include the Weak L~opoldt Cool ~ ~ ^ 3, i~ ~ ~ ~ ~ ]~C,U,< ~.2,~C ~ 1. Concorni~g consequence 2, ~t fl denote s anile set of ~ ~ ~ #~ ~ ~ ~ a ~ ... ~ .. S~lmer group SO by SO ~> = k~r<!f T add, ~ [age]) ~ ~ 7~(~) } ).~i'} Thus Setup ~ Sea. Choose a ante set ~ as board, but also complaining O. He su~eclivEy of ~ gaffs an isomor . . an-. ~c>~ t'' " A"' S~!~(\'.~;~/SeJ,..~.~ S.7,.~..~) tab!} .~`, . . . . . . . . . . 1 no 1somo~nlsm gas an 1nt~FOs~ :~te~relaTlon In connec . . ~. COD with hazards Main Coniccturc~ which asserts that [ha ... . chsract~risli ideal of the \-module Scab` ~ generated ha ~~ ~J~m~n1 ~ ~ ~ skied to 1he/-~d~ (-blncti~n Cry Mar a. The Distance of tb~dic (-<InOd~n ~ Anon ok under very restrictive hvpotb~ses. e. if ~ = ~ Aid ~ is . . ,. . a, . modular ~iplic curve. But if it exists 1bon it is Bag 10 c<~rslrt~c1~ :> ~ll~l~lp~ril~l~il~ive' ~lIlalo~u~ wll~h all Tr]tcrp<~>l~ltS<~>T~ `,. property involving rajahs of the H~sse-WcT! Action Cry gab the Euler actors far patios in ~ demigod. fine could am doEne an Lament ~ ~ a. 1I turns ^1~1 ~ = afar. Chary En generates the ~ar~ct~rk11c idea of ~ #ark. Tbus 1b~ 1.~1 . . . . .. . . . ^ main co~ecW~e ~ equlv~nt lo ~ nor alto ~sserdng {bat the cb~racl~ris~c ideal of Sank is g~n~r- at~d bv by. , . . . :oncerln~ consequence 3 some r~sthc1~0 These ~ necessary.~cr~lsanex~pletosho~tb~k 61\ = Sand = 5.L~13 bothccIEpt~ cur/ otconductorl] suchthat L(~) btdviaL<Thcothcrt~o ~HipticcurvesotconductorlI roiso~cnousto ~ andcont~ina ~ralio~Ipointoforderi. = }~ ~ ~ ~ ~ x Ga\~.Let~denotelhe TeTcbmul}crcharaclerof}.Th~ can decompose Seek baby action of a: I` .} Sea = t ~ S~36<~}< ;....-, . ,. Unec~nd~1ermin~tbc ~Fuct~re~a )-moduleofe~ch Actor. TheresuJtisthattbc pontIyagin d~alofS~l~;~isomor- phicto:A/32! >~ 0.0~ 1 the maximal idea ~ A/ (which ha6indexijiff ~ 2,~nd ?/~7 k? = 3.ThusSe~ basa !-submoduIeofindex~ ~ 5 th~k~rnolof pI~ectin~ to the ~3f~c10r. It is i~tcres~g ID note that F~s~a? ~-inv~ianl far S~G(~\ ~ no~zerointhe above e<~mple.~{~zur era gave > = it= s case he sh dlh~1 ~ = 1. The behavior oflhe ~-invarl~n under Owns has been studied by Scb~eidor<73[andin moreg~ncralcon1~xtby P~rrin-Rio~8~1 U<n~theirresull~ 1be Adoring conjecture would pr~dicttb~ halve of a. Con ~ ~n~ C.< ~ 5~hei~nousoD~ti ~ ~( ~ >>313~ ~ baa Sel,..1(.~)~ O We All Saw ~rmul~t~ ~ general Vernon of tbc Wean LcopoIdlCo{ccture~which~\esapredic~onoftbe Crank of, and,asaco~equence,~/~.~0 Bra very Sonora Oal(~/\O~m~dulc i<. The previously stated Y~iOn is Bespeak c ~ if = C>/~,cD which ~ ~6~/~0 ACTS tlivially(<llld ~ =1:~beselo:f~p~inle~of~Iy~i~ OV~T'# oared. Pious ~ora(/adon~ and specT,lcas~s have been consid- or~d bySchncider(7). Gre~nb~ (9< Cot sand BicConnc~ (~lO).:llld Pc::l~ill-Riou (11~. The~f<~r[s me grill grille third is inspired b~thethesk ofBicCon~elL Let L be a Anita demand . , Oak C#<ep~senta~on space Or Ga\~/~), Are ~ is Enitoselotprimes~f~ conlainin~1bc prim~sovcr~ and a. Into baa OaTois~i~variantZ~-l~tEcein K [~13 = dim. )~ = dim~,tP<) ~rtherralprimesof6~ There Pa denotes 1he(~1~-~\e~spac~s ~racompl~xcol~u~ado~ above ~ Let T7 = P/~.I.et<~/\ 5~y~p-cx1cn~ion.ltisk~o~n1h~1both 6~/~ i/) and Ok, T0 are counted Nerved A-~lodul~s<~h~r~ ~ = Z~[[~]] ~ = Go/ a~dtbat cor~nk>(~T(~/(, go)) ~ corank<(~27~/~. it) + 2. Chary ~ = ,~ + >~ ),.(See reE 9. pr~pos)io~ 3. The E~l~F-Poincsr~ char~cteris~c Oaf over As -8.) Foray prime ~7 (at a. ~c lot !~(~.j7) = I#ll (A 772{~ ~)i, /! am fibers ~re~cb~.~runs~verth~primosof6~ Logon Ens CAD p ~ ethe R(l~win~ Issue. PROFOS11~)~. ^~ ~/ ~# 22(~/~ ad) . t..~.. Our~cr~on ofthe Weak L~opoldt Conjecture ~ the ~L . . 1~ ~! {1.~. e ^ CoNjE6ilJR~ 77~ ~ 22(~/~, ad) ~ ~ /~< ~j . . . SS<~777()~. fine cab sham chat ~ ~ dogs not Spa compj~teIyin ~/~ l:lle~77:~(~., A) = t). ~o~o~or.l)~ri~[lles CSti1 So)] C$:~)mp~lS10~1) {1] a{~Oxte~io~/~.Forex~nlple,1hearchimcd~anprimesof [lsplb comple1el~ If ~ is an ima~inarv quadratic geld. . , ~.. ..

CoHoquTum Paper: Greenberg then every n~imcd~'n prime ~ of ~ not deeding ~1 split completely i:n <>lie 7~exte~ns:i<>I1 offs A. [T~b~is is obviates because G~/~ ~ ~ gad tbe doco~osLion subgroup far ~ is isom<>rptic 10 7~.~] If ~> is filbert ~ ~/~, then ~> splits completely in ibe ~/tic~otomic Pension of ~ ~ is co{cclured that far am otb~r Nonextension of ~ at most oar prime of ~ can split completely. (One can paw that at mast two can.) T discuss s~vora1 special cases. First assume abed #/< is 1bc c~Iotomic ~-ex1cnsion^ Thud 1hc Abed co{~cturo sages that #~/~) ~ ~ As. ~ because no~archimedc~n primes of ~ cannot spat completely in /. If ~ ~ odd. then ~#, ~ = 0 far (~ and hence ~=~ none 11 ~ (~/~e and in (A /24~-corank Sloan d~im~/~/~l,~1i,J, Where ~) ~ ~f~f(~!, Ha. 1~( the special case There #- = ~1~]. were ~ is an Reeler vary eta/\, ~l)/~.~)<iv ~ ,4~t.~/~<l<~>>l. the group of C011 elected component ~L can be nontragic ~ ~ ~ a. Let ( /\ be~ny7~cxl~n~ion.Considcri! = 3~/Z~a~d = #~ ~? ~= /716~/~ if) = O~m6/, C#/7 Chain ~ = G~l<~/~),>~d~notin~ ~sbc~for~t~be imp ab~li~tn prow cxt~nsio~n Off at< tlnr~tmifi~d oul:si(le 3.1~ tilts case ~ = Rand the abcvc co<~cturestatesib~t ~1~/~ ~)s~houldll~e !~co~rallk'^~-~.~. J~k>~)stlouJd ~q-tl3~1 ~ 2. Tag ~ ebb Weak Lcopold1 Co(~ct~ro bathe Z~-~en~n S~/~ as mated Ladler. Let /~/\ ha any ?~-extensTon. Consider if = ~ ~ C#(I)/Z~(Ij.L~t? be ~ ~nitcs~tconlai~ig~aD prover (1 Ha. T1~0I1 itisno1~difficul1l~o~prov~the ~vc;l~k~[~eopoldt Conjecture ~rAfand \~/~.<Tbisproofis~ivc~inreE 3)ln this case !36~ {0 has posed )-corank if ~ ~ ~ nabs arblmede~n prime which Sp~tS COmplOlC~ iD 6~/&, Thus /36~/~ ~ car have po3/ve )=corlDk. #~ =~. ~/ = 06~- dean eland Crony -tension \~/~.Tbc Weak Lc~pold Co{~c~r~sta~slh,1~/~,if) = 0~ \~nvoddprime. There ar~som~ known c~ses.Forexan~pl~.if~ tsar elliptic curvo/~,6~/< isthmi cyclotomic 7~-extension.and \/~ is abolia~. then the conjecture is sciCed [f ~ bus complex multiplication and good.ordin~ryr~ducUon am [Rubi~127. There he proves ~i~zur? co#~ctur~ in Ibis cant if ~ bus complex mu!<pIic~tion~dgood.supersi~gularreductIona1> (~c(SoilJ~lell).alld.m(~>rc~llcr~lly~i~1~76~lodu~1(lr<TIldbEls?~0d r~ducdo~ ala (~>to). Ab ~ftheseresultsusea nonvan~hin~ tbcorem of Robr[~h Bertha Hassc-W~il!-~cdon. lot ~,.2 ~=k~r<~/~-...~/j-- ~ ~.Af)~. t..~i The Weak L~op~ldt Co{cctur~ ~rTfsnd 6~/~ then asserts ha. !~) = 0. We manta ~t~cquKalen1ver~on (blspiredby ~cCoJl~ellj.L~t /? = HO~T~,(~ ~(T)) 8~) ~ * = Homage ?~ L~ti78 = Am/. Deane ~. a. A) = k~r(21(~/#, Ha) ~ ~ ~(6, ~)~ TO ~sacons~que~cc ofTatrY globaIdu~lLythcorem.~ne ca~sbowlb~l~,\.~nd-~l(~.1,Jf~jEa~th~ssmC Crank. The Weak Lo~poldt C~nicclu~e then asserts . i.~) ~ {-cotor~on. LO d~noteth~ Aged bald Martha kern~l~fth~chon of 6\ on Tf~.Let ~ = G~/~.Th~sthe~c~on of 6~ on Off Octo~sthrough as. Lag (~ denote the maxim ~ aboard pram exlO~sioD ofF~.>hi~h ~ u~rami[~da1~11 primes offal. Then ~ ~ C~1(F=/~) ace on ~ = C'. Lam /~ >~/ 7~) \~`~. 0~4 )~(~> 1~1127 . . . . . S~adlOl~eGroup,~ 1S aclosedsub~,oup.~ndon~basa~ex~1 ~quenceI -~77 -a ~ -+ ~ -a ~ Car ago bas )~ n^~dcdon julep . . 2~.~.\,AY't')-~ [lolll<~-{~# . AIN'T. Tbe keyed of p is a subgroup of 7flp) Ha), which is A-co10rsion. We assume now 1ha1 \~/~ ~ ebb cYclotomic 7~-~e~sio~.Thentb~cokernelotp Logo #-co10r(0n.Thus the Weak LeopolUt C~{eclurc would then be ~quiva]enIlo assuring tb~tlIo~,if*j ~ A-co10rsion. ~ theorem of H~rris(13)slateslh~t/~isatorsion-m~duIeovor<~[[0O]]iG a~ert~ins~se,~h~re 60 ~ ~suT1abI~p~llsub~ro ~ of O.If fireplace ~ by~nite~ensionco~ta~edi~ basso Bat ~< c!strivi~l} onif$~]),th~n Is pro ~roup.#ssu~l~lh~l ~/~0 = 0, which otcou~e ~ a ~lFkno~ coniectu~ of Iw~sa~-a. Tllis~nlcarlstb:~t }ad ~ Gal,/ is finitely generated J~-module,~h~re (~isth~rlaxi~31~b~lia~pro~ o~lonsion off un:~miEcd ~verywbero (denoted by L~ care lier).Bystu<Jy~in~tllc lTl~p }~/-~7y<,, -a /~x ~be~r~/~<isth~ u~mcntador idea of 7~[P~#l and by using ~ vcrdor of aksyam~ts~ler~rll~,<>ne finds t~hs1: (< blast be :~ finitely cra1cd [module. Butlbe Weak Leopoldt Correct . .. . .. . . 1uFc1~ (ano10rthecyclotomic}~-extensio~/6l would then ~lJo~ because Hon\~}~if~T Squid cons~quer1> be c~Eni13> acted as a {~-module and tbero~re A-co- lo~ion. ConlinuTr~ to assume 1bs1 6~/\ is the cycIotom~ic {~- ~e~sion.]el)I$(r) donotc1be ah Tate 3~31; Bore ~ 7. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ^ equivalent firm of1be Weak Lcopoldt Conjecture Rand Wily Maya ~ Z. Hare ^ ~=~ ~` bichh~shnL~-cor~nk ~raTIf.This ~rmul~lion~luslrales ^ = #~w Aid /~), ~ ~ 7, are sp~ciaILaii~s of a representation G~1T(~/~) -a (;~.~(A). {~hic~llis fit deforln~lli<~>n of ''ale cyclotom2~ damson as de(~din rag 14~. The Weak Le~pold1 Conjecture Rind Bran Try 7~-e~cnsion/~/\ h~st~oconsequenco~ which monologues imparts ofI~sa~as Dleorem stated Or. TO Grstisibe obvious consequence that ore could then d~t~rl~in~ the Outrank of /. I0 and hence of 2~/~.if).i~ arms of1h~ EuIer<Poinc~r~ ch~r~cterisLc ~ fir i! and tbc -coral ~flheloc~l Haled coho~ology~roups/36r. i?) Oboe ~ ~ ~ ~hiobspbtcompl~slvin a</ Thes~c~nd .. . . c~ns~quellce~islh~i~llo~r~s<~. i.. PRUPOS#1~: /~ FEZ ~ ~< (~6 ^~6 >~6 7 of <~) /. Zags 17--1(~/~^, ha) ~zez~ ~Z(:> >~e' i `~77-ZO)Z/~ #~ 377776 I. l~Tdlikelonc~ discossbri~flyS~lm~r~roupsassociatod 10 modular Army To iJustr~tr.co~sid~r ~ = i;~1 ~(~)~. Chard : \ R~man{,nSlsu-funcho~. Wolet k drools A{). thc>-sdicr~pr~sontatio~assocI,t~dto 3.l~e1 ~-~ P7{,~terc ~ = () iS [L (~-i[lv~lr~i:~ll~ I. ret ~ = {a. a}. Assume ~ ~ odd. Than the Scalier group fir Af OVOI 13C cycloramic ?~-exteDsion of ~ hasty Plowing deEni3~n. iSf(~^,) = ((/,, Ad} i` (,;, ~)/~..<,), ~=~(~=~(~# ~ .. = Lamp ~ ~ cyd~tom c ~ (I(t~ {~ ~ ~ # layer. For Bay gaily ~1~nsi~n /. ~ (~.~7) d~nole~1h~ ima?0i~ ~l(~.A!)0f#;(~> L),th~ ~sSbsp~ccof By! (go, a) donned by Bach and Knin the so~d ordinary case

l l I28 ~llo~ium Paper: G=~ ~bicb means ~ ~ Add. one can degree ~ as Ago. It is using 1be result that !~) bus no proper A^-modul~ of know Act 1hrrc cants ~ one-d-~nsional ~-~bspsce ~finite index find hence !~j conga bc Snitch A very of k Which ~ 0~ Invariant and scab that #+ ~ unarmed genera] result of this nature is armed in rc[ 9 urger r~tb) ~, ah Ads of 6~. St ~ denote 1~ image of ~ under the map P ~ A. T[]el1 it turns taut t[la In, = I;., ~) = Imp, ~j ~ 7713~ #) ~ ~ = ., #(~.~b~d~ <~/< Ha. This has heed proven by Pc~rrin-Riot1 if ~II7~> If ~ ~ :~` then !~) ~ A<cotorsion (proved by Kato). We consider two ordinary grimes:> = 11.~ = 23. ln red lit I hoe calculated tbc structure of i~) fir thee primes tin as ~ A-mo~k Ark ~ 11 < ~ groups i~) ~ ma/\ in both cams. The idea behind tbc cumulation ~ lo use Collie congruences boolean modular farms: ~ ~ R(mod I1# gem thy modular farm of weight 2 ass~ialed to <(i 1~. Ad ~ ~ 6(mod 237 fibers 6 is the weight j modular firm associated to ~ certain dihedral b~o-dimonsionaI Latin chars actor. Olle clan user easily vcl~iEcd ~ctt~ba1 !~[~1= i~~1.~bcrc one dc~n~stb~ S~ImeTgroup Tribe Knin O~lois~modtlle A7[~]in it Any an~Jo~ouslo1~he definition of !~<~),usin~thesubgroup \[~]ofi!L~j.~One needs mad byp~lhescs on i? lo verily this Act) One can c~lcul~le the Sclmergroop over ~ ~rP>6\~11)j~nd ~rP>(p)<moduIo ~-l~thccs). This allows one to show that in both cases . ^ ,~ ~ j, ~ a, . ~ . 3~hasorJ~r>. ~neconcludesths13~) ~ ~ - - _ ~ f _ restrictive bypolhcscs, and much more gencraLv in reef. Heaver. as Indicated Drier. thcre arc cases where such a result firs to bc truc. lh3~o~ ~ panda pond ~ ~Sd~Found~ ~1. . . 6. 7. at. . 9. ~10. = ._. Lusaka, K. (1973) Gus. my. 98. 246~26 2. Greenberg. R. (1978) Em am. 47, 85-99. 3. Coat^. ~ ~ Gre~nbe~, R. (1996} ^~< 6~. 124, 129-174. 4. Datum. B. (1~83) in ~'z~ <~`~ ~Z/~73~/i)~<Z/ (~S #~7 ~ (61- S^n~ ^~^ ma), an. 1~ Ore, R. in preparation. az-ul-. B. ( ~1 972) 7~. Ace. ~717~. ~18, 183- 266. ScbDeider. P (1987) ~ ~ ma. (c SZ, Ti9-I~. Pc~ri~-Riou. B. ( 1369) ma. Iran. ^~ ma. 17, 34~3i8. Creenbc~. R. (1989) am. ma. ^~ am. 17. 97-137. C<>ales, J. ~ ~cC<~nncl~l. C. (~1994).! !~; ~. age. 30, 243~69. 1I. Pc~n-Riou. B. (T^) /~ 229. 12. Rabin, K. {1988) ~< ma. Ha, 701-713. 13. Harris. ad. (1979) a. amp. 39, ~177-245. I4. Cre~nb~ry, R. (1994) ^r ~ ~ ma. 5S, 193-223. 13. Crecnbe~. R. {1991) (# (~ (< (Cambridge Unix Press. Cambridge). Vat. li3. pp. 2~1~1-234.

arc. \~:~^~;~` $~) ~ Vol. 94. pp. 11129-1JI32, October 1997 Colloquium Paper , it.. n) ~r ~, ~- ~! ~ ro~~ If ;~c 0~~ ~ ~~' 7~>', D'~ ~ ~) ~, i ^~ ^~> A~ ~- 7~< 7~{ ~! '6-e fit J~~ ~ I ~ A, ~ On the coemcients of the char~cter1st1c series of the ~oper~or ROBERT ~E. MEAN ~I)~p~-l:n~e~ll: <~1 Y>ttll~ll~atics, lily <~:~:1 C~lif<>r~i~. Be~:k~l~y, Cal Aim SS4~) ABSTRACT ~ concepts proof is given of the ~c1 that the coons of tab cba~~teristic series of the ~ope~tor . ^ . , , acting on lima 1es o1 overconvepen1 modular firms ale In 1be I. . . ~ ~ ~ lntroduct~ in this <10clll~llt. 1 ~1te~pl 10 ' explains MY the iS<]rmul:1 fir the char~cteris~c power series fir the Moderator acting on ambles of complain contin~us~-sdic modular ~rms~se~ section B4 of ~E 2) looks tbc ~ ~ doer In other words. I Me a c~ceptu~1 proof of thy part of tbcorem B6.I. whence is add. . . . . . . - w~ ~ Cadent trom Abe explant formulas (see appendix I of ref. 1) and which asserts shag 1he c~cfCcicn~ of Dais series (e fill talc I~ts~a ebb a\ = ^[[Z>]]. ~1~ also prove abut delis . . . .. 6erl~s~na~tlc~llycontinuesto~lar~erspsce. bis~ssasserlcd byte 1beorem Add ~ ~ot~videnttrom Abe ~rmulasflbav~ notprov~ntbisssserbon~be~> ~ 2~.1uselbcoperato;caJed Insertion B4 ofref.1.~blchislhe L>-oper~toron weighty ove~onv~,g~nt firms twisted by a ~unEyofEis~nsleinscries E<soesecdon Ibelo~.The keypointislbalthe<-~xpan6i~ co~fCcicn~ oft ~ ~ C 3.lh~ ~ crumb ~ provetbatthe Dacron Fig ~hose~-cxp~nsion ~ E(~/E(~)h33[n ~ ~ <FEZ) . .... . . . angry ~ ~ 1ne connected component oflbe ordin~rylocus cont~inin~tb~cusp ~ in)~<q)[asortofafinoid~-~xpansion principIo (~e Theorem 2.I belong. The operator ~ Bus on .\ ~ <~(Z~-) an<1~ifit male coolly con1~i;lllcust~hs1 ~)tllti basic~lly-d<~il,t~tl1:il~s~no1.1~nl ~rce~dint<>som~tecl]llic~1:itTes to get around ~ dificu~y in seconds S and 4 I complex tbo proof in second 5, and in sacks 6. 1 prove theorem B6.2 of red 1. Den ~ is odd, ~bicb assorts 1bst this chaTacleristic series 'co~nlrols fours of Hiker TV. ~:~.#q = 4~ = 2~> 01he~isc. Let A = ^[[1 + q7~]1 ~ . . . ~ ~ ~ ~ ~ ~ a ^ good reduction. lot ~ = ~ x Andy denote the < of ov~rcon~ergent humid sr,>Cc [unctions ~ ~ over ~ rhea section Ai of rail 11. If ~ is the rigid space of cot us characters on 1 + A with values in Cat i1 is c~rm~1 over ~ to the open unit disk. 1 can gild Jo think of A as Maid . . ..., ~ tuDchons of ~ dawned over ~ bounded ~ I. lf y is tab fined unit disk ash par~mel~r ~ let J°~\o[~? drools ,#~(y\~)r).~}~.Ide~nti~ill~.~) ~it~ht~ll~pe~Fl-urlit~iisk~.~ell]s~ naked ~ ~ Z~[~\D.ll~ (~ each O ~ ~ ~ 1 sad \{ &~2 A.s~t ~ :# ~ . _ . I ~ DIN ~ax<<l>~ll>> 1 , ~ . {~ 1 0 13~97 try 1 ha \{tti()llEl1 ^.~!I'1'T}' ,~f Sciences ()()27-8424/37/34 ~ 1 129- l$2.{~{~)/(} [IS is :>V<~!~t~l~ Saline at ::/,' p~l~s.~-~. ~7 E|~l.tbLIstbenorm oblsincUupon mappingan Emend of ~ ~ ~ J01(~) ~ndib~ntak~ ~ gun Foam offs ima~.T>en,If~ ~ ~ ~ j,l~g<~) ~ Owed on~c~ne~si~ aback ~ <~ ~ i <~ ~ I aria ~ ~ ~ . . ~ . . ~. . illl.plics I~7beth~ maxim deign #.Suppos~< ~ ~ ~/ ~ ~ - ^ ~tag ! ~ acidify ~ ~i~rl{~l(3g'(~). - 10g~(I/f)}~ f ~ 777 ...... . ^ ..# _ << ~ k~ ! ~ ~ ~ 1~ ~ ~ ~ CORAL 1 AR Y HI . 1 . ] . ~C /71Z(~ ALLA /~ 3~)~/~. 1 deans Spay to be the subbing Of ![ ~ ] consisting of cI~mentsoflhe:~rm .y ~ /.\ , { ) ~r~hichlb~rcexist~an~ ~ Oin Rsuchlbsl\~ E72< ~ri~r~e a. Th~n./p) ~ Spat if and only ifthe image of ADO in °lM/~1 1^ ~ 3~< ~ game f ~ 1 > ~] 13 ~? . .. . [1 flus LEERS 1.2. X?~[~J(~/.~) ~ ~T!1! 2. ~ -Expanse Principle 1~1biss~1ion.I ~ ~ Drove: ... THEATER }1 G-~# (~ ~# ~ ~ Its ~) /~) {Z/-~}~! (r~?Z/# /~) ~ {~?/ 37 Hi (23~/~)~) ME\ 2.2. 7~ #~? ~ #~!~! /~(~/~7z f (ala-! 7? amp ~1 ] ~ ~/ 7~< f ~1~{~)) ~ ~ ~) ~ ~ i~) . .. . . ^^ Eat Z ~ tag reduction of Z Id 7) be the dolor of dared zero on 7. 3!~] - (!~1 chard T~r . a, ^} ~ tbc sat of points at ~ (tb~ supersin~l~r points) on a. Ion ~~ is pr~cip~1 Or some positive integer ~.~>pos~ a; is minimal. [~L a ~nc~on on the completion of Z Lab divisor a, >2 [I] is a If separated m~rph~m such that ~0) = 7. We ~ ~32~ = ^~ ~ J°~[1]~# ~ ~ J°~) Id ~ = amp. tbou~b1 a[ as the <~ of the dosed s~b~h~m~ ~= of I, 10 conclude there is .... . . ... ~ ~13~ onto an ov~rc~n~er<~1 ~ction~on Z which ages a F~71~ morphism of degree ~ Mom /? only ~[1]? ~(lh~ pr~perly~-j(O} ~ I. r777~!< <<< 7/~7 ~/~7z>~2 Let ~ be asinlh~tatemen10fth~th~rem.L~t f be Asia theorem ma Suppo~e~hasd~gree {.L<t Zip do Mace map Tom J(/T)to/~[l]t~ Lel? bc1hc ~a~d~rdpar~m~t~r on Al.R~ardl~sap~ram~t~ral =.th~ ~cttha</islo1 r~ml6~d abase OimpIi~sthat /~extends r~t~raUYt~ ~ maD [Ann Z.[[3l ~ Z~[Pql F>nc~.\e ~< Bake 11 <an .......

llI30 CoIloquTum Paper: Coleman ago= ~ ~ ~ .< ~ am. . 77z as ail where ~ ~ Ha. In ~CL ~~ = 0 ~[ ~ ~ /. Far r ~ ~o~7i~ ,0~= ~ imp. - ~ ~ ^^ ~ ~ (Z~-l/~[71)? 1:(:) (-~[~'l ii I) ~ alla ~C ~x1~e~n.d 73~ ttcco~rd~ill< W. Than. , .. ~ ~ ~ , ~ = ~ ~ ~ ~ ,! /~! , ~ \ _ \ ~ ` 1 _ ~ ~ ~^^ /^ /~? \ ,7 ,/ keg. ^ the >~^ than ~r eacb an, the coe~de~1 of ~ Senile sum so lies in A. ~ also Ken I ~[11~1/ If]. S~lI1CC (11~SS EMS true ~1- :~11 ? ~ ~()(7t) ~c conc~lu(i~ Ago T(Z~/~} There ~ <cncratcs the d~criminant ideal in Z ~, of >°<z?~/~t>. Since ~ is separated. ~ <7). ~ principle :il~1 ~11~ Arm: (X)a(X) ~ AXE ~ a(Xi ~ {[~] ~` / = ~= ~ \ /f BEST {a 1 bathe d~<r~e~<thereducho~ of~># module Ahab defined because ~ ~ O.Lsin~lh~d~i~o~I<or~h2~e m ~ ado )~ = amp ~ + typo ~dx~r >60 ~ ~ O ~ a polynomi~lover2#ofdc~ree strictness panda /~[~]~.[Wc~rs~kllo~ wccal><J<~1b~is~il~1l/z,;~)0 ~ (,~.[heT1 h~equ~hon~? ~ ~ ~ = ~ ~ (~<Olmplies/~pO e SPIT / ~ all + ~ /~. r 1n~ Ad. [~e second sag must be 0 Once it has degree Dicta lass than a. Since J^/ is ~ integral domain. me conclude e am= > ago. ah 1~l~l]l~l Ells frt>~l cello Ltct Levitt .\ iS c[~>set1 ~11 4 1]~ ~ ~ . . . gore frilly 1.1.I. may True, u~iqu~. ... . . . . ~ = ~1 + ~(~S + ~ ~ + ~) -where ~) ~ /<J(~[f~l)[~] }. The 1~ happily the [ammos to ~) = Aid deduce the theorom. {~t ED do the Cement of T[~* such that BEE]) 73~. Accra. Al ~ am: HI. ~1 <>r<~>vc~1 i~ corolJ~l-y B4.1.2 <~ rear. 1, 1:~llore exists ~ frigid [t~]B~lytiC il.~.ncl~ioil. 77'fl on /~1 <~verc<~>rl ver~e~t relative to go]. such that Age, <) ~ ~) far ~ ~ id 1 dcdoce~ 6~(~)~01^.1..A~RY 2. 1.1. 77Z~ /~! /.S FEZ ( /~/Z! Em) ~ ~ i ( Z //(~) #~) ~ 7 O~ Zag ^~ q~ ~ E(4)/E(#P~ PR~POS1T1~\ 3J. S~ ~ ~ ~ 2~! ~r~ .\ !~.~!) /~! Z? ~ 7 I; ,.,39 {~) /~/ ~ 1~ ~ = Z//, 1! SZ 3<~-!P/~} / )~3 ~ 7= {~te<~, ~ Afar ~ ~) ~ ~~ ~ ~ ~ ~ <~< an= ~L 27 D~32 #, Ha, {~/ ( 1 -- ~) ~ Z [ [ 7] ] ( For ~ ~ ~ let >~ = raw ~ 1 and far a subset ~ of let <:~ ~ ~ bo the prJec~or Kilo the DBASE ~ spanned by 1~:/ S \1 as detained in 1~l~nla A1^# of rof` 1. Shell by theorem ELI and lemma >1.6 of red T. #6 - ~ ~ ~ d#I - P~. .\ as ~ rawer war Anita subsets of a. Nan since det(1 - 6 ) ~ Idiot ~# Bow awes 36 Wry Ed as matrix Bail respect to tbc basis {~:/ ~ i} has nutrias in [< ~ sea that It - Ems ~ I) e 1.[7]. Since ~ is a c<>mp]~1e stlbrill~ of at. 1~he propose Allows. > wet be able to apply 1bis to tab opcr~tor because L~E~.^ 3.2. !~ ~ /\ <' ~/Z/~7 /.~IZ<~ /~71~f</ {// <' ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ of // ~= ~ /~ ~ yam . ., (!~IZ/~71S .\ ~' //~ ~! {' /?~ ~ )~ 8 /~) /~ !~> //1~/ (Compare 1):ropositio~[l ~1 of gaff ~ 3.) a. . .... .. lulls ~1 be ad immediate consequence of Corollary 4 2.I. . . . . . ~n1CE 1S a more precise version ~df~e~.~e~-r~ Lea Err 1~:> fin old: of 1~.. Rota. however tab Air ally Oven ~ one mind hoe to replan ~ ~ a finite extension so Ha tog such Baa 1be Moth ~ of unit am contained ~\ Suppose ~ is ~ bash wide open defined Car ~ Pith minima underwing android \ such that ~ - ~ has ~ co~ct~d components E~ .... ~ tsar rag 4j. Suppose T~ Addison that =~ ~bb~ on ~ and preserves ~ For 1 ~ f ~ ~ and ~ ~ I ~ ~ ~ ~ ~ ~ ~ ## = Em. ^~# ABE I)\{O+ be ~ uni~rmizing parameter such Bat the subset of ~ 2~<r~ Suppose in Addison 1ha1 there Dig r<~ /} ~ ~ sob that ~^ = r<c age) Obis we can ~rs~g~ ~ using app~pr~te prJec10rs like Eq. 1 below corresponding to rho Ears in ~ of ~ I ~7 ~ h~30 ~.~ Lag #=~- US AD underwing adenoid of Each ~ ~ strict n~ighborbood and ~ preserved ~ ~ .. . .. . Tb~a~ine ~ h~s~poin~sa1 ~,~....~>correspondinyto A= # E ~/ 0U -I = -a) 1 61~) ~ if: - >~) = <. Fat ~ be a parameter at A. which gas to ~ add far ~ ~ a. let JO ~ ~ bo such that age- r/~P-#~) ~ -~ Lots be 1he den F>~2 ....~]/~:/ ~ j} and let ~ ~c1 an to that 3. Contin~ous\irsus Completed Conunuous Operators ~ ~ #(~ for ~ ~ A. Supposed h~compl~tesubri~ofJ.7>and ~ ~r~Ba~a~b rll<~<l-<ll~s<>v~r.~llTd (.:es>~ct{~ly`>ll~tl~:~> ~ Maze Isis . . . . annual ^~ ~omom~m. >~ let \~) he tab ~Ienlenl Aft. ..... \ . . ?3 ~ rafts ~ ~ i, .~}

Colloquium Paper: In It booms that dc~(A(~) = #~) Add \~6 {~) = *I Let the summit of ~ generated ~ i<~) ~hcr~ f rem aver If. < ~ ~ Then ~ Riemann-Roch ~ ~ a: = \/ Tar some positive galloper \. Moreover, 6// is limits dil~ns~ional over OF and ~ ac10d on ~ ~ Eat ~ be a bask of ~) eacb element of , . . . . Inn IS an Sector ~r the Lion of ha. (en the SON ~ = ~ U ~1 ~ ~ ~ ~0 ~ j ~ ~ ~ ~ ham) ~ ~ /~ = I) ~ ~ ~ ~ Ed /< = ~ Ed '>act} = { ~!:-~) '1 C = 1~-' ~ ~ - I((~)~ ~ [], I0I gee [1] E=)= 1h~1 if () ~ 0, ()) = ()). [2] ~/ ~ ram, ~ = (~ Ed ~ 1 ~ ~ ~ an. ~ ~6 in F[~! id. ~ <~>e~ls>~.o<~( Which Uncrate Ha idea consisting of < such that .- = 0. 61 \ ~ a 1~ ~ < to ~[~6 ~ #~1~. In Thy= ~ 1'1~ ~ ~ )-0 . a,. >~.~3, - . determine ~n abbe Abet ~ Which libs ~ and so there is an allomorphism Tom ~ to >~ Peak colon such tb~1 ~c paybacks of as, at, and ~< are 11~ of ·~) f~ and 7~. For ~ e ~ can the aping of (~) in ~ Dada ~ taking He appropriate product of these pullbacks age. 61 ~ = I: ~ 2~ =~1 got.= erg ~ ~ ~ ~ ~ ~ ~ L 27(~)I/1~ ~# 1 {~7 ~ e ED raze fir i777~> /~3 !~ ] . a. ~ ~ ~ BET ... ..... ..... 1 ~r _ ~ r ~ ~ Fr ~ 1 fir c'` arm !~3 1. h~6 1 Adam P~# = r <~ . . . i = ~ ~# ~ ~{~<,~)~}>/(I{(~.~)}) 3°~) ~ 76 ~ of He >~~ ~ # ~ ~ ~ ~^ k~)~`r ~#r~ ~ Fact ~ ~ as) . .. . . . .. 7! (~-~C (~ a. ,` , a.. . . . . . I: 1 As Pops ton is ~lm~]cd~lte Then ~ = 1 so suppose ~ <is 1. Lye (it tie tile ~tbovc co~l~lplele sul~:sl~lyeb,el. We kilo>. for <~ LC~/=~^ (T##3~# have to prong: (0 far ~D < ~ ~ there awls a ~ ~ ~ such that - 6^ ~ J~ ~ ~ ~ Ed ~ ~E ~ F0(i aced. Cop. (~) fallows after Baking :1 fail etch if n~c~s~< 0O Ellis Cam proposition 6.3.4/1 of reL 7, and ah abbe dc~=ipholl of ~# Ed [naI>, (~) (as Bell ~ the s~colld pelt of tile proposition) Will gallop. <~cc No It a -is~[ll~rp~l]is[<l ['/~6' amp. To sac the Her. first Rota that ointments in Junta) ma be In ~ He ~ #~) ~ #~ ~ #~ ~ O amp and Oh Duck Be gee Afar mapped He r~duc~on of atom, ~ homomorphism #2 J~ Ha. ~ ~ 2393 I1131 at. Bang Be previous lemma. Be see that far ~ close to 1, this ClOfS t~broug~b a subjection onto ~ Which its ~1 O-~borno~mor- phism by cons~uJion. Nap ~c produce the inverse 10 this bomomolp~m. For7 S ~ lot ~ = #(ad Consider the correspondence /~) ~ ~ mod '~? from r 10 ~ Rod Ace. It suffices lo sbo~ that far ~ sun close 10 1 this cxlcnds to an algebra homom~F- phism ~ ~ amp. Cat ~ he the Ibsen of ~ consigns of Remand of the farm ~ ~ (~ such that #~ ~) and 1~ ~ = Up #. if ~ ~ ~ ~ wit sag death = an. Ton is g~ncr~tcd ~ ~ [n11e set of relations of the farm ~ v' 'T - (') gels (Tb~sc relations ma include single monomial rel~tions.j For each octagon of ibis farm, there must be ~ relation ~ the Cam ~ `-~ ~ an+ ~ #=0 ~ ...~ ,. . Her :~7 d~g(z} t::~' . _. an Fag. If ~ and ~ are linings of the coe~d~nts and ~ and ale tag leftists of Abe ~mo~nomiaIs ~ find z Obtained by lift /~) 10 \~) far ~ ~ 7~ Then. because ~ fibs ~ there must be ~ relation of the arm ~ ~ + ... ~ go.. ~. - . ^ ~ as ~ ~ - ~ ~ a=: degf~z) A: where ~ is a po~omi~ in {P ~ ~ Huh coe~cnls ~ ~ sad L n . ~ ~ <~ ~ '. >~; do ~.dc~{~ )~ = a. adz few :~7 dew} <~; Sacs rag far ~ ~ ~ is ~ product of element of the farm ~ ~r , . . ~ ~ · · ~ ~ , . . #~ ~ m OIL ~ ~ ~ ~ ~ ~ ~ ~ a pol~omi~t wc she that far ~ Rosa to J ~p have ~ homomor- phism Tom ~ onto 636 Hub tangs i~ to ~ as desired. For a chancier ~ ~ #~^ ~) and an ~ module ~ on whicb ~ ~C1S, set #~) = ~ ~ . . .. ~ = ~ ~^ mall {(} ~ ~ 7}x ~/ Art). ~# Is e Homage. As) zzz) ~ C ~ /.< my Oaf {I () :.Y ~ ~ } is 67 )(~2 age 6(~), /7z~7 I) ' F r si) . ~ ~ AT ~ J . <~ o~7z<~'zo'~/ amid ()() 3. End of Proof Fix a posting integer ~ prime to a. Le1 ~ be connected component of the ordinary locus off) coin I 1ha cusp ~ and ~ be the opener of/), r~ _ ~ ~ ~ EVE- r~ Where /~ ~ the Weight zero Sop~ra10r, which ~ an operator ^~ ~3 ~ ~C ~2 c~= ~ ~C tile s~ln]c 1-l~1me in reck B4.2 of radii!. 1. We have :1 cultural actors of (ADZ)* on \1(~4) via diamond op~ra1~rs. {note that Qlis is i~tc~tionaI~ trivial Ghan ~ = 2. un~un,1eRj 61 ~ be ~ disk around zero contained in ~ Ed ~ a Edict amnoidnei~bbortood~f\\t~bIeundcr the Scion of (//~Zj~ suchthatf~convc~son /~.(lhk CXiSl~ by CoroJ~ry2.1.1< Lel~:(Z/~Z)2 -a Z>b~characI~.Ey Corollary4.2.1(and

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Yol.. 94 pp. I I 133-1 1.:1.37. ~ct<:>bor 1~97 ,-. ,, . . ~ ~ . ~ ^~ ~r ~\ ~ ~' ~ ~~/f/~' ~~ <~'f~ ^~ ~) (gear ~~, ~d ~ ~ ~r ^2 ~> 4~ ~- /~< 7~^ ~/ ~ WHIZ J~~~ ~ fig ^~ Am. ~ . ~ , ~ ~ ~ ~ . ^ Vega IuncOons and t1senstein series on class1~1 Bong 2., ^ OORO ~HI~ D~par:m~t of ~<ltJleIIl<~cics. Fiilo Fall. Prirl~clo~:l.-~..T~l~ive:si1y. Pr~in~ton. AT (j\544 ABSTRACT ~ cons1~ruc1 Clan ~E~.l~r product Mom the Hecke eigenvalues of an Isomorphic farm on ~ dassic~1 group ~- prow Us nnu~t~ conG~Uon ~ the -~e c~- plex plane abed 1be group ~ ~ unitary group over ~ Cat geld and the eigen~r~m is bolomorphic. We lasso prove nnaly1ic continuat10n of an Eisenstein series on another unitary group, on1aintng 1be group just mentioned defined with surb an ei~u~rm. As an upplic~110~ Of car metbods~ ~ prove an explicit glass number formula tar ~ thy definite bermidan farm over ~ Cat held Seaman 1. Open s reductive algebras group ~ over an g~br'L number geld. me denote ~ Ha, 6~, Aid ash IS ade~z~lion, ~ arbimcdca~ #10r of 6^, Id the non arcLim~dea~ factor of ~^ We take an open subgroup ~ of O^ of 1bc Arm ~ = ~ web a compact subgroup ~ such that n Ca 1\ maximal comet in #. ^~>n~ ~ sac tape of reprcsen1~don of ~0 n Ha. ~ cab deE\c attomor;hic .. . . forms ~ ha as usual Far s~plkir ~ consider bag the farms ivy under 7\ n ash Lag Recks oar ~ g\> ~ Id. ash ~ in a subset ~ of 6^, ~bicb ~ a s~mi0~up chin ~ and the loca[~lio~s of ~ far almost elf nook . . . . ~ 1lmcocan palms. Thing an aulomorphic farm f sum that Few = Calf Aim ~ comp2x number A(# far every ~ ~ ~ and Heck idea character ~ of a, me pal ....- sr~. r ~> = ~ >~<~0~.~O<~-~. [l.l] a\/ arc ^~# is Me denominator ideal of T and #~) ~ as coral \^ our first mall result is that if ~ is apoplectic, orthogonal fir uni1~ry. 1hon all. <~ ~ >> H ^[~<j~(P~ 1 . [1.2J Chain age. ~) is an expJ2idy deemed Product of <~ anchors depending on ~ ~ is a pol~om~1 Determined far each ~ ~ h Bose constant term is 1, Add ~ runs Mar ~1 the . . . . . pFlme meals of the basic number field. This is a purely l~ebraicres~ltconcerni~ only n~rarchimcd~n primps. Let Z ~ ~ }death ~gh~hand (de ofEq. L2.~ our second maiDrcsub,~obtsi~aprod~ct ~ #)of<~m ma ~ Coors such 1bat ~2 Cal ha continued 10 the whole -place as ~ Tner<>n]or-~pbictu~ctio~ ~illlfilliteJy T!lil~\ poles^ Thea ~ is s ~ni1~ry~roupofa~arbilrary<ynalur~d\1~ibulio~oYcra C\i E~ld.and fc~rr~sp~dst~ holomorphlc firms. Boa theso problemsare dos~J~con~cc1cd ~ithth~1he~r~ . a. . . · . ~1 senses ~ an group ~'i~ ah ~ isen<~eddcd To dcscrib~tbeserics.I~t 3'd~not~lb~svm:~e1hc~c~ ~( which 6'~ct~ lh~ntb~series~a Lichen of ~ <) ~ 3' x #, . . .. . . Ban Do Elves (lathe classicalstyle)~rll~he farm ^? ~ ~ No! academy ~ #~ ^23843/~7/94I 1 133~/0 F~N,\S is ~tv>~il~l~:>l~ t~:i~l~ {ti T~t1~:,//~!.p~l~s.,irg. ~ ~ Inn. [1~] Ha, 1. wl~erel~is~lcollgru~1lccsubgrou~p of 'and Lisa para-b<)lic . ..... . . suo<roup o1 ~ which 1S as~midkoct product of a un~otent group and ~ x (ALL ash some ~.Tbc adel~dve~ion of8 . , , ~ . . . Woe cXpllC~> describ~din Section 3. Boa ourlbird main resultisthaltherc exists an explicit product I' of Cam ma actors and an expect product i' of !-~nclionS Cab that ~ >~> age ~ <}a< ~ r ~> can b~condnued ~ Keyhole ,-plane ash m~romorphic Unction Huh finitely made poIcs TboughEbc above rcsubsconcorn bolomoFphic amps our mc~odisappEcableto the u~itsry~roup of3totalL dc~nilc Bertha brag overt C\ifield.l~nt~hiscase,~e c<\ ~c ~T1 c~plic)classnumber formula ~rsuchaberm>isn Wry Hick tbc ~urtb main FOsull ofthispapcr. Section 2. FOI 1~ associ~t~edn~ ~)hid~nti~ clomcnl ~c the ~moduI~ of all ~ x ~ matrices ~ a~tdesin ~ TO indicate that ~ union ~ ~ Urea ~ is disjoint, he virile ~ = hi. Let ~ be ~ associates ring ~itbid~t)~ ~l~m~nt~nd an . . . ~ . ~4uuon ~ b~a~+dx~ ~, ~ !3 ~ pay = \\ alla ~ = #*) li~issquare~nd i~]ve~rl:il)le. G~iVO~ll ~ fin ideally generated ~fc~3module ~ wedc~ote by(~<P)thc~roup an \~l r<~r~ aloft< Evoke }on Loathe Rename v~ denote by ~ ~eima~eo<{ e ~ undoers ~ 07.~1'). Oliver ~ ~ i1. by ~ ~hormSti~ll~ Brie on ~ he understand ~ biaddL\~ map ~:F x ~ -a ~ suchth,1 ~6 >~2 = C# # ~ ~ ~ = ~~ / ssumin~lha1 ~isn~ndogencrate. ~c put #=~=~=D ~ <61~=~# ~ GR~n<# ~)and(~40.wecandcEnea~-h~rmi(~ arm ~ . ~ ~ . ~ (~+)> +.' +>') = ~s a') + )> a') At, i' ~ [) hi. a' ~ Ha. [2~] ^~(~3 = ((fad are ~ondo<~ncratc. ~ c~ van ~ x O~ as a subgroup of We ~ (a ~ oF 0< x ~ vl~d ~ an event oF ~a = ~ ~ ~ am /~+~=~+~ ~. ~2 ~ ~ J Wc shall always use {~ ~ a. and ~ in this sense. Wo und~rslsndth~t ~# = {O} and (0 = 0. Hoer me no ~ and a nondo~enerate ~ on ~ assuming . . i- - .. . . . . 1:~Et1 ~ AS 8 (l~l~YtSjOll 1'1~# ~1~)S~ Cl]3l-{tEtCT~iXtic iS {li1~lUll: alum 2.Lst~beaS>subl~oduI~ of ~ ~h~hislo1~13 ~-\otropic.bY winch he ~ ~ a{< ~ = ~ Than he can and ~ drowns ^# ~ /~<~# . ~ ~ ^^ 11~3

1I134 CoHo+< ~ Paper:Sh)~ura Apollo 09#, ~)suchtha1<p) = S.ln thy setEng.~e define 1hr p~r~boEc subgroup 7} of Q~ Ire to ~ ~ 7< = i~ ~ 6~!7~ = 7} [2~4] add ^ STY ·<:F~ -a 6# find \6 !< -~(LL60 such ~ ~ zag - z~) ~ Son ~ = ~ ~) Ez ~ Z.~ ~.6 Taking a fend nonnegadvein~gor ~.~o put /~=~# )~=~-# \V~ can normal yvi~v L< X 6< ~ a ~ up of (ad Inca D/ = P~.~^np~ /~eE~s~^ k ~ ~ and ~s every Cement of ~ ~ the farm (~ ~ >) ^~ ~ r~~ = ~E K~ 6=3 ~ 22~2 7=/ I t ~- · . ~ ~ ~J Obs~r~l~plhr ~ ~ total} ~-i~otropic, ~c can drone a;. P~oPosmo~ 1. #< \~r ~ ~ ~ ~- of \~.Y~z~j<~/e~( ~ ?7z~z . ~ ~ ~ ~ #~# x 6# an] (~Y ^~('//f \~) <~Y ~17~! ~ ~ - - ~ ~ ~ ~ 7)3 60 x 0~] = u~((( x 1~)D I<) [2.71 == 1,: i! . w . In ~ct,wec~ive~nexplicb sot otr~pros~nlal~es a! tar Eq. 2.6 and abo an explicit sat of r~prcsen1~1iv~s Tar # x O~ ~ ~ ~ ~ ~ ~ ~ ~ , , < .. proposition plays an essential role in the amasses of our ., . . . .. . ^ ·O^^C+~^ 0~0 ~ ~ ^^ ~ ,, # hat {~ ~ ~ a' $} Ha. . Section 3.I~ this section. ~ is aJoc~l~ compact cold off characl~ris~cO with respect 10 8 discret~v~luatioT~ ~uraimis oestabl~h the Euler ~c10rO#ofEq.1.2. We denote Wand qtbevaluabon Gn~andi~ maxim~lideaB~cput~ = [~:q]a~d N = ~-~ iffy ~ ~ Aide ~ ~3 rx wi1b ~ ~ Z.\V~ ~sumethst ~-~>P~2 ~ tang= b <~=~g=<n~ A. Weconsid~r(6 Cousin S~cGon 2 Pith ~ ~ &< Aid defined by a<< pi = >~$ E~ ~ ~ ~ ~ Ada a made ~ oT 0 Arm ~ (I (') ~ ~ I) ~ 1 #=l o ~ I 6-1I O ~.. ~L3=~ Aft. I] 1 Ha= ~ - A. W~as~um~th~183 anisotropi~d~holhat ~ = al and ~ = 2 ~ ~ = 1~ Q<~J s ~ L 81= ~ and ~ = ~8 ~ ~ ~ ~ F12tq Thus our~roup ~ ~ ortbo~onaL ~ mpl~c~c~rllnitar<.Th~ ~ ~ . , . ~ elem~rt8 of Eq.3.2bc~n be obtained by putting ~ ~ ~ - ~> ash ~ suchlha1~ = g[~1 \VeincIud~tb~ case ~ = gin Air d~cussi~n.If~ ~ O.~imp~ India 8; 13iS iS al~ays~o and ~ = -L \~ bavo ~ = 8 + < = ~ knows by {~? standard b~isof\~.~c put / . 2~r~ :~ ~ ~ at'< ~ ~ ! ^ ' ~ ~ . ~ I:::: ~ I. a... #=~(~+~+~=~^~ 2~. \~/.~c~. act. ~ #~7~> r=~=~=~ Then ~ = (~.\Vc choose {~+;~lso that ~ ~ {~Irr+~ , a. .. . . . Awn me can bnu an Demons o1 ~ subhtbat ~ = 8 ly + ~ Iamb, [3.3] Pat ~ = ~7 = IN ~ Alga = ~38~. [3.41 We can Urge every clemcntoff{Tnth~ Arm .' , (( ~ {) a= O ~ ~ .~=~.~\ O O ~ amp= -art c = ~ - -~# ~ s ~ ~1 < = ~ ~t Copy Bum ~ Hand ~ ~ ~ ~ = ~ ~ :~c ll~vc ~ ~ ~if' = {i , ~ . ~ ~ \(cco~sidertbe H~ckeal~cbra ~ ((L~)consktin~of Exam ~ [nke~m~>r,E E web ~ ~ ~ Andy ~ (u.{\n<~}b Thelma of mul1~12~6on defined basin ~E I. Takin#r ~de- ermirales7~ .., we define a Opined map -~/ C[~)~!,,,,,/~/Z1 ....~7,31 [3 61 2~0~; Aeon ACE Ash ~ ~ ((L/~0 q>.~11~h uppertr~ia~n~ular~ Otiose di~ollale:~]lr:ies:~re All, ... =^ ash ~ e Z.T~en ~c put ~=~ ~=H~ · ~^ ) ~ :: ~ '1 ~ex1~cconsid~rth~ Creche algebra ~(C ~)cons>1~f aD Arm ~ Ho And 2~)h~ ~ and;# ~\Ve Hen . ,.. . . jeans ~+ Par map as Allows; g:iVOll ~ To with ~ ~ O~.~c CITE pU1 (a ~/ ~ ~< (ad ash ~ ~ ~ of Aria Eq.3.S. Within put . . Tam = ~ ~~ ACT) = ~0~) ~.91 abhors ~Ois~ive~by Eq.3.6a~d)<islh~-blockin Eq.3.5. We Cell 1~:r<~>vC t~t)B1 tlliS :iS ~11 defillC) Et~Dd gives fit. ri~l~-~i3CCt-i<)n. Owen ~ ~ ~7, me donate by ~) tho ideal of ~ wbicb is 1bo inverse at the product of ~1 1be elementary d\Lor ideals of ~01 contained in ~ me put than ~) = I) Wo cad ~ primi1\e if rank0 = ~in(^ a) and ~1 the ~l~m~ntsry divisor ideals of ~ arc PROFOS1TION ~ ~ ~ Hi LIZ my. 3.S. S~Se ~/ # . , . ~ ~ - ~ ~ ~ ~ ~ ~ > ~ ~ ~ = ~1 ~ ~ _ . ~ 71~ . .. .. act) - j {a - j } ) ~ death) ~0~ * )t at= ~} We nab deans a Army Didch{1scr(~ S 5v 33) ~ :) (}(TV) ~ ~ = 6~6 [3,10] .= ..! , ..... . l his is ~ Armed verger of the Euler Actor of Eq. 1.2 ~1 a And . , If ~#, ^~# ~ ~ ~# ~ ~ #~ T~EO~ ~ \~Be ~/ 6# ~ Q(~);~( = [{# n #. ...., . ... .^, ... . ..... (a = ~ C~ = r~ T0~

I I I36 Colloquium Paper: Sbimura These operators form s commutative And of Norma operators ._. . . . on ~:,f.~(~)' For ~ ~ 6t ~ denne an ideal ~) of ~ ~ ~ {}0 = .1. .1 ~()61 )) F4-131 `.ei here~>jisdcb~d~sinSechonSwitbresp~ct 10 an<~as~ of#~;..Clcall~y;~)d~p~-ndso~llyon(~. LetfLe aD Clem~ntof23~6D)tbat~acommon~n~nc (onof~llh~)witb:~},~dEtf<~(f~i~< ~ C.~rnaL>ckeidcaIcharac1~r~of~suchth~t I7i = 1. d~Didchletser~sS#.t ~ by ad. ,) = ~ \(T)\ f got to T)) . [4.14] Tact\{ /o whiffs a* is the idea character sssoci~led ash ~ and Jo ~) tab norm of an ideal a. Venom ~ ~1 the rcs~[~ion of ~ to 6, and by 8 tbe Hecke cb~rac1~r of ~ correspondin~ to th~ qusdr~1ic extensi~n ~. For a~ H~cEc cb~racter ~ of F~ pu1 ~ <ts, () = q [I _ <~ << ~< ~ -~] - 1 [4.13 j p.~. From ^~ ~ and Eq. 3.13. ~e sec that D . ~ 33. ~ <) I <~ - ~ + J ~l3~l) /... ~j . ... = ~ I ^[~*(q2(~)-~l-i [4 161 (l ~ c ~)bapolynomia}U~ofJe~rce~hosecon~tanlterm i9 L wllOrOq~Funsovcr~llth~pri~cidcalsoF&-prim~>t<)c.letZ(~} r ~ denofe th~ ~nc~on of Eq. 4.16. Put /71 - i . ..... F~(~) ~ /(~1~< 11 F(i - (). [417J A: () TF1EO~E~ 2. \~/ //Z~/ /~{~) = >; I>I ; ~i7/ ~ ~ Z ) ~ ~ R~!~C) /~/ %~ KE = ~ F~! ~ = ~ + 277 - ~ ~) .~.... . bt{~. f . >) = 1 T 7~ + (~/2 jj /~3, f . ~j . =~ . (~#j = [~## ~= ~ ~ ~e can ~c an ~pIidt~ dcEned [~ite s~t of points in (~h he possible poles of ~ belo~. {ti~ tbat ~ ~nd q1 are ~;~= 1 >0~= 1+ ~- ~ ~ - ~ be resuLs of 1hc ~b~e t~c ~nd ~ho of 1bc t~e of ^~ ~ belo~ ~ere oblained in rein. 2. 4. ~nd 5 for th~ ~rms on 1be ~mplect~ and metaplectic yroups ov~r ~ 101a~y re~1 numb~r b~ld. Tbe Eul~r product of lype 2. bs ~ralytic co~tinuation. ~nd i1S reIationship +)b tbe Fourier coe~nts of f hsvc been obtained ~ Oh (6) ~r the ~r~up ~s sbovc ~ben ~ ~ ~. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ #. ^# ~ ~. 2~ ~ith (C ~ of Sechon 4 ~d ~ ~ ~ <5n~ simp~ 1 ~ #, we c~n c~sider 1bc par~o~c subyro~ ~ of 0~. We put F =~=~e#~ ~$ ~ = ~ (r~> + ~ - t~+~+~) + ~, [5.11 #. W~ can dcEnc the sp~ce 3~ ~nd bs ori~in 1# in 1h~ samc manncr ~ ~, 6~. We ~en pUt ~ ~ ~ ~ ~ = ~ ~ ~ ~ e ~ = #=~- (C~^ 15 31 Hcr ~ is 1bc ~mcnt of P~d(~) d~6n~d ~r< ~ (~)~. ~r ~ ~ E.. W~ defil~c an R~[lle(I functio~n /r on 0 1~v ., ~) = l#~A~ ~r~-~ ~ ~ ~^ Taking f ~ Sl>~) and ~ ~s in Sechon ~ ~ d~Enc ~:6 ~ C = ~: ~ = 0 # ~ ## >T = ~ ~? E and ~ ~ ~# n ~ ~cn ~e put #(~) = \(~{)0.) 'I\c(~)(~$)) !{l'(~'77~> 'IC~ 0)q ~ = ~ ~ ~ ~ ~e ~ q 6 C by (~=~0~=> #~(~ ..~ ~ = \~. [5.61 7~(~) ~ (~(~)~{~)1 c(! - ~ + /2 + ~ ( (~ + l~r) -- - --- 1 2 ~ E ~ S \ ~-, ~ S r i ~-'^ _ ,, , ,, _ _ 2 ~ Tll~Ts ~is jrctllliIl~fu1 Tt ~(~) ~ >~227l>I7~ ~ 2; ~illl ~< ~ R~. \~en ~ = 0. ~d lh~ conductor aF ~ di~ides ~ W~ 1~kc such a ~ in 1be ~IlowIn~ 1h~orcm. The serie~ of Eq. 5.6 is 1he adeEzed version of ~ coll~ction of s~er~1 scrics of 1h~ 1~e in Eq. 1~. ^~ 3. > ~ ~ ~ ~ 2 ~ ~ = ~ ~ 73(~} ~(i I~(~>~.~) 1 ~1# - ~+ (~2)). F. ~ + i<. -~.l\ . > <. -~.\ ' ~ ~ ^ r>~= ~ ~-~-I~)~-~-~ r ` 4 / ~, ~, ~,. )~ /~r ~)~r ~/ ~\ ~= 0 ~ _ 1 ~ 7~> J ~ ~. . . , <-- 1 \ ~. . T:(.< + <~,/2~)} 1 Z <.{~2> /.~) / ~ 1 .,..,/ 1 ~\ ~f = I . _ ~ _ . - ~-,.) _ ,- ,,- ~ . ~ ., /'>, '! '/(~ ~ \)~0'\; ~ \'~#

Colloquium Paper:Sblmur~ ^~ saga ~ ~ ~ - Ace, ~ ~ ~ <. Wec~n~ive~nc~plicJJydc6~cd ~nitese10fp~i~tsi~ which the possThlo poIes~flhe above produclbe}on~. Secllon 6.Lel ~ be an ~bbr~yreduc~v~ algebraic group over Q. When an opcnsubgroup ~ of ~co~1aininy O~ Old ^l~n~ =- =~ n ~ Rovers ~ 6^. Wesssume~l:hal~acis O:D a ~ m~m~e-tric also a~sumc~batf~S has anus measur~,~>t~n volt, Respect ~ a fixed Visor measure on /~.Takin~ s complet~se1 aFr~pF~se~lat~es ~ far Q\~3,>c pal ... = ~ ~ ~ ~ n ~ ~ assume ~ ~ Rat n ~I;6 Eni~.(lc~> fin do~notd~pend o~theclloic~ omit. Wec~ll<~6 2)l~llc~D~assofO with Aspect ~ [( ~ 6\ ~ go ,~S ~ bee Ends pon10T~r~ 1 ~ ~ ~ ~ ~ (. 00 = () = )# [F23]-je . => .? .... at.. Ace. \~) Dim/. act. #~ P#(7~] 11137 ~<~# an)= (~btl'~j,>~(~I'') l~r~v0~ry} ~ ~ . F7,1] The principle ~ the same as in Eq. 4.6~ Add so ~ is su~cien lo paw the ~ssc~ion of #~ ~ with ## ~) in pace of (# J. In pa~=l~. ~c can Lee ~ ~ be ~ = ~ x 1^ ash ~ ~ a. Define ~ ~) as in Eq. 2.~. Then tber~ ~ an isomorphism of ~ ~ ~ ~ ~ >~ ~ ma ~ ~ ~ 7 standard par~hoIic subgroup ~ of 6~+~. Verdure, <c can iden3# 3~ wag the space ha aid a= ~ ~ Cal )~6 - :)isposLive dc~nileI. [7.2] We can ago drone an Eisensleln starts ('# ~ <) Arc and ~ ~ C. Wish is defined ~ Eq. ~6 with (~+~< a, 1) in place of ago. Far, fig. ~F~kil]g E' ~d (air, <~) ~ ~ (with ~ e ~) in place of ~# <) salad a, me call define a ~fuuction a. a) of(i,~) ~ (~ x Cinthe~amc m~r~sin Eq.7.1. b ereis ago an icon ~ of 3< x Shinto (0 compel ~1c Huh the embedding O~ x ~ -a C(~+~ We put ~ () 6~= 8t~ :~-~0,~~) ~ ~ 3;\~ ~ 3~) [73] :~re~cry ~nclio~> on #~.~bere 8~^ J) ~ Sure ~clorof sulomorphy associated ~ithibc ~mbeddl~ ~ Take ~ Hecke eTgen~rm figs iD Secho~4~nddobne) byth~prInciplc~fEq. 4.~. TheD.employ~ill~ 2~.~7,~e cs~ prove We call sb<~w t:b~ll~ ~6') = [~:2']<~C') if a' is <~ sllbgrotlp of a. If sawn appr~im~bon golds ELF 1hr semi~mple Scar of 6, Anion happens Outbox [Fig n Ian vote L # ) ~ depend only on a< sothal ~ 0~/ 6~t ~ ~)'~) . , . == n ~ Ad= C}6 eve~ry~,in Rich case ve~b~tv~) = =~l6,`/~.If Seth stabiEzerofalsldce (in av~ctors~aceo~ Rich ~ act~then =~tL33L) ~ ~ nut of classes in He genus of (. P ere~re.~60 msybeviowed~sar~#nedver~ ~ of1bc class numborinlhissensc. . . Coming back 10 the unitary~roup 5~ofSoc~on 4 wacky prove the ~lIo~in~thoorem. ~1 11~E(~)RE~ 4 iZ-<~ ~77 ~ ~ <~( 7 -Z-~' ~ > go/ /~ gage >~3 fag // ~ -I- ~3 ~ ~ ad; {~ #? ~ !~ ~# ~ > 2~? by 7~ a j~# ~ i /~.~.~ = 2. ~ F l</ ~ 1 (. A) = 2; I I (A A) ~ ~72 go ~ #=I ,...... ~ 1 I >(b)~2(2~) -a Age ~ .. ~ ~~) = [~:Vt<~3 ~ \<e)~(bj-~ ~ ~< ~) a. , ~ . . . oddest can ago r Ad') b ~ ~ ~ ~ ~ - p C ~ ~ ~ P ~ ~ ~ >ah an arbitrary inle<IJI ideal c. Than BUT ) = 2-~0~? fibers T ~ the numb~rotprimesin F r~mi(6di~ a. Seeded ~ L~lusno~sk~hth~p~ofofth~abovc theorems. lh~tulid~1a~ils ~illt~e~iv~illlef.7. Wrlir~l-t<~c {S C (it so ? =. 3s##xc~ tL~.~;~08~<j~. [7.4] hollered = ~ x ]~.~isacert~ti~a~l~rna ~1~]r~1nd(~ = I\. T~hec(~}mptltSitiollissi~rni~la~rt<~.-btlt:~l~oreinV(~[vedt~haD.th[~10f~lCE 4 Ration t.S~ce ~e~\dcn~uleofF/~anb~seen Tom h~r~s~l~ ogre 8,~eca~ derby 73~ from Eq.7.4. Take ~ = (I. Then ~ = ~ :~ll(l ~0,~) = 6(~z~).'lhen talc ana~ticnat~^ of ~ ~ ~ Wand consequent 1h~1ofZ#`L ),ca~ be derived from Eq.7.4.Ilo~v~r.b~r~ He brave to sssu~< ~a1~) ~ >~+~>rR-~-~L Dab ~ ~ B/,>~ ~ = 0. ~dthec~nduclorof~ divides ~ Tb~la~crcondihon oncis a miner matter butth~condidonon << ~ ess~tist To obtain Zig t ah AD arbbrary~,~ bavetor~pl~c~by S~< wh~Ic (~ ~ asericsoFlyp~ a' with 2~ - gin place ofL Aid ac~rl~i~ din~rant~loperaloron as. As ~r7~~ ?.~c take awls ~ = ~ ~<icbs~rvc Plate constant Anchor can betake~asfif~ Taco ~ ace Ida space #consists ofs angle point. [h~i~t~grsl~ ah ri~bt~balld (de of Eq. 7.4 ~ mere} the vaIue (~# ;. We can co~]p~leitsrcsidue~tt~ = ~ explic~Tl~ly.C~nlparingil~itllt~lle residue an the TeR-b~nd side. ~ bblain Ron ~ when ~ sadists Eq.4.10.lf~ ~ odd.~e can remov~th~ condidon by c~mpuLn~ ~ lineup Ada\ of ~ eland .... ~.. .. a. i. a. . . 6. 7. 1 . Slits, G. ( ~1971 ) /~l<~:~(<r/~ zag 73~ If 7~> q 3~ 6~< (Panama Shorts and Princeton Unit Frost Pl^~i~c~t(~). f~lbl. ~111. Soc. J~p>~. No. 1~1 Sbi~ur~, C. (1994) ~< Amp. 116. 331~76. ShinTura. O. (!964) Age. Amp. an. 369-409. Shimur~ S. (]~93) ~r If. 119. 53~. Shimur~. G. (1 99i) ~r If. 121, 2 [-60. Ch. L. (1996) HAD. thaws (Anton ljn~e~11< Ton). Chimera. O. (1997) ^# ~ ~ ~ ~ (Am. Lamb. Sock Providence. R1), CBYS Sprigs 93. in press. Sbimu,~ G. (1983) #\ Ado. ~ S~ 4J~476.

Vol. 9~. pp. 11138-11141, October 1697 CoTlo-~m fear J~-~ARC FO~^E US do P~d~-Sud, ~aLb~maDq~, Belmont 4~, Delhi Clay Ida Frank Comets -1~ ^~sent~hons (ma. 1 and 2; gap. HI a. ~ ~ ~ a ~ ~ ~ ~ ~ ~ ~ ~ I/. {or cacb prime rumba /, He choose an algebras closure Van of Q~ toaster Cab an emb~ddi~ of ~ into 0< and He gal ~< = ^~) C 6. Me choose ~ prime num- ber ~ and a Unix extension ~ of 7. ~ ~ ~ a ~ ~# ~ a ~^ . . .. Bomb ~ vectorsp~ce equipped ah a}ine~ gad co~1i~uons . . . analog ox. of ~ ~ gad ~ be ~n~ ch~^~^ {~) ~ is polentisI~ 6em~l~ble ~1~ (~e ~1 writs pS1 far ~ ) +\ .l.(..r,>. [Tho second condition ~ ~ liesib~t ~ ~ de Wham, honcc Hod~e-T:~l:e.~nd He clan defilleits!~\e^~z7~z<~ a' = Oak] = Ok] wh~reC>~isth~usualTate twist o~[1he~-~t~liccompl~li/rlof toes {~ezJ' ~ ad. I1 plies also abet one can associate to ~ a represc~t~tio~ of the W~i6Delignc~roup of~p.heDcoaco~du~tor \~.~b aT>~r<:>f~l . ~ sulfa ~aproper~ndsmootb~rietyov~r~ Add \,j ~ / then 1ho >-adic representation ~36~ IRAQ)) an ~ arc m~,rlc. Granted tbc smooth base change theorem the ~p~s~- t~al~io:l~ is urlrslllified <>tlt-s~<l~ of ~ and the prairies <ail bad r~dudiSoofi.FaI6n<~3jbasprovedtbatth~F~pres~nt~li~D crysl~[inc ~inthegoodr~ducUoncase.llseemslbsll#Ji <4)basnow prov~dib~Lin case ofs~tabl~r~ducd~n.1be r~pr~se~l~ion ~sembtsCle.lbco~ncralcasecanbededuced .. ~ Bow Ts{(sr~sultusin~ d~J~n~'sti)~ork on Lions #~ at< G./~ ~ rO~ ~ 2 ~ ~/C 72~. '~er~>zi'-~/zez<~ ~d~,'~.~.~/: 77!{T~ {< ~ -I`, <~S ~ ~ ~ /~ /~ ~ ir/~! (?'~ S0471 age. Q~,(~), Eves more should 5~ true. Loosely speaking, say TV a g~ome1hc irreducible (-r~pres~ntalion ~ of ~ ~ ~ Cram [hero ~ a finite 2~s~ebra ~,g~n~r~led by Hacks operators~cdn~ on some auIom~rphicrepresent~tio~ space equipped with a continuous homomorphism a: ,. coI~palible>~it~ht~h~:<cl~io~l1of11~ Hack oper;i1$~>rs,'' Sucllt~h:!ll:~l~1.~..is~isom$~)rp~hic10t~hColle Begat Mom p as ~ map ~ ~ ad. Then any g~ome~ic Hacks Paw\ is Vile I Watt 1l~l,:~://~.I:~3ts.~3lg. represent~1ion of ~ should coma from algebraic Homers and Any ~eomethc Inducible representation should be Hedge. ~ this moment, ibis conjecture seems OU1 of Cam. ~cv- erlheless, far an krcUuciblc two-dimcnsion~1 representation of 6, to be ~eomc1hc Locke means 10 bo a Tare twist of s r~pre~nt~GoD ~ss~at~d lo s module farm. Such ~ r~pre- sontatio~Js khan to coma from ~l~cbr~ic geometry. Observe tb~1 the head ~ #Ls? proof of ~ ~ a theorem {6 ~ 0.2) asserting tb~t, if ~ is a suitable geometric Heckle (- r~prcsent~tion of dimension 2, then any ~oometdc En r~pr~scnt~tion of ~ Hick ~ 'dose epochs to His also Hedge. ltsee~s clear that Walsh method should spplyin more g~ncralsl~atio~slO prove tha1~s1~r~n~ asuilableFicke 2-r~prcs~nlation of any enclose enou~b'geomeldc reprc- sent~lion is again Hecke. The purpose oftbesc notes ~ to discuss possible ccncrslizations of the notion of 'close it. enou~h~and ~epos~bi[} of~endin~localcomputadonsin GaloR cohomology which =~ usedin Wildest theorem STOIC . . ~ . .. . . . UelaUs should be sylvan elsewhere ~ . .. De~rm~fiona (at Lc1 ~ be the And of integers of L. a ~ ~ ~ ~ = ~ ~ Ida In. Denote ~ ~ tbc carry of local noethcri~n co~l~1e (]- Title residue Field ~ two ~i1~1 simply call 111~ obiecl6 of this category ~-al~ebras). Let j be a profinbe group and ~) the category of -modules of gaits length equipped ~# ~ (near and con- sinuous action off Considera S1<C1\ MU s~bcatc~ory ~ of stable u:~delsub<~\ecls.cltlol:i~nl:s,and di~>clsums. F6rX ~ Iran/- ~ ~of7LanJ-modu~ offense typ~quipped w~b,EDe~randcon[Duousactio~0f~ Wesay ~ . /~repreSenlatlonsof7,r~o~ectsof~.TbeJ-r~pr~sOnt~t10~s of/lyin~in ~ farm a AH subc~1~<ory )64) fifths category 60 OfJ-reprcs~nt~ti~nsof7+ Washy Zis7~'i1`~1is~1:li~>fr~e) {tS Al ~4-~>du~le. Fix atfla1!~k~r~pres~nIationof7Tvin~in ).Foran<~ in {.l~t<6i) = ~ ~ ~b~1he SC1 ofiso ~ realism classy flat /-r~prcs~nt~1ions ~ of? such that 77~! ~ a. Set #~64) = (~64) = the subset of /64jcorrcspondin~lo r~pre~e~l~ . ~ ~ . ~ .~.~ ~ I. .~ ~ a. PROPOS1~0~. (~# gag )) = ~ ~) ~ g {an) ~ +#, (Tab And ~ = /~> wh~hr~prese~1s Fat ~ ~ q~obc~tof sharing ~ = Reprinting F.) F1xalso a ~13~-r~presenialion ~ of~l)Tiny~ ~ndlyi~in A. As class defies an element of (~) C F(~. back au~mcnlaJons~:~ -+ 0~ and ~> -+C\. ^s = ~&= ~=~ = ~ Ace. we hoe canonical bomorpbisms #~+~)/~+~)~(~. 3)~:~. gage) U tj tJ (at ..~+-~...~)./(~.~ . ~ To= (; I}: I;. {}'{~{-~}) Chose Enough 1~ ~ P~pr~sent~fions. We fix ~ geometry L~r~pr~s~ntation Of ~ (moray Ileck~r~pr~seDtation'). J1138

Colloquium Paper: Fontaine ~< << ~' ~y ~ ~ <-~7j 11]39 Wo choose hat ~-slabla L7-l~l1:1icc [7 of ~ and assume '~ = ~(~ ~ ~ (~ ~< ~ I).} f'>= 1 ~bsolutck ineducable fh~ncc ~ is ~ Tori ~soluleh irreduc ~ We ~ ~ a ~ ~1 ~ ~ ~ cat ad a ~1 subcategory ~ of ~# stable under subo>ccl~ quo- ie~nts, and <Jirect: Stl~lS. For am ~-representador ~ of I, me say ~ ~ / -ale lsldco has in I. We say an -presentation of ~ is ~ ~ by, ~) if it ~ . ~ ~ ~ , ~ ~ u~r~lco outbox ~ ~ ad lies in I. Boa me assume P' is of lynx (~; ala. We say elan (- represent~lion a' of ~ is (! 7~ 7~ r << (~) giver] ~ Nestable latch a' of K. than C'/~' ~ z<; man. ~ ~ denote ah maximal Knob Tenon of ~ con- l~ined~n Q id ou1<dc of i. dc~Fmallon theory apples ~=~ whose ecu are as aback, vised as repros~ntations of ~, are fill I. But i:~c Abet to de~fill~ition of Hi. <~-close 10 Fto be good far our propose ~itIsc~rucTalt~hal~t~l~cale~o~ry)<i~s.s~,i.e.. Wo Could Uke Olson be abletos~ysometbin~ abouttbe conductor of an ~-represe~t~tion of 6) Tyingin 63. Since #,~I{~)j ~ the k~rncloflhe natural map ~ ~ it, b~11eralsoif Be are ablate compute ~ /~#,g{~. Inlborest~flh^c not~s.^e Big d~cusssome example of such se~lslablr calegories a,` E~amplesofSemPS1able acts. [~ ~ ~ 7: The category )< (application oftlO);cr,crys~ ~ll~i.ne). Forany O~-aI~ebr~ /,consid~rth~ category AfF64) Those ounce are -module Of ~f~nitetype equipped 60 a decanting Paradox #~<k\cd by Z) ~ ~ ~ HI ~ ~ . . . - . . ~. ~. ~ by sub~4<m~xtdc~ dam sunu~ands ~ ~ modU as. ash #~ = #~S ~ 0 ~ = 06/ ~ ~ (~) far DIM ~ 2,ao gained map ~ :773~-~(such1bat ^ a= ~ ~ land if ~ {~ { Witl1a:1ol~vio~lsdcEn~il:ion ofl~he~m<~>rphisl~s,TfF643is~ ,4-l~il]~r:~boli~lJ~csle~orv ~ , S ~ ~ aim) ~ubcsle~oryof1hos~7,suchibst13? i! = ifa~d!~+!i? = O.lf~ < ~ we dodge ago T~](~) es the ~llsubca1~0ry of TOF#~64) Show Coeds am 1hcse T} ash no nonz~ro subo~c~ ~ ah 133+1! = 0. Asj3~lls~lLc~t1:~0ITesofifF(~. ~['7`~!64~>d ~!I~.~!~) ar~stablcunderlakinysubo5cct~ quolien1< dRectsums.~d extensions. ... if Zag denote 1bc>-~dic completion of1be normalization ofZ#in (~.thr any .. . ... /~ = ~ I. ~# . . . . . equipped ~hanacllon afC#and, mo~pblsm ofFlob~nius ~ : Boa -> ~< There ~ a c~nonic~1 map J~ -a 2~ whose kernel ~ ~ divided po~ridcalj. Moreover. far () ~ ~ ~ - 1,~) C a. Henc~,b~causc~< h)~no>3o~ion. C C2ll defier stlcJl aisle, ~ )~1 ...>,4<,~, (tS Reilly tile rcs#icdon of ~ lo /( divided outLy if. Fori7jn}~-~-l~6i) ~thenc~ndeEneL/~64~,? !0 as U~ ~d~-n~xIJ~c#JO~ ~> !t ~bkh ~ U~ ~ m cfU~ ink of Be F/#o~ ~ F>-0t F~ O ~ ~ ~ ~ - L \V~ can deigns ~ :~? tab ~ an J-module ofEniletypc equipped with ~lIncarand continuous~cdonof63. WogelinthR ~yanJ-[ne~ Factor 'at ~ #< #~ ~ ~ ~ ad. ~ ~ ~- SKI ?-I).(~)J~) is fully faithful. We call /;~)1:he essential Me. PRoPo~CoX. [~ ~ ~ ~ I- ~ ## ~ \! at? ~ 7~ ~) 773~ ~ 36~ C0ZZ)~! ~< i: (a) P22 r'>~'z~.e., a! ~ (~97 ~? ('~77~C~' Ha ~ Ha) = ] ); 00 a) = 0 {r ~ 0 ~, ~ -a + I; <off) ~ /~ a ~Z{~!~) i[~/ Ha ~) [~> + ] j .... . [~71~. ~7~, (] 1 ), /\ ~ C/~#p~7' ~) i~()~777 t>~} 61~' (\ ~ # O ~ ~ ~ - ~ lips Zeal z <Hera q~ ~ At < ~ define ~ ~ He Cry ~~ All, boseokects~rercpr~sentationswblcb~rcisomo~phiclotbe gen~lEb~rof~ (~beandf~tgroupsohem~overZ>.~ ~ 2, ~ ~ ~ ~Usubc~te~oryst~bleunderextension~of~(tbis is the Bestial ~i~ge of Affix ~1 0j(23~. (~) De~rmabors in ~ don) cbs~c No~ ~e: if B~ are E-reprcsentations of ~JyTng in )/ and ~ one can End IatUccs ~ # L ~ ~ of r ~ ~ ~ ~ >/~, <~ ~ = ~)~r~Z~=~ Em= ~ Computation of 2~. This can be translated in terms of the category ~) ~ Am+ 1 ~) in ~0~), dame Am, ~ as bring the /~ derived Anchor of the rancor IRE) (~~ - a, mesa groups a~ He cohom~Iogy of the comply Ago e ~) 3~ = TP!I'&6 J1sb~pd~ ~ i0 = + #(~ H~nc~.~ Lisa C#~stablcIathce~fan ~-repr~s~ntatio~ Fof 6~ bang ~ ~ Ended Brady? ~ ~ fir ~ (/k), Huh obvious rl01:~t1:ion~ ~e~et/~<,<($>,~{~) ~ ~! >~I.~,)~.~) BRIG. 92.) = ~ ~{Q~.~r, ~ )) and ~D = <~ + ~ #~( = 3~# ) Rbis~cneraliz~s~ ·csultof Ram~krishn~(93]. Special Case. Of specie inlcre~t is tab case There i(p,gl(~)) = {.whichis~qui~alentto ~crepr~entshThtyof the actor .Intbisc~s~. ~ <~((G.~1(~&)) ~ (P#)~1 and ~ >((p,#1 ~ )) ~ ((~)~.\iorebter.b~causeth~re ~ no ad. tab d~rm~lion problem is moots, bench ~) ~ (~[f~> ad. < ~1] !~ ~ ~ 2< <~ (the naive ~e~crahzatlon of ~ to 1bc . ~ semlsla ~ ~ cask. Forgery -algebra ~.~ec~n dchn~thec~te~ory {~64) boseo~ieclsc<:)nsi6t~0f~l ~pB~tl (my, Ha) 11 Tf(~}bj~ctofAfF64) and ~ :if --i~suchtha (if) V(133~) C ~< fly, ~ ~ = ~\ Web an obvious deLnihon of the morphism, this is an abo#~-Eno~r category and Am) can be id~nti~od to the #~)~) ~ , . . ~ ~ . . bang an oDvlous d~I]nlDon of the c,tecorv ~F\~l------77--l-~()l64) T~h~r~is~ln~sl1:tlr~1lw~tytooxle~nd plO PI ail (7:if/`~I-7'iJ(~>J)~) -a ma){ ((a ) ~__. ' ' ' a! ~ Till owl alla ftllly i`~il:~ll~ft:~l. We call ?;(~)~he es~llti~lI image. There is again a simple ch~ract~r2~1ion oflhe category );~6~)of~-r~preseniationsofO~lyingin ~ss~suilabl; ~II

^~oquSum Paper: Nine a~kr~bna. R. (1~93j mu. ma. 87, 269-286. ~10. Foe. jog. ~ Lucille. G. (1982) ^7137/'I/~ t~ ~- ! 1. Foam:, jog. ~ grassing. Ha. ~ 1987) >~< aims I -I #~r ~ ~< (~ ~e~ 67, AT 79-2()7. ~{ 4. ~12. Bre-Llil. C. ~ 1995) (~1~ ~ ~3e'>~-~z'r~ ~17 ~. I. ~ czar. S~ a. 2^ V) (7 >>7j 1 1 1 4 j p ~13. Brctji1. (if. (1996) (~ #~# <:~r 76~$ ~r 6~/~77Z~ en ~ a-. t}~li~v~rsit} de P~ris-S(d, nrenr-~-lI ~ ~ 1~. Conrad, B. (1996) Cafe ~ #~ ~) S~r - ma. Pb.D. (Pdn~lon Un~e~i#. Pr~ceton~

Vol. 94 p. I1142. October 1997 CoJl<>q tiT ugly Paper r ~! ~ ~/ ~ f0~ aft ~r 0~` ~d ~r Brag, >, o~ ~ ^~r ^~r ^~ ^47~) 48~ ~D 7~! !~^ ~' ~ art ~# ~/ ice 7~ aft Ace. GPRD WINGS ~-~p:~lc~k-lnscitu1 Tag ~tlle~llaIik (~)oltj:ri~d-CT~r~ll-Slr~sso 26. 33225 GonTl. C~r~m~y ABSTRACT %Ve explain ~ tectonics result abo~tp-ndlc cObomolog>~ndappkittotbestudyofShimur~varieDes.]be teEbniCalresulf~ppliesto~Igebr~icv~ricties~ithlorsion~h~t cohomolog>,bu1 ~rsimpllctty~onl>~atabelianvarieties. Suppose ~ ~ an ~b~Ii,~ variety OVOF ~ ~ J-adic discrete valuation And Peepers ~ residue gelds Tap ~ Up{) denote Me ma~imalunramiE~dsubJ~g,P~ ~ Baaed ~ ~ Babe Dacron b~lds.lf ~ ~ ~ uni~rmiz~rof ~ then ~ S8136cS as ELenslein equation ~ =) = 0,and ~ ~ ~[7]/~7)# Lc1 ~r denote the<-adic~l~ complied ^-hulI of #~ gong Act. associated lo ~ share at the gage cohomolo~ 21~)~ ~ ~ 3'<) andthr ~ys1~line cobom~Io~y . . Add) = Yr~/~ hich~fterinvcr6~> allows ODC 10 recoveronecohomolo~y Tom Me odor. AnJ1ale Tatecvcl~oide~rc~risa Galo3-invaFianl~l~mcnt , ~ ~)/ A cr~st~1nc T't~cyclc ofdo~r~crisan element , ~ ., {~r ~ ~(X ) hichlIesi~ the ~ - ann~dbued by ~ ~ ># 8yFonlaiDescomparhonthc ~p-v~ctorsp~cesofElaIcand cryst~l~nc Tote cycles ~~isomorphic. Wesh~w: Then. ~r ~ ~ - 2 ~ ~ ~ ~?~< /~)6'> [4] tS1 ~ stage oftho Hodgo Duration andis r ~ ~ t Ij - - - - -- -- - --. - ~ _ . . Brag: At. /S If, The proofusestechniquesdcvelop~d previous. >~V2sTu(2jb~susedthis:cs~I1toshowthatcc~ainShimUra v~riotios cI{s~ibin~ ab~lian varieties gab hi~b~r-ord~r Tats cycles have go6d >~duc3~n. Ha obtains smooth models far 1h~]~by~loT:~Il~t~l-izi~n<1he modul~i-spacc<>fat~ol:i~llvarietTes iT1 He generic fiber of the Shivery varied. To congas thy [21 . The Age cobomology ~ at ~ ~ #~ /p-modu~ limb co~inuousacUoDoT 6~.~tit ~/~)~ ~ ~ ~ ad gee ~:~llod~le will] ~ F~r<>l~ell~ius-~rd<>m<>rpllis~ a. l`~hos~ bile rela1cd ~ Fontaine s 1somorphism , . 713~L4) ~ ~.~> ^-~:,(~-:~ 33~.$. [A ~ ~ ~ Clue ~ ho ? a. lI142 orma#/ation one usesthe valua13~ crkerio~1o~etber web the Thor appIi~dfothe Tam cydesde~nin~the Shimnrs ~ ~ . . J. P~rltillgs 0.(~1994)~/~-}~Z>~r (~'z~> O1;~,F)~> ~{~/P)7~ ~s.p~-cplint. VBSiti, A. (1995j If/ (~-~717~7 ~6 An' \/~ZZ(~! ti7/'/~Y ~7~7~7~.pl^~pri~lt.

Vol. go, pp. I T I43-11!46 ~clob~: 1997 ~q~um Pat FRED LI~OND Dope of ~Pu:r ~lbelll~lic~ ~ll<l ~lh~ll~tic~tl S:~aT:isl:ics. I6 ~i!1 lore. U~live~rsit:y a:! Car>15ri<1g~. (~:aml~ri<:l~ (3B2 !6B, (J:~lil:cd \~lgtI~1 ABSTRACT We discuss the rei~1ionship among certain q~neralizations of results of Hida, Ribef: and Miles on cons u~ ~ ~u~r Irk. ~? ~t as Or congruences in terms of Be value of ~n {-Nnction, and R1bers result is related to 1be bob~vior of the period Off appears fibers. Blest tbeor, leads to ~ class number formula Cling He Clue of (be [~ncUon 10 1be size ~ ~ Clog coho~lo~ group. The beb~vior of the period 1` used to deduce tb~1 a formula at ~nonminim~1 lever, Is obtained Rom one at Minima 1~ ~ dropping Euler actors Mom tbe {- h,~ction. ^~ Sample of a congruence botwe~n modular Arms is provided by tbc arms #. .... Ha= Bomb and <(~)~ >> C2~, ~1 ~1 of Hews IT and 77, respective>. whose ~1 few Fourier coemdents are Rued in Table T. One can shag tba6 in By. #~ mad 3 for a] ~ not divi~bJ~ by 7. (See Theorem i] below.) We shag discuss the r~la~onsbip among He ~#o~in~ Free results conccrnin~ congruences to ~ Prearm ~ of ~el~ht 2 find Laos a. We assume that is ~ number [rid continua the ,. a. . .. .. . . coc:~clents al ~ gnu restrict our attention to c~gruonc~s mod powers of a prime ~ divldin~ (. warmup of Hide (1) mo~suFi~ congruences to ~ in terms of ale value of an [-function. result of Ribe1 (2) thal es1~bllshcs the existence of certain systemic co~yru~ncos between f find arms of level ~ (such as 1be one ^^e). theorem of Wags (3), completed by his Bog with Taylor (4~. which shags that ~11 suitable dc~rm~1ions of Galois representations associated to ~ actually arise Tom farms congruent co 6 a. . Hida s formula, though not par of the l~icaI ~ruclurc of red 3, provides some insight iota tbe role played in Wb~s'p~of a certain ~eral~atio~ of Rivets reset. This ~cncraliza- bo~ can be interpreted as the invariance of a period appearing in HIda s formula. Using this in~riancc. one shams that Is' theorem ~1 minima leveI ethos the theorem al nonminimal , , t^,,,^ ~ . ~ . hi. ~# <7: We arc concerned hare mainly Rib Right r:~isill~ tile level res~TI1. rallier trail his lowering tulle Jeve~l" result of ref. i. We remark that Hide also Fund systematic congruences bet~cn f and farms of leg. (c shad not discuss 1hesc. but Ecus an congruences between f and Arch of label ~ with ~ not divisible by a. ~ 1< ~ 6~ Neons ~> ~ ~ ^~31~-~. IMPS is ale <:,~Tir1~ sl llltl~://~.~T>l~s.~-~. TabTc 1. Wader coexists adz 1 2 3 4 i 6 ~,7 :1 2 -1 2 :I 2 a,) ~1 ~1 2 ~1 -2 2 -:1 ^1~OOn ~"4 ~ We5xa~pri~fa~dembeddi~ys) -and ~ --C.Supposc that ~ isa number bald containedin C Indwelt d~notclbc prime of(\ determined by our choice of~mbeddio~s. Let O denote tbeJoc~[z~tion offs Eta. \Ve suppose ~ f ~ a lam of wheat ~ level and Chaucer >> web coetEJents in /) Tbo Lichl~r-Sh~nura construction ~ssoci~testo f an /-adicrepres~ntati~n ~:0~) ~ #~3 sucb tb~1 iffy does not divide Ha. Ben ~ ~ unrated am and from has c~l]~r~lcterisl:ic polynomial \2 _ <~/,(,()\ + \~# [1] \Vc1~1 ~ de~olelhes~m~impEEcation ofthereduchon of/. If f and ~ are terms of I 2. tllen ~c writer --Cliff ~< equh~de~tto ~.BR1h~ Scrotal d~nsbythcorem and ~3 Br~uer#\esbbllh~orem,~e have f ~ ~ ~ and and ~) ~ <~) ~ra~primes~ no1d\id~gA~^361hecongruonc~b~}g modulolho film ~ idealofth~inte~r~ closure ofZ<in Q<. Weassum~tbroughoutth~ltisodd.~2docsnotdivide \> and~d~esnotUl~Td~ttccol~uctorof Of. WeassumeaL01b~t thereslrictior of into G~1~/~)TsirreJucible Chore ~ the quadrants subfield bfO<~.Tliscon~e~ienllo disdn~ubEt~o 5~1S otprimes Blob can crealetechnic~lp~obl~m~ s Wc}et!>de~otethes~tofprime~such1ba1 ~T<> is not minimal radioed in the sense of ret 6. We let ~y denoto the sat of primes ~ ~ ~ such that )~ = but aj°(O~ ~ ~ j ~ ~ not in ~ U a, Ban ~ ~ lo ~ if and only ~ the pears of ~ diver in the conductors of ~ and 2. an 1bc introduc10ry === Haunting Congruences We assume that ~ ~ dribble by \< but not ~ f~ and Jet >> = {~#h1 2 norms ~ such that ~ ~ ~ Ala and ~y _ . ~ ..... Off LO p denote Ha 0-subaJ<ebrs of Urea C {enor~t~d bv the set of 7~ Ark got dividing #, where to Anodes A. Ben T~ is ~ local and. frog over ~ of rank aqua lo (1~ cardinally . #. . . . ... At Ah.

^UoquTum Papers Diamond T~EOR~E~ 3.1. #> ^~! Calf ~ ~! Whiz ~P 2/~)z7 ~= , i~ ~, ~ Of Of . · . (b) We c~' :~z<~ ~: ~ <~ + ~1 )2 mod ~ /~. The introductory example is a congruence as in the tbc~rem. Wc take ~ ~ 7 and ~ dividing 3. Because <~<<j = ~2. we see share must be ~ farm ~ congruent to f filth ~ = 77 {because = 7 is impossible). Tbo direction (~) ~ (b) of the theorem Allis from consideration of the representation 2. We give thc idea of the proof in the case ~ ~ (: If tberc exists a ~ as in 1be 1beor~m. then the ratio of the cigervslucs of ~ (Froth) must be>~1mod a. Then one apples the ~rmuJa ~Z~(~j~ - >~j~ ~ 1 j: = - y<~0 - ~> t)# _ ~ ~ 1) We direction (b) ~ (a) is closed Elated to broom 4.2. aim shows that C;~> = ~ - If- ~# + 1~;# . . > is not in ~ Ad dog not divide Ha. RibeEs proof rages a comparison of cobomoI~gy congruence ideas. but his soap . . . ........ . Am ~ll1eren1 from the one berg. He compares cobomol- o~y congruence ideals at level ~ and #, Ash the result abet the factor of ~ - 1 does not OCTAL ~ pram Decorum i2, oar dcEnes s certain Abner . . . Mellon {:~1#,0~ ~21~> ad . fir Or ~ ~ dogged so that ARCH' wbcre ' indicates ~c are used a, . , ~ ~ Instead 012. Wc ma even normalize the map so that this r~s~icd0n. tensors with C, sends ~ to <~ Lea the map drops Euler Actors The key ingredient S~ the proof of independence . . ^ .. .. 1S lee 10110~iG~ ~c~De~r~l~ization by Miles of ~ lemma Off tribal: a, . . . 1 D1S 1S proved -using ;T result of lb~r~ chaise role in tile comp~rPon of c~homolo~v congruence ideas ~ Dentin in Rib~t"s work. alto billows At ~ T:lduces an iso~lorpllis~m ~ -+ A', and me conclude that ~ = 3' using )~).~0~) as fit basis fir #'. From boorem 4.2 me deduce: Lo~o~ 32. \~ ~/ I'~CU\. )/ ~ ~~ - i. ~ ~ s = ~ ~ ~ ~ = 2~S ~ (~ = (-by ~S 1), = 6) C{~6f~ II ~.1}-~ [31 . ^~\ ~ .- ~.... Reladon~ilhSelm~ Groups bang ~=ur~lhcoryofJe~rmati~nsof~aloRr~presenta- doll~one~ssoci~1esahn~and~univ~rsaldc~rm~1ion C~1(~V) ~ G-L2(~) . ... . . ot~ ml~lms~yr~mi5cd~u1<de%(seeref.6~.H~r~e~ork Air the compl~nCof(.~hich>~/c~ascontai~edin Suppo~nythaticonla~s<.~eobta~abomomo~is~ <fr~mp~andthoun~crsalprop~rty.Thc(-rlodule ~=k~r<~/(k~r~2 ca~bede~crib~dusing~aJoiscchom~Io~y.!nElc1~eh~, . . . . . canonical Isomorpnlsm PA Jr~.~.67^ ~ <7~) T114i ITom~{~fs,6/O)^ Hi{~.L~/Z<) [4] Chore ~ is~olten Tom ~d°~.The~roup ontb~r~btR some[im~sc~[ed~Selm~r~roup Tbesutscriptli~dicat~s tba1~r~ith<cohomology6>ses~r~supp~sed1~res~ict loclementsof~;~/ ~z,0~/Z<~<asdeEnedinrcE8~.Tbere 3~3,93.Th~unK~rs~lpropertyoftb~dc~rm~tionalsoyields a- ~ Mom ~<o~k~ Ti.Thc~resultofWil~s(3~~dits~encr~iz~lonin<9jis ,... . . and Tesul1 turns out to be relsted lo the comparison of the . . c~n~uence 1~ ~s wig Me Ei~ln~ idea Of ~6~, Rich denote Em. {Rec~D ~ ~ #< kiss gnaw 1~ a, then [~ nail ideal ~ ~ner~d ~ ha, and if the Ie~tb ~ infinite 1ban tithe ~Fitt~n~ ideal is trivia~I.) On the fine hand, an antsy coml~u- t~1Ke algebra ~~umen1 Ohms The ~ ~ .... ^ ~s C Am. [31 An tho other hand. a deeper commutatRo Debra Argument . .. ~ ~ saws ~1 equably holds in Eq. ~ ~ and only ~ the ~in~ . . . . . ~ ~ ~ ^b ~ ,~ ^~ . . 1~. Onc first proves the 1~0 ~ss~rbons in tbc case ~ ~ 0, so lo gel Carted one needs the Once of ~ such that ~ = 0. (h c`~is-tcncoisaversion<]~EScrr~'scpsiloll conj~ctLlfc. ~lndtbe most di~epinlheproofisRib~t~tbe~cmonlo~erinG th~l^~5~.~sumin<1h~t~a~oh~e~ = ~ T^lor~nd WDes(4)showth~tT~,co~pleleint~rs~clioD~and using this f~clWTles(3~)sbo~sthat<~is:~r iSO~DlO~F~p]]iSl~. T#eirproof~ use the~ener~#zationo{~a/ur~resul1discuss~di~Remark 3.2.~TlUf~r<>m which~eaTso deduce Of ~ = (/ ~ = <~) [6 ,~ .~ /> . . ~ ~ = ~ ~ ~ = ~ .. .. Co~hinin~lheinclu~onEq.3~i~ Scooted r~sullil~Frt>~1 Lois Co[loln<>J<~#y ~lr~tjmC~t. He ~FillUt~ll~lEq. ~=~=~ alld~l)),and1:h~I~cZ)j~=~.assumi~golllythati~n{1 (7 = 0. ~pplyin~t~h~r~sultoflenl~rk3.2.~<el~Eq.6~s~e~11 iT1 thltc~e . a, . Gem r>.7:Improv~elltstotllese argtlmc~ts.du~ 10 . ... .. . . . Alms, Le~str~. F~#ar~.~ndlhe~uthor(lO)cstabIisb(s), (b),andEq.6<mullareouSk(~rSl ~ri=0,thenin~enera) )houtapp~aUn~t~Re~rk1? ~ . . ~ ~ ~l:j? /'~ is slot empty. two ~ can ~orl]cl{Illes gel amply 72< fir : #31. but in yc~r~1 we appeal to arc 9 lo Act (a) and (b) in the cam of ~ = ~ ~ ~ ~- #~ if Y = ha. ~ >~1 TI1~EORE~ 6.2. (~ /~ {Z)(7~ /~(~/~.~! ~/ /~/ZZ/{77! ~ ~ ~) ~ ~j ~) ~ any. ~ ~ ~ CQ~! ~ /~ (~ ~!)~ ~ ^~> ~ U 7# ~ ~ 6 bok\. ~t~:Coatesalld Flab h~v~pointedoutlh~lo~ecan deduce Farm thelbeorcm ~ ~rmulaF~ladn<1he ord~r~f (~L~',~/Z<)to (~(~d~l).Tor~Jatclh~ ~rdersof ~ and .o~ uscsavarianlotproposLion #.14~)otrof.8.I~th~ csseof~corr~spondi~<lo~neElpticc~rvc.sc~s~ction30frCE jl ~rthi~ri~nlandr~[ 12 Sara discussion ofthcr~lslion . ~ ~hth~tam~g~anumb~rcoJec~rc(~

lI146 Co~oquTum Paper Diamond The author is graded to ha. Flach far comments on ~n earing draft. 7. 6~ ~sea) Ads supposed ~ the E~incorins and Physics Sconces Research Council {Oran1 No. G^4761~. ~1. Isolde, [1. (~198:~) /~< amp. 63 2.25-~26~. Roes K. A. ( 1984) in Arm #~ ~7 ^~^ Cloch, C. (PWN. Whose). pp. iO3-314. Wiles ^. ( 1995) ago. ma. 141. 443#i 1. .~. 4. Tamer. R. ~ Vilest ^. (1995) age. ma. 141. ii3-572. 5. Ribald K. A. (T990) Ad. ma. 1~0, 431-476. 6. Diamond, F. (1997> in #~r /~ ~) ~S [~ ^~ 'am, ads. C<>l:~cll, C., Silverman, J. ~ Stevens, G. (Springer. Few Y<>ck)^ iI) press. Dram. IT.. Diam<>Ild, F. ~ Taylor R. ( 1996) in C~'lt ! /71 AI2Z/~T~17# SPAT FIJI.. Bolt. R.. JaI:f~ A.. Ilr~pki~s. at. Signora I. Smock, D. ~ Yau, S. T. (Inter:nat~i<~al ~Press, Cambridge, aim), pp. it- l i4. Blocb. S. ~ ~to, K. (1990) in #~ ~# A, ^< ~ ads.. Cartier, P., TlTusic. L.. Katz. at. at. I,ITUmO11. G., mania Yu. ~ Ribet. K. A. (B1rkhauser Boston). pp. 333-400. Diamond F. ( 1996) /~^ Am. 1^ 133166. Diamond. F. (1997) #r Am., ~ pass ~11. Coat~s, j. ~ Syd~nbam, A. (~1993) in Air Camp i7~' ^~ ~) Gaff ,~! ~, ads. Coyotes, ~ ~ You. S. T. ~ntern~ional Pass ~bddg~. Act. go. ~1. 12. Each, at. (1993) in ~i~f~ ~ ~2 ~ #~ ^^ Z~-~ ad. David. S. (Bi~aus~r Aisle). pp. 23~. 8.

VoT. Ha. pp. TT14311148 Ocular 1997 . . Colloquium Paper ~ p-~ ~` ~~ of BEHOLDS SHEPHERD-B)RRON~ TO RICHARD TAYLOR? .. . . . . . (:a~llt~:~ldge uI~IversTty 16 ill 1~ll~ (~TllbIid<~. CB2 ~1SB~ United ~ingd~:l:: arid 1~at:~lle~l~alic~ ll~sT~itTll~. O:~>r~i UI1[VC]SStV. 24-29 St. (digs. Oxford, OX1 31.B, wind ~ ABSlRtC] ~ discuss proofs of some nag special cases of Serrates cofeature on odd, degree 2 rep~senlations of 6~. We strait c~l1 a simple Abel variety J/3 modular if fit is i~cnous mar ~10 ~ Scar of the Jacobi of a modular curb. If ~/~ ~ a modulsr Than variety tan ~ = Ending /~) is a number geld of dogree dime. Placing ~ an iso~enons (over J) Began variety He may assume that E~d64/~) = 0- ~ ~ ~ a at 0~ ~ ^ ^ t e cb~DX~ l~ristic /. then U~ acts an ITS] ~ Hi. so 1ha1 art is a continuousr~present~Jo~ Ace: ~ -(. WeshaUc~1 a r~prcs~ntation arising in this May modular. If ~ denotes complexco~u~at~iont~hc~ld~t ~.~) = -1.~i.e.,~.~iso~ld. Alec ~llowi~np two conjectures have beef c~lIe~nl~ly~ill~[lu- entiaL The bat ~ ag~n~rahzabon of go ShimurasTan~ama . . co~eclu~ ~ second ~ dug ~ Serb HIS # P ~ \~ ~ ~ ~ ## C<~j-s<<TlTR~ 2: arm Cal -a SLY /~ ~/~ Hartz) Sr~)z'<~2 ~ p ~ ~~ Vcryli~le~kno~abou1Serre~coJe~ure.but~dohavC the ~llo~i[l~ deep res~ltof L~t~ngl~t~l~ds(2ja~Dd Tu~llllelJ<3~. 1~EO~E~ 1: ~ am: 6~ -a ~.~) (~ 6~(f!~! ~) 67773 paw Recel~t~>rk<>f W~ile~4)c~>m~p~1ctc<iby Taylo:rarld\V~il~sti) and ~xI~ndcd by Diamond (6jprovesth~ ~llowin~theorem. T11EORE~ 2: art/ /Y ~ \~? I I} (~71~ 77Z~! ^~u /~) ~ ~? ~ ~ /~ ~ ~ ~ ~ ~ a. \~pp<~/~ i<~ He /~<(~;~.'e ~r2~O / ~ 2 \~) ~ 4~! !~#f~2 7~7e)~; ~<~\{~iT{>T)~>Z) /.Y ~/Z/~-~!) {r7~)ZZ{!~! ~713 {t).A ~ lnreE 7 ~coblai~a ~ no casesofSerrcscoJ~cture.ln fact me provothe ~llo~in~lhcor~m. ~z~z) ~ ~P(~1~1077~3 ~/z< p . , . . . . IhlS 1S snc~syconseque~ceoflh~t~otheor~msU1cdabove Widths ~]o~in~ ~l~bro-~omelricresulL BE a \/) awn ~ ~ a. ,. surface wesbaB mono a ~iplo(~.~./j where ~ Sian ab~l~n surge.: s~ Jvisa pri~cip~lpol~riz~tio~ adds: ~[~1 I ...... . . . . ij/2~1 ~ Plldt3), wll~lcl1 ll~ts~illla~c fixed T~> tt]e ~ll<~s~s'1~i ~ + art. (~:!~''/~r'(e'~//~) (:Z// t'2 [/~ !~`/Z 2. ~ ~ ~) -a /~) /~/Z /~ [\ 6Z { # !~71 S!77j){ a! (~1, hi. 0/~ i~< ~/ ~ ~ ^.2 ~) ~.~7 63 ~ ~2(>i) `91.~('//'1`~. (:; 19~7 by 'Id N~1~ll~t! .\c~l(~iel>~y o1 Sciaticas ()(~27-8~24/~7/?il~l <7-2$2.(/() INS ~ ~D ~ ~ ~:~ Pats ofiblstheorem ~ a ]~b1 gc~raUzat~n ofan old resultof Hcrmite(8~<sceaL~re~.9andl0.[Werema~ that tab an~lopousstatem~nt ~rroprese~tations Q~ -a ~/ 47) ~ false] InthL farm (2XCCpl ~rthesudec1Kib off oneofus(R T.)pointed ~ Quito WiI~sin T992ando\plained hog ~ could be usedio deducepartl ofThcorem 3 Mom tbc Shimurs-l>~\am~co~eclurotseeroF 43.ParlSsecmstobe n~w.Ttes~e~umentalsog\~sthc k<1owingF~suL Recap {hat \L2~) ~ J3] PROPORTION I: [~ \ \ ~ ~ ~C ~ { 5 ~ ~ ?~L A/ ~ ~~/ ~ ~) (~F ~# 723<jjr~ {~ Hem #~ Vi~)e)~d~ce//6\ V))~D ~7 = ~(V))(~[2]). \V~wig~o~skc~h ~eproofofp~t20fTbcorem 4 Veered 7 ~rlh~ d~taDs~ Lot ~ denol~th~ cubicsur~cc \} - ~7 ~ - 6 ~ ~ ~ ~ ~ - J ~ :~} '1 Nissan obvious~cGon of!~.Tho27lin~son ~ dKidrinto3 o~rbitsof~lerl~tb li.~.alld 6 un(lert~bo:~Clio~ ofX5.The~li~nes i~11lle orb[to~Flell~thliareal[detin~d oval ~.\Ve ~:il~llet / den~1etbeircomplemcnt.TboolherI2lincs~reo~ch defined ..... . °v~F ~(Vi). Thol~esin glob orb) oflen~h 6 are dT{oinL FJ ~ the open subspac~ of the coarse moduJi space of ...... . V5 Abelson spruces with ~UleveI2 structure which param , .... stows V3 abchansur~c~s which Arc not the product oftw~ [ipticcurv~s.[OvcrSthis~sdiscover~dby Hir/ebruchtsee far Limpid rag 113] We can 1~i) y and }a by p: 6s -- !~(F<) ~ .~<to ot~lai[1 (,alld Id. l~`~h~n-y~isstill~t cubic sur~c~b~causelheachonofJ<extendsto one o~1b~aT~biCnl ~ Itch CANT aft ~ a /> -a (ago }a AND cont~ins6 dl~oi~1[nescollcctTv~ly damped over ~(Vijard blo~in~thcm down ~c obtain ~/~\/3~(again because tbc . . . .. ^ action BILL D11s to a r~pr~entation7~-- ~3). Elf denotes the restriction of scalars from J(Vi) 10 ~ of ~ then Be deduce that </~ ~ a raJona} world. Were is ago ~ dominant r~donal map ~ ~ ~ ~ which on ~eomel~c points sends a pair 01, >~) lo the third pain of in~rseclion of the any >~: ~11b Ha. ~ deduce ~ha! ~ contains maw radons pointy Un~r1~na1cl~ a rational point ~ ~ ~ dogs (01 nUceSsOlily Sac dsc 10 ~ ~ Oberon ~c~ ~ whim is deEnod over A. llo~veF if i1 does then p ~ ~ ~ Over Ethers ~ no universal abeam surface. Hager there is a canonical >unbundle C/~ Ear 1be Za<3ki topolo~y)>Dd (x s~cUons ~1. ~ . !6, such Ha >> ~ ream then tho ~! abeam surface parametrized ^y 31h~ Jacobian of the double cover of ~ ramified cxacdy al ! 1~), i6~) Tbc action of 3< extends to 6. Chore pOr~lpl~S !~ ... , i6 1f8Gs)~Cly, SO that ~ <g a -bundle Ha/ ~ nab ~rth~Etale 10po}0gy~ ~ point of ((~)<vesr~e ~v<~^ ~ 4~l~llotlg~ll A`./! is at Split, OT1C Calls SllOW t)tt1 its pullt)~{ckl~ )~ is spat. Thus pointsin 86Y>(~)) do correspond 10 Vi ~be~srsurUccsd~)n~dovcr ?.Thisksufi~enttoprovcpar1 ~ ~ . X~ .^ . . .

11 T48 ColIo~olum Fader: Sberherd~B~rron and ~OI , ... . . 2 of Theorem 4. t~ shag that the pus back of C~ to ~ ~ we Erg shag that 11 Stands guide codimension go ad Lance is ~quivslc~nt t:o s consl~t:nt btl~ndJ~ (as By, its r:1tiollal) Tbe go End one Trio] point: on it above tabs botlndal-y of )~. Saga. J.~F. ( ~ 987) ~ art. 54. 179-230. 2. Lowlands. R. 6198Q~ ^~ ^~< 6' 6~(2) (Prince10n~7niv. Prep. Pr~>lonj. ~ Turner. ~ (j981) ^? am. at. ~ S. 1~-175. 4. ~< ^. (IBM) age. at. 141, 44#il. 5. 6. 7. ,) 9. Color. R. ~ gilds. A. (1993) am. go. 141, 5i3~72. Diamond. F. (1996) ago. go. , 133166. Sheph~rd-Barro~. a. I. ~ T~Ior. R. (1997) ~ Ace. amp. #~ in acre tag ~ ~o Hoe. C. (18i8) ^~# {~\ 46. Scan. R (1888) [~ ~1 zag a. grins. Price, a. G. (~`rO-bller. I,ondoIl j. Serre, j.-P. (~1986) in 6~-~ ~ 1Z3 (raps iIl-t~d ~1986, Springer). van der ricer G. ~ ~1988) /~' ~z-~ S:~ (Sp]-~i~ogc~r. Saw York}.

(NAS Colloquium) Elliptic Curves and Modular Forms Get This Book
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