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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

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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~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 ~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 . . . .. ? .'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.

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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 <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~rtof1beGberoverFp~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~~ ~ = ~ \ (~ 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.~ 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:to-t)~ 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 ~) ~ #' j~ ~ ~r^~ OCR for page 1
ColJoqulum F> 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^ decom~op~! 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~ckol^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 ~ (~('?(''~/'~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~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 {~. TOO, 431-476.

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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 VT. 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~. Exps! 6. Bu~ard. K. (1997j ~ {~. ~ 87, 391-612.

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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 ) be the dasse-Weil <-sparks of ~ over A. lf ~ is ~1~ abe~Jlall Wrapup ~ wraith ~) go its >-prin]~ry ~tlb~lul>. 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 >~ ' /~ ~z''''~, am //ztz' ~ (~ ~) ~.; ~ of )~ ?~) <~ ~ 6,~E~r~ -~ 2 ~'<~6 Tcl~-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(~)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

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1i11( C?~>l~lqu~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 OaloRtndal~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/~~ = ~ ~ 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 [ ~)impl~s Bat )~ ~ istor~on overhead ~ ~ ~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 ~ ?(~) ~ /~}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 )~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~rotlpT`~^,,>~>vcr fit' Len R:>r~llf ~ (), ED -~{ 6) ~ age},, ~ /7 I {~,< ,,,, ~ ) (at ) O~tb~o~rhand.tber~ul~ ~freE8sho~tbat7f is~is..-~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

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:~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. It' i...... ,sp: \Vhere th0 pr(~cct1ve I~t IS taken Wyatt] rCSpect 1;o the co ~ ~ ~ ~ ~ ~ it, ~ s ~ 3 ~ to ~ tic ~ ~ S~0 ~] ~ ~ ~ ' ~ ~ 3 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (t s. ~ ~ ~ C). ~ w ; . - i. ~.~ ~ `~>is'~s~> ~p O3.t I:-- Amps. ~3' I. {_~ ';; {~: C ~ i,\ Of '/~.~^~);s/7 or ~e 1~ ~:~ I(: c )^ cyst ,'~ j (r ~ w_ O~ / /~(34 ~ h's gs a~:~:~s t tf3~ ~0si L.,oox~n a;~< um`eg-~t tog' ~I~e ~ ~y ~ ~ ~ ~ ~ ~ 3 03 ~ ~ ~ s (/ \~t~ ~ ~ ~ (~) 3~ ~ j~ ~ ~f ~ ~ ~ ~ s; g () ~ 3~ ~ [0 ~ (;l t; t3 ~ ~ ~ ]~ p3 ~ ~ ~ ~ ~ s V ~>j~6g ~ pp0~'i~g11?. cx~:~$ ~t ti]0 {g=~t tg3~;l 0~) 0t ~. ~ t~ ~ ~ rg ~ ~ ~ ,' t.8 ~300 .~ &.~^ ~0 ,,~5 i;~.~:~< .~3.[ (i g 3 ] 0}~;\ t ~ ~ ~ ~; ~3,0i ~? ~.} . ~ f.p ~ ~ ~ 3, ~ .~3~ 63 y ~ ~ ~ t ) ~; ~ ~ ~ g g ~ ~ ~ ~ ~ ~ ~ ~ ~ ~,~, ~ ~3 \, ~ ~:} ~ .$~ 7) g ~3 . ~ . iS plai~ : ~ ~ y: .~ ;~ ~ ~ ~ ~ :~: <~ ~ ~ ~ ~ se cr ~ ~ g~ ~ ~ <~ ~ ~ 3 ( .t ~: ) s~ ~ :~ <:~; ~ r t::h ~ ~ wa ~ a`~ <~ aig<, ~ ~ ~'~ ~r (~ ~ ,) l ~ ~ ~2) ~ ~ t ~/)t ~f ~ ~ (3 ~, ~ ~ ~ ~ g, ~ ~ ~ ~ ~ ~ ~g ~C ~ '. .C :0 (.) ~ > ~ ~ ~ /,,, ~ '~ ~ ~;t t. ~ ~ ~ ~.0 i.~:~ X 6, ~ S, <~g `(-~5 r\~1g~t (~; tt?g~ p~t (~ :a ~3 ~ ~ ct ~ ~ ~ ~ ~ j) g -t / \\ g ~ Dg ~ ~: ~,? C3 g 0~.?t :~( 3g] ~gC'tl)~' 9~;~>, (~g `~ ~<~t $~(g27;~2 (A) ~g ~ )^ > ( ~) ~) ~ jr;~ 3 ~' ./,~'?'~(;t~;~s`/{~.~ 6C<,- j ~33 pi~, :~\i~/'/ ~ g.>~/ 8(,[ :~{ (~4 (' ~ 99 7) ~-?x -, .~ \\f ~ ~' ~ (;. ~ ~ ~ ~ ~ S )` tsg ~ .l ~ t33 g ~ (3 ~ ~ ~ ~ 7g~ ~. ~ ~ ~ i; ~ ~ y (~0 ]~] ~ i]] ]~) '3.(;3 . ~; ]] 0~ ~g ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ $?~ ~ ~ 2t ' l; t] ~S:~l,(4.~.l ~,i'p30g t06;.~> ~;t . J g .~ ,. ~ ~.,~ ~ ~ 33 . ~ ~ 4~ ~ ~ ~ ~ ~ ;r i] ~ ~ ~ ~ ~ f ~ ~ ~ ~ ~ C~ ~ ).} S ~ ~ ;} ~ ~ ~ ;~ ~ ~ ~ ~ ~ ' ~ ~ 7is ~J.~[ , ~ ( 6] ~c~ ~ ~ 1 ~ ' ~ ~ /; ^~;) ~ ~p x~) ~ ~~ () p-?~ ~ '~~> ~ ~/~$7 (; ~ {~ i'.- ~ (-g`_4' (~;/~.,/.~<,,~,)) A~ ~ (~:v~v7 ~) ~g ~ ~ .] 0~of. ]?~g~ 4Vi.$ji;~t~t,Sx03.] (~ Ji,l) ti;;,)],i,$~4~. t~r'~g t~h~ 3~g5~~ ~r 5pS32;g~t (~ g 8~ 9, ~ ~ t] 0~ ~ ~ ~g . ~ ~ ~ ~ ~ i.~ 3 oi ret. ~ (.3 ) ~3 ~ ~4~S g ~c~ .~;g ~ ~ c ~ ~ ~ ~ ~ ~ ~ ~ ~ L ~ n: ~ ~ A ~ ~ ~, ~ m :' ~ ;~ ~ g ~ ~ )] ~ ~ C ~ ~i ~ ~ ~ ~ ~ ~ ~ Yg ~ ~ g~ ~ ~ ~ ~ ~) ~3 rL' ~ ~ ~ c . ri, ~ ~ =,0] ~ ~ ~ 3t :0 ~ ~ ~ 5:0 (] ~ ~. ~g0~ ~ ~ g ~ 00 t~ }.~ 3 c,: r~ t. :l ~ c i;l /~ 3;01 ~ 63-( ~ /~^^) ~8 {)~~('lt`S)4~S;4,)40.t'_~g(~t ~g~].]83.(3~ ~ ~r~ ~ ~ ~ g .4 `,f.: \~\ 3. ~ g~ t; ~ ~ if :~. f ~ ~ ~ ~ ~,.t ~ ~ ~ ~i `~ p3 ~ ~ .> 3 g~ L~c g ~ ~ ~ m ?t, ~ ]] ~ c~ ~ ~ ~ }, ('? {~. 3 |0 /~g91 0~ t66 p.~:~; (;f~ ~.~f~3{4w~) ~ S \~: t3~. ~ .4 x t)\ ~74.~fi~ t: ~3t' ti~g~ ~t 5~g~f4]c~f~ ~ ~~ ~ 04. t3f`?~ 3~'tfw~348 (f~ 8~ 3~6 ].~-j \~0 f AC ~\ ~ ~ ~,, ~,~ ~ ~ ~ ~ ~ (~q ~ ~ ~ ~ ~ ~ .?'>) ~i ~ ~ _t ,Af ~p~> ~ ft~6 tti.Ci] ?~ ?~ (~.^.f.~ `..} (J ~; }) ~w (3 g(,374^ (;~) ~ ~ ~ i;~g~? (~f3 ~] ~ g] ~W ~f ~ ~f ~g. ~.3 > ~ \lY~ g ;4~ ~li~g0,g' g-~g]~^ t._~:~ .~"<; t.~36 t ~; }A;~.~f(;i tg.~lg~ (f~S4: )~ ~f~i3.~0': 03: -~'~f:", 80g;~. i~t i'i^,,: 6~3~ j~ ~igi('i)~ gt(-ft~cp f f/` f f S.A f c f f f . f , ? (;i~;.~ (f~~$ C _ fS ?, ~ ? f f A ..? (..9g,g j~`ft.~` 4. A t6.0 .S~ ~', ~'2. ~f~Cti6~{g5~ ~l t~g~ 80~3.. ;~f~3 ~`~fif\~'~ ~ ~,>~;~ t~f~ 4~> Cg~?g]3Ll~ 3,~6 (f9),i,~t,,] t~ [3<,~1~. ,l'3)~ {; \~5~0 j858'~?4~.~ i;g]~; ~:i(~;X) )2,5 t(13~5g`,.~ (~\t(~( /(~ 3j :;ye can pr`:~e ~g~2,~ W03 g g ~ ~ (~;~:~0 ~ 3t ~ ~ ~ ~ ~ ~t S~ ~ `\,t'\~; c/~ ?~] 8) 14~: ~ - 31 ~/ ~ ~ ~ \\ (;i ~l ~ ri~ Y3 y :~-at~ ful L<, ). P. 502'?^[ t\S: p((i)\' ~g~ (A)C, W 4 t]3] ~ *; Sg~ .A3t t~ f^~432 r~, ~f) ~F<.~3~f~il~Sg)3; 7f~ c; 3 -~) ~^rCt1?`f ~ ~ ~9~> ~)l ~fl ,7\']/fiit7~ i.' {. i 263~, l. f^~.4 f' S~ ~ 3 S~ .c ji ~, ):t/] . ~3, .f ~:3 ~(, i) ~ ;~ . ~' ~f iS iS S it ~ ~ ]3 3 ,/ ? ~ c -3 f ~_ 3fAf ~? ~f, f, , ~. . ^. Of ; 3 S i f. . fJf `, 3 3 :S (, ,3 , ~ tS ~,5 ~3 ~s~ C. ~ ~ ~ ~ ~f ~ S ~f- ; 3 ~ S ~ A A 3 3 3 f ~i S , ~ A ;> tA ~f 4r w, A ~? ,A ~ . . 7 :< ~ ~ ~ . A ~ ~ 6~[ {6 . , ~ c~; t ~ ~ `~8,,;~ ~t S, ~f. 6~l {. f ~. . ~ 7 . . ~ 7. ,, .3 A 3 f'' f -? 6` ~ ~ f ~ ~S ; ~ f S? i ff. g , ~6 ~A, ~ ~ ~ S ~ A] ~ 3 g ~ (. ) . ~ S. ~ ~ ~ . S S . A C. f ~ 3 ( .. r `N ~. c . ~ ? ~^ S f f ~ ~ ~ A_. ,~ () ~ 4 ' j~46 ~ S S' ~ ? ~ i.A f f f ,, . S; f A. I~ S ~ /- S g ~ 3 ~ (.3 i3L t~v~>^ J~ t( v31~ 3. 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,

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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~) = ~ LIZ1-> 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~~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~, 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

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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 ~) = ~ ma. The ureas} norms Arc contained in {{~}(~. ~ [elements of ~1(3 ~ which are unr~mjE~d oot~d~ of ~) ~ 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):

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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>~rf~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~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~: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 S1fr~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

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Colloquium Paper: Fontaine ~< << ~' ~y ~ ~ <-~7j 11]39 Wo choose hat ~-slabla L7-l~l1:1icc [7 of ~ and assume '~ = ~(~ ~ ~ (~ ~= 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 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~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; + ] j .... . [~71~. ~7~, (] 1 ), /\ ~ C/~#p~7' ~) i~()~777 t>~} 61~' (\ ~ # O ~ ~ ~ - ~ lips Zeal z .~ ~ 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

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^~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~

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Vol. 94 p. I1142. October 1997 CoJlq 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 ~ ~ ~ ~?~ [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~nfat~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~rl~ell~ius-~rdmrpllis~ 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.

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Vol. go, pp. I T I43-11!46 ~clob~: 1997 ~q~um Pat FRED LI~OND Dope of ~Pu:r ~lbelll~lic~ ~ll15ri<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 ~ ~ ^~31~-~. IMPS is ale 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 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 OCR for page 1

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^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 <~<~1mod a. Then one apples the ~rmuJa ~Z~(~j~ - >~j~ ~ 1 j: = - y 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~/c~ascontai~edin Suppo~nythaticonla~s<.~eobta~abomomo~is~ . . ~ ~ = ~ ~ ~ = ~ .. .. Co~hinin~lheinclu~onEq.3~i~ Scooted r~sullil~Frt>~1 Lois Co[lolnJ<~#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.~.7:Improv~elltstotllese argtlmc~ts.du~ 10 . ... .. . . . Alms, Le~str~. F~#ar~.~ndlhe~uthor(lO)cstabIisb(s), (b),andEq.6~1 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~ OCR for page 1
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. Cl:~cll, C., Silverman, J. ~ Stevens, G. (Springer. Few Yck)^ iI) press. Dram. IT.. DiamIld, 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.

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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~ (~TllbIidr~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<rkf W~ile~4)c~>m~p~1ctcZ) /.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~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~~\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 -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 ~) 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 ZaDd (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 ~ (~)) do correspond 10 Vi ~be~srsurUccsd~)n~dovcr ?.Thisksufi~enttoprovcpar1 ~ ~ . X~ .^ . . .

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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}.