)~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
OCR for page 9
:~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,
OCR for page 10
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
OCR for page 11
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):
OCR for page 30
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
OCR for page 31
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
OCR for page 32
OCR for page 33
^~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~
OCR for page 34
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.
OCR for page 35
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 36
OCR for page 37
^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 38
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.
OCR for page 39
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~
.^ . . .
OCR for page 40
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}.
Representative terms from entire chapter:
unit disk
