This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
0 at all r e al plac e s of K . ...
1
AB E LIAN
ll - 3 6
PRO POSITIO N 1 - The e i g env alue s of n (i) = , n) . IT (a ) cr (i = 1 , X . (a ) 1
cr cr
•
•
ar e the numbe r s
.
FNv
This is tr ivial by c on s truc tion , b e c aus e unde r T (Q)
�
- ADIC REPR ES E NTATIO NS
T m (Q ) .
a
is the image of
(F
-
are { pv } - units (i . e . The e i g- e nvalue s of 41 v ) the y are units at all plac e s of Q not div iding p ) . v CORO LLARY 1
CORO LLARY 2 - Let z l ' . . . ' Z b e the e ig envalue s of 41 (F v ) , n N Le t w b e a place o f Q dividing p , indexe d s o that z . = X . (a ) . -v 1 1 Then w (z . ) = I: n (i) . normalized s o that w (pv ) = v (pv ) = e v 1 --
N We have w (z . ) 1
=
w (IT cr (a) cr E r
n (i ) cr
) =
I:
cr E r w. cr =v
cr
n (i )w o cr (a) , and
cr E r cr
w . cr (a )
= 0
if
w o cr (a )
=
N
if
wo Wo
cr
cr
F
v
=
v,
N s ince (a ) = Ev . Henc e the r e s ult . COROLLARY 3
41 1 : Gal cKl K )
ciate d to 41.
-
�
Let 1 b e a pr im e numb e r and let Aut (Vl ) b e the 1 - adic r epr e s entation of K as s o
Then 41 1 i s inte g r al (d . Ch. I, 2 . 2 ) if and only if all the charac te r s X . oc c ur r ing in 41 are pos itive . 1
P r oof of C o r ollar y 3 . As s ume fir s t the
X .
1
'
s ar e pos itive .
Let
v • Supp ( m ) and let z ' . . . ' z b e the c o r r e s p onding e igenvalue s of l n
THE GRO UPS S
II - 3 7
m
F
a s in C o r ollar y 2 . C or ollar ie s l and 2 s how that the w (z 1. ) ar e v pos itive for all valuations w of Q ; henc e the z . ar e integr al ove r 1 Z . Henc e the 4> l ' s ar e integ r al. C onve r s ely, as s um e 4> 1 I S integ r al for s ome 1 . The r e exi s ts a finite s ub s e t S ' o f 1: K , c ontaining Supp ( m ) ) such that if v t s ' , the e ig e nvalue s of 4> (F v ) ar e integral. Choo s e a pr ime number p
which s pli t s c o mp lete ly in K and i s such that p == p im p li e s v v � S ' . Let w b e a valuation of Q div iding p . The valuation s w . a,
a E
r , ar e
pairwi s e ine quivalent.
a
Let
E
r ; and let v b e
t h e normal i z e d valuation of K e quivalent to w o a s o that ==
w o a for s om e
X, >
Let z " ' " z b e the e igenvalue s of n l 4> (F ) . By Cor ollar y 2 , w (z . ) = x'n (i) . S ince the z . are integ r al , a 1 1 v thi s shows that the n (i ) ' s ar e all po s itive . x'v
O.
a
PROPOSI T IO N 2
�
Le t X
=
T
E
-
Let v
X (�
i
Supp ( m ) and le t X
be a characte r of
) b e the r e s tr iction of X
to
1ln
and let
j (X T ) b e the inte g e r define d in 3 . 2 . Then , for any archime dian ab s olute value w of Q extending the u s ual ab s olute value of Q , i
we have
P r oof.
If x = a
1: [ a ] + 1: b [ 0] a s in 3 . 1 , we have a aer ae r
w (X (F ) ) N = w (X (F N ) ) v v
=
IT w .. a (a ) a
a
.
IT w o a (a ) a
b
a
,
AB E LIAN 1 - ADIC RE PR ES E NT A TrO NS
II - 3 8
and
IT W O (a) a O" 0"
= W (N (a»
a
remains to s how that x =
= Nv 0
0"
0
c = w,
w
b
iN/ 2 , whe r e
0"
(d .
B ut y henc e
x
2
c = c
W
0,
c O" b = IT w o c oT (a) T T 0"
0"
we have y = x,
Le t
3 . 1) . Sinc e b + b
b
It
= 2a.
is e qual to 1 .
IT w oO" (a) cO"
we have x. y = 1 with y =
w
= Nv
IT W cr(a)
b e th e I I F r ob e nius I I attache d to
and , s inc e
aN
= I,
=
and x = I ,
s ince x > O . Exe r c is e s 1 ) Check the pr oduc t formula for the e ig envalue s of th e cP (F ) . v (Us e Cor . 1 and 2 to P r op . 1 and P r op . 2 . ) 2 ) Show that P r op . 2 and Cor . 1 and 2 to Prop . 1 de termine
the e igenvalue s of the cp (F ) ' s up to multiplic ation by r o ots of unity . v 3 ) (Gene r alization of C or . 1 to Pr op . 1) . Let (p 1 ) be a s tr ictly c ompatible s y s te m of r ational 1 -adic repr e s entat ion s , with
exc eptional s e t S (d . Chap . I, 2 . 3 ) . the e igenvalue s of F
v, P1
,
1 f:. p
v
'
Show that , for any v E �
K
- S,
are p - units . v
APPENDIX Killing ar ithmetic gr oups in tori
A. 1 .
Ar ithmetic g r oup s in tor i Let A b e a l ine ar algebraic gr oup ove r Q ,
and let
s ubgr oup of the g r oup A (Q ) of rational po ints of A.
r be a
Then r is
s aid to b e an ar ithme tic s'ubgr oup if for any algeb raic emb e dding
THE GRO UP S
II- 3 9
m
A C G L (n arb itr ary) the g r oups r and A (Q) n G L (Z ) ar e c om n n mensur able (two s ubg r oup s r l ' r 2 are s aid to b e c o mm ensurable if r l n r is of finite index in r and r ) . It i s we ll -known that it 2 1 2 s uffic e s to che ck that r and A (Q) n GL (Z ) ar e c ommensurable for n one embe dding A C GL n . ----
Example Le t K b e a n umb e r field and let E b e the gr oup of units of K . Then E i s a n ar ithmetic s ubgr oup o f T = R / � ) . K Q m If
T i s a torus ove r Q ,
let T
O
b e the inte r s e ction of the kernels
of the homomorphisms of T into G O
m
The torus T is s aid to b e
"anis otropic if T = T ; in te r m s of the char ac te r g r oup X = X (T ) this means that X has no non - z e r o e l e m ents which are left fixe d by G = Gal (Q / Q) . T HEOREM - Let T b e a torus ove r Q , and let r b e an ar ithmetic o s ub g r oup of T . Then r (\ T is of finite index in r , and the quo O O tient T (R ) /r () T i s c o mpac t .
This is due to T . Ono ; for a pr oof of a m o r e g ene r al s tatement (" Godement ' s c onj e ctur e " ) s e e Mos tow - T amagawa [18] .
CORO LLARY - Le t T b e a torus ove r Q ,
and let r b e an ar ith
metic s ubgr oup of T . If T is anis otr op ic , the n T (R) / r i s c o m pact.
II - 4 0
AB E LIAN
1
- ADIC R E PR ES E N T AT IO NS
Exe r c is e Let T b e a torus ove r
with char ac te r group X.
Q,
a) Show that T (Q ) = Hom
Gal
- * (X , Q )
_ �c
b ) Let U b e th e s ubg r oup of Q g eb r a ic units of Q .
.
who s e e lement s are the al -
Le t r = Hom
Gal
(X , U ) .
Show that r is a n ar ithmetic s ubgr oup o f T (Q) and that any ar ith metic s ubgr oup of T (Q) i s c ontained in r A. 2 . Killing ar ithm etic s ubgr oups Le t T b e a torus ove r g r oup; put Y (T ) = X (T ) � Z Q
Q,
and let X (T ) be its characte r
Let
A
be the s e t of c las s e s of
Q - ir r e duc ible r e pr e s e ntations of G = Gal {Ol Q) thr ough its finite quotient s . For e ach
h.
E A,
let Y
Xo
b e the corre sponding is otypic
s ub -G -module of Y , i . e . the s um of all s ub -G -module s of Y is omorphic to
o
Xo .
O n e has the dir ect s um dec ompos ition
whe r e 1 is the unit r e pr e s e nta1 : on of G ; let Y b e l the sum o f tho s e Y whe r e for all the infinite 'rob enius e s c E C Xo 00 + (d . 3 . 1) we have Xo (c ) = - 1 ; let Y b e the s um of the other Y ' Let Y
We h ave
= Y '
Xo
T HE GRO U P
S
II-41
m
y O = yG = { y e y ! gy = y y-
=
y = Y
Note that
{y e Y ! cy
o
=
-y
if and only if
for all for all
g C
E G} E
C }, 00
is ani s otr opic .
T
��
and H = { l , c } , then , s inc e T (R ) = Hom (X (T) . C ) , H we s e e that T (R ) i s c ompac t if and only if Y = Y If
c E
C 00
PROPOSITION - Let r be an ar ithmetic subg r oup of the torus T , and r its Zariski c lo s ur e
(d.
1 . 2) .
Then:
(* ) [Sinc e the torus T /r is a quotient of T , we identify Y (T / r ) with a s ubmodule of Y (T) . ] Pr oof. Suppo s e fir st that Y is ir r e duc ible , i . e . that T has no p r oper s ubtori and is 1= o. If Y =
yO,
then T is is omorphic to
This shows that Y (T i f )
=
G
Y (T ) , hence
m
(* ) .
and
h enc e
If Y
=
-
r
Y ,
is finite . then T (R)
i s compact. Sinc e r is a dis c r e te subgroup of T (R ) , it is finite . Hence Y (T i f ) = Y (T ) and (* ) follows . =
Y + , then T (R) i s not c ompact . C ons e quently , r is infinite s ince T (R ) / r is c ompact by Ono ' s the or em. Hence r is If Y
an algebr aic subgr oup of T of dimens ion > 1 . Its c onne cted c om ponent is a non - tr ivial subtorus o f T . This shows that henc e Y (T/r)
=
O.
Henc e again (* ) .
r=
T,
II - 42
AB E LIAN l - ADIC REPRESE NTAT IO NS The g en e r al c a s e follows e a s ily fr om the ir reduc ible one ; for
ins tanc e , choo s e a torus T '
to T which s plits in dir e c t product of i r r e duc ible to r i and note that r is c ommen s urable with th e imag e by T '
�
T of an ar ithme tic s ubgr oup of T .
Exe r c i s e Y
Let y by G .
E
Y.
Define Ny as the me an value of the tran s form s of
a Pr ove that N is a G -line ar pr oj e ction of Y onto y O , hence Ker (N) = Y
C HAPTER III LOCALLY ALGEBRAIC AB E LIAN R E PRESE NTATIONS
In
this Chapt e r , we define what it m e ans for an ab el ian
1 -adic
repr e s entation to be locally alg ebraic and we p r ove (c!. 2 . 3 ) that s uch a repr e s entation , when r ational , c om e s fr om a linear repre s entation of one ()f the g r oup s S
m
of Chapt e r
II .
When the g r ound fie ld is a c ompo s ite o f quadr atic extens ions of
Q , any r ational s em i - s imple
1 - adic r e pr e s e ntation is ip s o fac to
locally alg eb r a ic ; this is p r oved in § 3 , as a c on s e quenc e of a r e s ult on tran s c endental numb e r s due to Siegel and Lang . In the local c as e , an ab elian s e mi - s imple r e p r e s entation is loc ally alg eb r aic if and only if it has a " Hodg e - T ate de c ompos ition " . This fact , due to T ate (C oll e g e de Franc e , 1 96 6 ) , is proved in the
Appendix , tog eth e r with s ome c omplements . §l.
T HE LOCAL CASE
1. 1 . Definitions
Le t p be a p r ime numb e r and K a finite e xtens ion of Q
let T
=
RK I
Q
(G p
ml
p
K ) b e the c o r r e s ponding algebr aic torus over
AB E LIAN l - ADIC REPRES E NTATIO NS
III - 2 Q
(d .
Weil [43 ] , Chap . I) . L e t Y b e a finite dim e n s i onal Q p -ve ctor s pace and de note , as us ual , b y GLy the c o r r e s ponding linear gr oup ; it is an algebr aic p
g r oup ove r Q , and GLy (Q p ) = Aut (V ) . p ab Let p : Gal (KI K) � Aut (V ) b e an abelian p - adic repr e s en tation of K in y , whe r e Gal {K I K)abde note s the Galois g r oup of the * ab is the maximal ab elian extens ion of K . If i : K � Gal (K I K) c anonic al homomorphism of local clas s field the ory
(d .
for ins tanc e
C a s s e ls - F r �hlich [6 ] , chap . VI , § 2 ) , we then get a c ontinuous homo * morphism p o i of K = T (Q ) into Aut (V) . P
D EFINI TIO N - The repr e s entation p is s aid to b e locall y alg eb raic if the r e is an algebraic morphism r: T � GLy such that * -1 P 0 i (x) = r (x ) for all x E K clo se e nough to 1 . Note that , if r : T
�
GL y
s atis fie s the ab ove c ondition , it
is uni q ue ; this follows fr om the fact that any non -empty open s e t of * K = T (Q ) i s Zariski dens e in T . We s ay that r is the alg ebr aic P
morphism as s oc iat e d w ith
p.
Exampl e s and dim Y = 1 , s o that p is g iven by a p ab c ontinuous homomorphism Gal (Q I Q ) � U whe re U is p P P P the g r oup of p - adic units . It is e a s y to s e e that the r e exi s ts an elem ent li E Z s uch that p o i (x) = xII if x is clos e enough to 1 . p The repr e s entation p is locally algebraic if and only if II b elong s 1)
to Z .
T ake K = Q
This happens for ins tanc e when
y
= y (f.L) ,
cf. Chap . I, p 1 . 2 , in which c a s e II = -1 and r is the canonical one - dimens ional r e p r e s entation of
T =
G lQ m
p
LO CALLY ALGEBRAIC R EPRESE NTATIO NS
III - 3
2 ) The ab e l ian r e pr e s entation
as s oc iate d to a Lub in - T ate forrnal g r oup (ef. [ 1 7 ] and [ 6 ] , Chap . VI , § 3 ) is locally alg ebraic (and al s o of the form u
�
u
-1
on the ine r tia g r oup ) .
PROPOSITION 1 - Let p : Gal (K / K) ab � Aut (V) be a loc a lly alge b r aic ab e l ian r e p r e s entation of K. The r e s tr iction of p to the ab ine rtia s ub g r oup of Gal (K / K ) i s s emi - s imple . Let us identify the ine r tia s ubgroup of Gal (K/ K)
ab
with the
gr oup U K of units of K. By as s umption , the re is an open s ubgroup U ' of U and an algebr aic morphism r of T into GLy s uch K -1 that p (x ) = r (x ) i f x E U ' . L e t W be a sub -vector spac e o f Y stable by
p (U ) ; it i s then s table by p (U ' ) , hence by r (T ) . But K eve ry linear repr e s entation of a torus is s e mi - s imple . Henc e , the r e
exists a pr oj e c to r T . I f w e put 1T '
=
1T : Y � W which c ommute s with the action of 1 -1 p (S ) 1T p (s ) , we obtain a p r o 1: (U K : U' ) S E UK / U '
� W which c ommute s with all p (s ) , S E U K ' q . e . d. Conver s ely, let us s tart from a repr e s entation p who s e re
j e ctor 1T' :
Y
s tr ic tion to U K is s emi - s imple . If we make a s uitable larg e finite extens ion E of Q ' the r e s tr ic tion of p to U K may be br ought p into diag onal form , i. e . is g iven by c on tinuous characte r s *
X
: U � E , i=l , . . . , n . We as sume E larg e enough to c ontain i K all c onj ug ate s of K , and we denote by r K the s et of all Q - ern
b e dding s of K into E . Rec all (ef . chap . II , 1 . 1 ) that the [ a) ,
a
E r K ' make a b a s i s of the char acte r g roup X (T ) of T .
PROPOSITION 2 - The repr e s entation p is loc ally algebraic if and only if the r e exis t integer s n a (i ) s uch that
AB E LIAN l - ADIC REPR E S E NTATIO NS
III - 4
x . (u) = TT 1
aE r K
a (u)
- n (i ) a
for all i and all u c l o s e enough to 1 . The n e c e s s ity i s tr ivial . C onve r s ely, if the r e exis t such in n (i) of te g e r s n (i ) , the y define alg eb r a ic charac te r s r . = TT ( a ] a a
1
T , henc e a l ine ar r e pr e s entation r of T / E . It i s c lear that ther e -1 is an ope n s ub g r oup U ' o f U ' such that p (u) = r (u ) for all K
Henc e it r emains to s e e that r can b e defined ove r Q p (c f. chap . II , 2 . 4 ) . B ut th e tr a c e a = � r . o f r (loc . � ) is 1 r s uch that a (u) E Q for all u E U ' . S inc e U ' is Zariski - dens e in p r T , this implie s that e is " defined ove r Q " hence that r r p c an b e defin e d ove r Q (loc . c it . ) , q . e . d. p
u
E
U' .
-
Extens ion of the g r ound field Let
K'
b e a finite extens ion of
K,
p'
and let
b e the r e
s tr ic tion of the g iven r e p r e s entation p to Gal (K I K ' ) . Then p '
i s locally algebr a ic if and only p is ; mor e ove r ,
if
this is s o , the
as s oc iate d algeb raic morph i s m s r: T ar e s uch that r ' K'
N Q r , K' / K
, : T' � K / K from K ' to K .
and
norm
=
N
-+
G Ly , r ' : T '
-+
G�
whe r e T ' i s the torus a s s o ciate d with
T i s the algebraic mo rphism define d b y the
All this follows e a s ily fr om the commutativity of the diag r am
LO C AL L Y ALGEBR AIC R EPRESENTAT IONS
III - 5
i
K'
':c
--+
ab Gal {K / K ' l
and fr om the fac t that the kernel of N K , for the Z ariski topology .
/
K: T '
--+
T i s c onnec ted
Exe r c i s e Give an example o f a loc ally alg ebr aic abel ian p - adic repre s e ntation of dimens ion 2 which is not sem i - s imple . 1 . 2 . Alternative definition of " l oc ally algeb r a ic " via Hodg e - T ate module s Let us r e c all fir s t the notion of a Hodg e - T ate module § 2 ) ; here
K
[ 2 7] .
i s only as s umed to be c omple te with r e s pect to a dis
c r ete valuation , with pe rfe c t r e s idue fie ld k and char (K ) char (k) = p . Denote b y C the c ompletion of
(d
�
= 0,
of the algebraic closur e
K
K.
T h e gr oup G
=
Gal (K /
K)
acts c ontinuous ly o n
K.
This action
extends c ontinuously to C . Let W be a C -ve ctor spac e of finite dimens ion upon which G acts c ontinuously and s em i - linearly acc or d ing t o the formula s (cw)
s (c ) . s (w)
(s
E
G, c
E
C and w
E
W) .
G � U b e the homomorphism of G into the g r oup p U = Z of p adic units , define d by its action on the p v_ th r oots p p of unity (d. chap . I , 1 . 2 ) : Let
X :
=
*
III - 6
AB E LIAN 1 - ADIC REPR ES E N T A TIO NS s (z ) Define for every i W
i
=
{w
=
E
E
z
X (s )
if s
E
G and zP
v
=
1.
Z the s ub s pa c e w i sw
=
i X (s ) w f o r all s
E
G}
i o f W . This i s a K -vecto r s ub s pac e of W . Le t W (i) = C (8) K W This i s a C -v e ctor s pa c e upon which G acts in a natur al way (i . e . i b y the formula s (c � y) = s (c ) � s (y) ) . The inclusion W � W extends uni q uely to a C -l ine ar map a . : W (i ) --:> W , which c om 1
mute s with the ac tion of G .
PRO POSITION (Tate ) - Let .u W (i) be the direct s wn of the W (i) , i
E
ab ove .
Z.
Le t
Then
a
a:
.u W (i) --:> W b e the sum of the
is inj e ctive .
a. I
1
s defined
F or the p r o of s e e [ 2 7] , § 2 , p r op . 4 . i CORO L LAR Y - T h e K - spac e s W (i
E
Z ) are o f finite dimens ion .
They ar e line arly independent ove r C . DEFINITION 1 - The module W i s of Hodg e - T ate type if the homo morphi s m
a:
11 W (i ) --:> W i s a n i s omorphism. iE Z
Let now V be as in 1 . 1 , a vector space ove r Q , of finite dimen p s ion . Let p : G � Aut (V ) b e a p - adic repr e s entation . Let W
=
C Q!)
Qp
V
and let G act on W by the formula
III - 7
LO CALLY ALGEBRAIC REPRES E NT AT IONS 5
(c
� V) = 5
(C )
� P
(s ) (v ) ,
5 E
G, c
E
C,
V E V.
DEFINIT IO N 2 - The r e p r e s entation p i s of Hodge - T ate ty p e if the C - spac e W = C Q9 Q V i s o f Hodg e - T ate type (d . def. 1) . p E x aInple
Let F b e a p - divis ible gr oup of finite he ight (d . [ 2 6 ] , [3 9] ) ; it . ) and V = Q Qg T . The gr oup G let T b e its T ate Inodule (lo -c . cp acts on V , and Tate has pr ov e d ([3 9] , C or . 2 to Th . 3 ) that thi s Galois Inodule is of Hodg e - T ate type ; Inore p r e c i s e l y , one has W
=
W ( O)
ED
W (l ) , whe r e W
=
C
�
V as ab ove .
its T HEOREM (Tate ) - As sUIne K i s a finite extens ion of Q (i . e . p r e s idue field is finite ) . Let p : G � Aut (V ) b e an abe lian p - adic r e pr e s entatior: of (a)
p
K.
The following p r opertie s are equivalent:
is locally algebr aic (d. 1 . 1) .
(b ) P is of Hodg e - Tate type and its r e s tr ic tion to the ine rtia g r oup is s e Ini - s iInple . F or the proof, s e e the Appendix. § 2 - T HE G LOBAL C ASE 2 . 1 . Definitions We now go back to the notations of chap . nUInber field. Let
1
-
1
- adic
i. e .
K
denote s a
b e a pr iIne nUInbe r and let p : Gal (K / K )
be an ab elian
II ,
ab
-?
repr e s entation of
Aut (V 1 ) K.
Let
v E
�
K
be a plac e
AB E LIAN 1 - ADIC REPRESENTA TIO NS
III - 8
ab b e th e of K of r e s idue characte r i s tic 1 and let Dv C Gal (KI K ) c o r r e s p onding de c Olnpos ition gr oup . This gr oup is a quot ient of the ab (the s e two g roup s ar e , in fac t , is o l o c al Galois g r oup Gal (K 1 K l v v m o r phic , b ut we do not ne e d this he r e ) . Henc e , we get by c ompo s i tion an 1 - adic repr e s entation of Kv p : Gal (K 1 K )
v
v
v
ab � D -4 Aut (V 1 ) . v
DEFINI TION The repre s entation p is s aid to be loc ally alg eb r aic if all the loc al r e pr e s entations p , with P = 1 , are locally alge v v br aic (in the s en s e defined in 1 . 1 , with p = 1 ) . -
--
It is c onvenient to r eformulate this de finition , us ing the torus
0 -torus obtaine d fr om T by extend ing the g r ound fie ld fr om 0 to 1 0 . W e have 1
= K &? 0 . 1 1 Let I b e the id� le gr oup of K , cf. Chap . II , 2 . 1 .
whe r e K tion K i: I
�
*
� I, followe d by the c las s fie ld homomorphism 1 ab Gal (K I K ) , define s a homomorphism
* ab i : K l � Gal (K / K ) . 1
The inj e c -
III - 9
LOCALLY ALGEBRAIC REPRESENTAT IO NS
PROPO SIT IO N - The r e pr e s entation p is loc ally algeb r a ic if and only if the re exi s ts an alg eb r a ic morph i s m f: T / Q
1
� G LV
-1 s uch that p o i 1 (x) = f (x ) for all
X
E
1 *
K 1 c lo s e enough to 1 .
(Note that , as in the local c as e , the above c ondition determine s f unique ly; one s ays it is the algebraic morphism as s o c iate d with p. )
Sinc e K I1O Q Q 1 = TT K vl 1 v T/
Q1
=
,
we have
TT Tv
vl 1
is the Q 1 -torus define d by Kv v tion follows fr om this de c ompo s ition.
whe r e T
cf. 1. 1.
The propos i-
Exe r c i s e Give a cr ite r ion for loc al algebr aic ity analogous to the one of P r op . 2 of 1 . 1 .
2 . 2 . Modulus of a locally algebr aic abe lian r e pr e s entation ab Let p : Gal (K I K ) � Aut (V 1 ) be as ab ove ; by c ompos ition with the clas s field homomorphism i : I � Gal (KI K) ab , p define s
a homomorphism p o i : I
�
Aut (V 1 ) .
AB E LIAN 1 - ADIC REPR ESENT A TIO NS
III - I O
We as s wne that p is loc ally alg ebr aic and we denote b y f the a s s oc iate d alg ebraic morphism DEF INITIO N P
-
Let
is de fine d mod
m
I Q1
�
G LV
•
1
b e a modulus (chap . II , 1 . 1 ) . (or that
m
T
One s ays that is a modulus of definition for p )
m
if (i ) p .. i is tr ivial on U V, m (ii ) P I) i l (x) = f (x - l ) fo r x (Note that IT vl l
In
U
v, m
is
an
E
p 1= 1 . v
when ---
IT
vl l
U
V, m
ope n s ubg r oup of K
�
=
T
/ Q1
(Q l ) . )
order to prove the exis tenc e of a modulus of definition , we
ne e d the following auxiliar y r e s ult: PROPOSITIO N - Let H b e a Lie gr oup ove r Q 1 (re s p . R ) and let a be a c ontinuous homomo rphis m o f the id�le gr oup I � H. *
(a) If p 1= 1 (re s p . p 1= (0) , the r e s tr iction o f a t o Kv - v v * i s equal to 1 on an open s ubgr oup of K . v * (b ) The r e s tr iction of a to the unit group Uv of Kv is equal to I for alm o s t all v ' s . Part (a) follows fr om the fact that K
*
is a pv - adic Lie gr oup v and that a homomorphism of a p - adic Lie gr oup into an 1 - adic one is loc ally equal to 1 if p 1= 1 . T o prove (b ) , let N b e a ne ighb orhood of 1 in H which c on tains no finite s ub g r oup e xc ept { l } ; the existence of s uch an N is c las s ical for r e al Lie group s , and quite easy to pr ove for l - adic one s . By definition of the idMe topology , for alm o s t all v ' s . B ut (a) shows that , if
a
(U )
is c ontained in N p 1= 1 . the g r oup v v
LO CALLY ALGEBRAIC REPRESE NTAT IONS i s finite ; hence
a (U )
v
a (U )
v
III - ll
= { l } for alm o s t all v ' s ,
q . e . d.
CORO LLAR Y - Any abe lian 1 - adic r e pr e s e ntation of K is unr am i fie d outs ide a finite s e t of plac e s . This follows fr om (b ) applied to the homomorphism induc ed by the g iven r e pr e s entation , s in c e the the ine r tia s ubg r oups .
a (U )
v
a
of I
are known to b e
Remark This doe s not extend to non - abel ian r e pr e s entations (even s olv able one s ) , cf. Exe r c i s e . -
PROPOSITIO N 2
Eve ry loc ally algebraic ab elian 1 - adic repr e s enta
tion has a modulus of definition. ab Gal (K I K ) � Aut (V 1 ) b e the g iven r e pr e s entation and f the as s oc iated morphism of T into G L Let X b e the V1 I Q 1 Let
p:
p '1= 1 , for which p is r amified; the v cor ollary to Prop. 1 shows that X is finite . By Pr op . 1 , (a) , we can s e t of plac e s
v
E
choo s e a modulus
l;
K
m
'
with
s uch that
p o i:
I
�
Aut (V 1 ) is tr ivial on all
, v E X . Enlarg ing m if nec e s s ar y , we c an as s ume that v ,m Henc e , m is a modulus of p o i 1 (x ) = f (x - 1 ) for X E IT U , v m Pv = 1
the
U
definition for
p .
R emark It is eas y to show that the r e is a s mall e s t modulus of definition for p ; it is calle d the c onductor of p .
IIl - l 2
AB E LIAN 1 - ADIC REPR ES E NTATIONS
E xe r c i s e ...
',
Let z ' . . . ' z ' . . . E K o F o r e ach n , let E b e the s ub l n n field of K generate d b y all the 1 n - th roots of the element 1 1 n-l zlz 2 · · · z n a) Show that E is a Galois extens ion of K , c ontaining the n 1 n - th r o ots of unity and that its Galois group is is omo rphic to a s ub g r oup of the affine
group (� � ) in GL (2 , Z /
1 nZ) .
b ) Let E be the union of the E I S . Show that E is a Galois n e xtens ion of K , whos e Galois g r oup is a clos e d s ubgr oup o f the affine g r oup r e lative to Z 1 . c ) Give an example whe r e E (and he nc e the c o r r e s ponding
2 - dimens ional 1 - adic repr e s entation) is r amifie d at all plac e s of K . 2 . 3 . B a c k to S
m
Let
m
b e a modulus of K and let
be a line ar r e p r e s entation of S
m
/Q
Let
1
ab
1 a s 4>
(chap . II, 2 . 5 ) and the c or r e s ponding pr op e r tie s of 2. 3) . The conve r s e of The orem I is true .
£
1
'"
£
1 (chap . II ,
We s tate it only for the
c a s e of rational r epr e s entations : ab T HEOR EM 2 - Le t p : Gal (K I K) � Aut (V 1 ) be an abelian 1 -adic repr e s entation of the numbe r field K . As s ume p i s rational
(chap . I, 2 . 3 ) and is locally algebr aic with m
as a modulus of de
finition (cf. 2 . 2 ) . Then, the r e exi s t a a -vec tor s ub spac of e V0 V 1 ' with V 1 = V 0 Q!)a Q 1 ' and a morphism rP o : Sm � G LV o of a - algebr aic gr oup s such that p is e qual to the 1 - adic r epr e s en tation rP
l
as s oc iate d to rP o (ef. chap . II , 2 . 5 ) .
(The condition V 1 structur e " on V 1 ' ef. Pr oof.
Let r: T p .
c iate d with p
(>
/a
1
�
=
V0 @
a
Q
B o u rb a k i
GL
V1
�
means that V 0 s a " Q r Alg . , chap . II , 3 ed. )
1
b e the algebr aic morphism as s o -
We have
i (x)
=
-1 r (x ) for x
•
E
K n
�
Um
=
IT vi i
U
V ,m
III-14
l - ADIC
AB E LIAN
Define a map
rj;:
I
�
Aut (V L ) by rj;
whe r e x
L
is the
d i ate ly that rj;
R E PR E S E NTATIO NS
(x)
=
p o i (x) . r (x )
L
th c omponent of the id� le x. One che cks imme L-
i s tr iv ial on
Henc e r is triv ial on
Um J•
=
Em
and c o inc ide s w ith
K'" n
b r aic morphi sm rm : T m / QL
�
Um
on
K
�'
1 s uch that the f N . are 1 N locally algeb r aic . This implie s , c f . 1 . 1 , that p i s locally alge b r aic , henc e (ef. 3 . 2 ) that p its e lf i s loc ally alg ebr aic , q . e . d. Exe r c i s e As s ume that K i s a c ompo s ite o f q uadr atic fie lds . Let X be a G r O s s enchar akte r of K and s uppo s e that the value s of X (on the ide als pr ime to the c onductor ) ar e algeb raic numbe r s . Show that
X
is " of type (A) " in the s en s e of W e il [ 41] .
method than a!:; ove , with E replac e d by C . ) in a finite extens ion of Q , s how that
X
If
(Us e the s ame
th e value s of
is " of type (A o ) "
.
X
-+
lie [ no
a s s u m p tio n o n K is n e c e s s a r y , th a n k s to [ 8 3 ] . J
APP ENDIX Hodg e - T ate de c ompo s itions and loc ally algebr aic r e pr e s entations Let K be a field of char acte r is tic z e r o , c omplete with re s p e c t t o a dis c r e te valuation and with perfe c t r e s idue field k o f charac te r is tic p > O . In thi s Appendix w e deal with Hodge - T ate de c ompo s i tion of p - adic abe lian repr e s entations of K .
LO CALLY ALGEB RAIC RE PR ESE NTAT IO NS Se c t ions A l and
III 3 l -
g ive invar ianc e pr ope rtie s o f the s e de
A2
c ompos itions unde r g r ound fie ld extens ions . Spe c ial char acte r s of Gal (K / K) ar e defined in A4 ; they ar e clos ely c onne cted b oth with Hodge - T ate module s (A4 and AS ) and local algeb raic ity (A 6 ) . The pr oof of T ate ' s the o r e m (c f. 1 . 2) is g iven in the las t s e c tion. AI . Invar ianc e of Hodg e - T ate de c ompos itions Le t C be the c omple tion of K (cf. 1 . 2 ) ; the gr oup Gal (K / K) ac ts c ontinuous ly on C .
Le t
X
b e the characte r of
Gal (K/ K )
into the g r oup o f p - adic units define d in chap . I, 1 . 2 . Let K ' / K be a s ubextens ion of K / K on which the valuation -; of K is is a finite extens ion of an unrarnified
dis c r e te ; this means that K' one of K .
A
Le t K '
denote the clo s ur e of K '
in C .
Let now W b e a finite dimens ional C -ve c tor space on which Gal (K/ K) acts continuous ly and semi - linearly (s e e 1 . 2 ) . As befor e , � n n we denote by W (r e sp . W K ' ) the K - (re sp . K ' - ) vector space defined by n n W = {w E W I S (w) = X (s ) w (r e s p .
w� ,
= {W E
w i s (w)
=
X
n (s ) w
s ' Gal (K/ K) }
for all
for all s
E
Gal (K / K " ) } ) .
n n Identifying the ' Let W (n) = C :
*
G � K , the r e i s an
c (q, ) LX ;
when
K
is
loc ally
c OITlpac t, this c (q, ) ITlay be c OITlpute d explic itly, s e e A6 , Exe r . A3 .
A
2.
c r ite r ion for loc al tr iviality
F r oITl now on , E denote s a s ubfie ld of K having the following prop e rtie s : (a) E c ontains Q c OITlpact) . .(b )
p
and [E : Q J p
< 00
(s o that E i s loc ally
K
c ontains all Q - c onj ugate s of E . p We denote by r E the s et of all Q - eITlbe dding s of E in K . p C ons ider a c ontinuous char acte r 1/1 :
-
Gal (K/ K )
�
E
*
with value s in E . For e ach cr E r E this give s a char ac te r * cr * * cr o r/; : G � E � K of G = Gal (K / K ) into K .
III - 3 9
LO C ALL Y ALG EB RAIC R E PR E S E NTAT IO NS
PROPOSI T IO N 3 (1 ) r./J
T h e following two pr o p e r tie s a r e equival ent :
-
i s e qual to 1 on an op en s ub g r oup of the ine r tia group
of G , (2 )
ao
(1 )
=>
r./J
1 for all
--
a
E r
'
E
P r oof
(2 ) i s tr ivial fr om th e r e s ul t of Al
(s inc e we know
that admi s s ib il ity c an b e s e e n on an o pe n s ubg r ou p of the iner tia g roup ) . (2 )
=>
(1 ) .
W e us e the l o g map defined in A2 .
take s value s in the g r oup U
E
of unit s of E ,
Note that r./J
henc e log r./J : G
�
E
Le t I b e the ine r tia g r o up of G ; the s ub g r oup n log r./J (1 ) of E i s c ompa c t , and h e n c e i s omorphic to Z for s ome p n . If W i s the Q - v e c t o r s ub s pa c e of E g e ne r ate d b y log r./J (I) , p we s e e that l og r./J (1 ) i s a latti c e in W , and d im W = n. Note that i s well define d.
s aying that r./J is e qual to 1 on
an
1
is
� E is a local E S upp o s e thi s i s not the c as e , i . e . s upp o s e that
e quivalent to s aying that log r./J (1) i s om o rph i sm) .
open ne ighb ourhood of I in
=
0 (s inc e log : U
Cho o s e a Q - l inear map f : E � K s uch that d im f (W ) = 1 ; p s uch a map obviously exis ts . B y Galois the o ry (indep e ndenc e of
n > 1.
char acte r s ) the s e t r exi s t
k
a
E
and we have
E
i s a b a s i s of Hom
Q
p
(E ,
K) . Henc e , the r e
K wi th
f o l og
r./J = � k a o l og r./J = � k l og (a o r./J ) . a a
AB E LIAN 1 - AD IC R E PR ES E NT ATIO NS
III - 4 0
But b y as s umption (and Prop. 3 of A2 ) , the additive l - c o c yc le log (a D rJ; ) : G fo r f .. l o g rJ;.
�
K is c ohom o l o g ous to O . Hence the s ame is true
N B ut we may as s um e (r e plac ing f b y p f ,
with N
larg e , if n e c e s s ary) that the r e exists a c ontinuous homomorphism F : U � U s uch that f o log = log o F . We then have E K log (F o rJ; ) = f o log r/J and h en c e (d . P r op . 3 0f A2 ) , F" r/J - l , L e . F o rJ; is admis s ible . But F o rJ; has now the pr ope rty that F o rJ; (I) C U is a p - adic Lie gr oup of dimens ion 1 (product of Z p K with a finite g r oup ) . This c ontr adicts a the orem of Tate ([3 9] , § 3 , Th . 2 ) , henc e the r e s ul t
A4.
Th e charac ter
.
X
E We keep the s ame hypothe s e s on K and E as in th e p r e v i o us ab s e ction . By c las s field theo r y , the g r oup Gal (E / E ) may be id e n "*
tifie d with the c ompletion E
of E
*
with r e s pe c t to the topology of
open s ubgr oup s of finite index. In particular , we have an exact s equenc e
wher e Z
-
IT Z 1 denote s the c ompletion of
t o p olo g y of s ub g r oup s of finite index
-
(d .
or Cas s e l s - F r �hlich [6] , Chap. VI , § 2 ) .
Z
with r e spect to the
for ins tanc e Ar tin - T ate [2]
LOCALLY ALGEBRAIC REPRESE NT A TIO NS
III - 4 1
Let now 7r be a uniformiz ing e lement of E. The imag e o f 7r ab in Gal (E / E ) g ener ate s a s ubgr oup who s e c l o s ur e is is oITlOrphic ,.. to Z , and this g ive s an is omorphism:
ab Gal (E / E ) � U E b e the proj e c tion as s o c iate d with this 7r de c ompos ition (the Galois extens ion of E c or r e s ponding to Ker (pr ) Let pr
:
i s the c ompos ite of all finite abe lian extens ions of E for which is a norm , cf. [6 ] . p. 144 - 14 5 ) . O n the oth e r hand , the inc lus ion E
phism Gal (K. / K)
�
X
E , 7r
(abbr . G
=
x-
1
7r
K define s a homomor
Gal (E / E ) , henc e als o a homomorphism rE : G
Define
�
7r
X
�
�
Gal (E / E ) ab
E ) to be the comp o s ite homomorphism ab Gal (E / E )
�
i U UE � E '
U E . Ob s e rve that the r e str iction of X E 1 to the inertia gr oup of G is x � r E (x - ) , and hence is indepen dent of the choice of 7r.
where i (x)
for x
E
PROPOSIT ION 4 - Let F be the Lub in - Tate fo rmal gr oup ( [17] , 7r s e e als o [6] . chap . VI, § 3 ) as s oc iate d to E and 7r. Let T be its
Tate -module , which is fr e e of r ank 1 ove r the r ing 0 E of integer s of E . The action of Gal {K. / K) on T is g ive n b y the character : G � U , defined ab ove . X E E
AB E LIAN l - AD IC R EPR ESE NT A T IO NS
III - 4 2
This follows fr om the main th e o r e m of [ 1 7 ] (s e e al s o [6 ] , Th . 3 , p . 14 9) . CORO LLAR Y - If E
=
7r
Q
and p c o inc ide s with the char ac te r X
=
th en the characte r X
p,
E
defin e d in chap . I , 1 . 2 .
Inde e d , the Lub in - T ate g r oup i s now the multi p lic ative g r o up G
m
and its Tate module i s the modul e
R emark If
* . 1 o c a11 y c ompac t , we may 1' d e nh' f y G ab to K K 1S .... an d th e X
c har acte r
E i s g iven b y
"* N ,, * K � E
whe r e N
T (f.I.) defined in ch ap . I , 1 . 2 . P
=
N
K/
E
pr
7r
�
•
U � U ' E E
i s the norm map .
[ This follows fr om the func
tor ial p r op e r tie s of the " r e c ip r o c ity law " of local c las s field the o r y . J
U
In
K
AS .
particular , the r e s tr ic tion of X to the ine rtia s ub g r oup E ab -1 (x ) . of G is x � N
K/ E
Character s a s s oc iate d with Hodge - Tate de c omp o s ition s Retaining the notation of the pr e vious s e c tions , let p : G � U
b e a c ontinuous homomorph i s m .
s pac e ove r E ; we make G ac t on V by (s , y)
Henc e V i s a G - m odule .
�
E
Le t V b e a one - dimens ional v e c toI
p (s ) y ,
Le t W
=
S
C 60 Q
E
G, y E V. V,
p
whe r e C
=
�
K as
III - 43
LO CALLY ALG E BRAIC R E PRESE NTAT IO NS
befor e . =
d a
x
This i s a d - dimens ional vector spac e ove r C, whe r e
[E : Q J . p
Eve ry element x o f E de fine s a C - endomorphism
of W by a
X
(!:
c.
l
�
y. ) I
=
!:
c.
I
� xy .
I
C .
,
1
E C , y. E V . 1
We g et in this way a r e pr e s entation of E in the C -vector space W ; note that th e action of ax c ommute s with the action of G. Let a E r and put E
Wa
=
{w l w
E
Each W LEMMA I - (a) --
s table by G .
W , a (w) x a
=
a(x)w
for all x E E } .
is a one - dimens ional C - ve c tor spac - e
(b ) W is the dir e c t s um of th e
(c ) For e ach i s omorphic to C (a o p ) .
a E r
E
'
W ' a
s,
a E r
th e Galois module
E
W
.
is
a -
[ For the definition of the " twiste d " module C (a o p )
s e e A2 , Remark 3 . J P r oof.
The as s e r tions (a) and (b) are c ons equenc e s
known fact that C p r oj e ction s C
�Q
�Q
P
p
of
the we ll-
E is a pr oduct of d c opie s of C ,
the
E � C be ing given b y the elements of r E .
For (c ) note that the s ame de c ompo s ition holds for V
K
=
K � Q
V,
s inc e
P
henc e for e ach
a E r
E
K
c ontains all the Q - c onj ugate s of E ; p
' the r e exists a
W
E W
a
c onta ine d in
V
K
.
AB E LIAN 1. - AD IC R E PRE S E NT A T IO NS
III - 4 4 =
F or s uch a w , s ay w s (w)
� k.
1
=
� k.
=
� k 1.
1
= a
= a
and this im.plie s that W
�
p (s ) 0
a
09
�
(k . E 1
y.
1
K , y . E V ) we have 1
s (y 1. ) p (s ) y .
1
w s ince w belong s to W
P (s )w
a
i s i s om.orphic to C (a o p ) .
If P I and P ar e two characte r s of G into K 2
t"
then we
if P and P == P I 2 c o inc ide on an open s ubgr oup of l 2 the iner tia g r oup of G .
shall wr ite P
THEOREM 2 - Let P , V , W b e as ab ove and , for each a E r E ' let
n
a
be an inte g e r . The following ar e equivalent : P =- IT
(i) n
(ii )
-
1
n
a oX E a
a
for all a E r E for eve r y a E r E the Galois -m.odule W a o p "'" X
(iii) n
a E rE
a
to C (X a ) .
a
is is om.orphic
[Re c all that X is the char acter define d in chap. I, 1 . 2 , and that X E is the one attache d to the s ubfield a E of K, as in A4 . a Note that, s ince X E r e s tr icted to the ine r tia gr oup depends only
on a E ,
a
(i) is m.e aningfu1 . ]
LO CALLY ALGEBRAIC R EPRESE NTAT IO NS
III - 45
CORO LLAR Y - V is of Hodg e - Tate type if and only if the r e exi s t n
a
E
Z
s uch that p _ - IT a a E rE
-1
'X
na
aE
This follows fr om (i ii ) and the fac t that W s um of the W ' s . a
=
C
�
V is the dire c t
Pr oof of The orem 2 We p r ove fir s t : LEMMA 2 - (a)
X
E
(b ) If
Pr oof.
Let
7r
-
X
a E rE
is not the inc lus ion map ,
b e a uniformizing par amete r of E , let F 7r ,
Lub in - Tate g r oup a s s oc iate d to E and module , and
aX
V 7r =
T
7r
�
Q
P
V 7r
S ince
7r
E
-
1.
b e the
let T 7r b e its Tate
is a one - dimens ional ve ctor X
: G � U E (d . A4 , P r op . 4 ) , the ab ove c ons tructions apply to V 7r and X . By a E the orem of Tate ( [ 3 9] . § 4 , C o r . 2 to Th. 3 ) , W = C � Q V has 7r 7r p a Hodg e - Tate de c ompos ition of the type s pac e over E , and G acts on V 7r thr ough
W 7r whe r e dim W
7r
(0)
=
= W (0) 7r
d - l , dim W ( 1 ) 7r
=
fine s c anonic al is omorphisms W 7r (0 )
ED
E
W (1 ) 7r
1 . Mor e pr e c is ely, T ate de =
C Ci> K Hom (t ' , K ) , whe re t ' E
AB E LIAN 1 - ADIC REPRES E NTAT IO NS
III - 4 6
i s the W (l ) = 7r
F (d - l) - dimen s ional tang e nt s pac e of the dual of 7r (C � Q V (f.L ) ) riJ t , whe r e t i s the one - dime n s i onal K P P
tang ent space to F , and i s th e
Q
7r
p
V (f.L )
P
-vec tor s pac e of dimens ion
1 define d in Chap . I, 1 . 2 .
Note that C
�Q
p
V (f.L)
p
is . i s omorphic t o C (X ) , henc e one gets an
i s omorph i s m
The s e i s omorphisms c ommute with t h e action o f E . S ince E acts on t by the inc lus ion map a E � K , this l: of W is W (I ) . Henc e , us ing show s that th e c omponent (W ) 7r
Lemma I , we have C (X )
Clj,
7r
7r
�
C (X E ) ' and this implie s X E -- X , whenc e (a) . On the o ther han d , the s ame ar gum ent shows that (W ) 7r a
•
a 1= a
'
ar e c ontained in the oth er fac tor W {O J of W
l hen c e C { a o x ) -- C {l) , E
7r
7r
{whe r e 1 s tand s , of c our s e , for the unit
charact e r } , and this prove s (b ) . W e now g o back t o the proof of The orem 2 .
The equivalenc e
of (ii) and (iii) follows from Lemma 1 . T o show (i) � (ii ) , note * -1 take s value s henc e a o X fir s t that X take s value s in aE in E
p =
1
*
aE
and the s ame is true for the characte r
aE
LO C A L L Y A L G E B R A I C R E PR ES E N T A T IO NS
Let
T E L
E
.
We have =
To a
T 1= a
IT
aE I
Tc a
-1
n
oX
E
a aE
2,
applie d to the field aE , we s e e that -1 - 1 l' f T o a is not the identity o n aE , L e . if K j T h e mo d u l ar lnvarlant · 3 3 of E i s •
E L LIP TIC C UR YES
IV-3
if
Two elliptic curve s have the same j inva riant if and only they be come i somo r phic ove r the alg e braic clo sur e of K.
(All thi s r emains valid o ve r an ar bitrary field, exc ept that, when the characteri stic is 2 o r 3, the equation of E ha s to be written in the mo re general for m y
2
He r e again, 0 i s the point at infinity on the y - axi s and the cor r e s ponding tangent i s the line a t infinity . T he r e a r e corre sponding definitions fo r A and j , fo r whi c h w e r efe r t o Deuring [ 9] o r Ogg A:
[ 2 0] ; not e , howe ver , that the r e is a mi s p r int i n Ogg ' s fo rmula for 3 the coefficient of 13 sho uld be 8 instead of - 1 . ) 4 -
1 . 2 . Good r e duc tion Le t v e l; K be a plac e of the numbe r field K. We denote by ( r e sp. m , k ) the cor r e sponding lo cal r ing in K ( r e sp . its o -v v v maximal ideal , i t s r e sidue field) . Let E be an elliptic curve over K . One say s that E ha s good r e duction at v if one c an find a coo r dinate system in P 2 / K such that the cor r e sponding e quation f for E ha s coefficient in
0
v hence and its r e duction f mo d m de fine s a non- singular cubic E ( -v v an elliptic curve ) o ver the r e sidue field k ( in o the r words , the v di s c r iminant � f) of f must be an inve rti ble element of 0 ) . The v -
-
IV - 4
A BE LIAN l - ADIC R E PR ES E N T A T IO NS
curve E
i s called the r e ductio n o f E at v; it do e s not depend on v the cho i c e of f, provide d , o f c our se, that A ( f) = 0':' . v One can p rove that the a bove definition i s e q uivalent to the following one : ther e i s an a belian s c heme E the s en s e of Mumford [ 1 9 ] , c ha p .
VI,
over Spec(O ) , in v v who s e generic fiber i s E ; thi s --
s cheme i s the n unique , and i t s spe cial fibe r i s E . Note that E v i s v defined over the finite field k ; w e denote it s Fr obeniu s endo v morphi sm by F . v On either definition, one s e e s that E ha s good reduction fo r almo st all pla c e s of K . If E ha s good r e duction at a given plac e v, its j invariant
mv and its re duc tion J mo d -i s the j invar iant of the r e duc e d cur ve E . v The conve r s e i s almo st tru e , but not quite : if j belo ng s to
i s integral at v (i. e. be long s to -
0 ) v
0 ,
the re is a finite extens ion L of K such that E XK L ha s good r e duction at all the place s of L dividing v ( thi s i s the " potential v
good re duction" of S er r e - Tate [ 3 2 ] , § 2 ) . For the proof of thi s , s e e Deur ing [ 2 9 ] , § 4 , nO 3 . Remark The definitions and r e sults of thi s s e ction have nothing to do with numbe r field s . They apply to every field with a di s cr ete valuation. 1. 3 . Prope rtie s of Let the Galois
I
VI
r elated to goo d r eduction
T1
and VI by :
be a prime number . We define , a s in chap . I, 1. 2 ,
mo dul e s
ELLIPTIC C UR VES · whe r e E
in
I V- 5
i s the ker nel of i
n
E(K )
�
E(K ) .
W e deno te by Pi the co r r e sponding homomo rphism of G a l(K / K) into Aut( T i ) · Re call that E , T and V are of rank 2 1 i 1n n o ve r Z I 1 Z , Z and Q l ' r e spe c tively . 1 Let now v be a plac e of K , with p I: i and let v be some v extension of v to K ; let D ( r e sp . I) be the cor r e sponding de com po sition gr oup ( r e sp . ine r tia gr oup) , d. chap . I , 2 . 1 . I f E ha s good r e duction a t v , one ea sily s e e s that r e duction at v define s an i somorphi sm of E onto the cor r e sponding module fo r the r educ e d n 1 curve E . In parti cular , a r e unramified at v ( chap . v o f T corre spond s v, P1 1 to the Frobenius endomor phi sm F of E . Hence : v v
I , 2 . 1) and the Frobenius automorphi sm F
det( Fv and de t( l - F
v, P
1
)
' P1
) = de t( F ) = Nv v
= det( l - F ) = 1 - T r ( F ) + Nv
v
v
.....
i s equal to the nwnber of kv -points o f E v . C onye r sely : ,
"
"
CRIT ERION OF NERON - OGG -SAFARE VIC . If V1 i s unr amifi e d then E ha s goo at v for s ome 1 I: p v , -- d r e duction at v. Fo r the proof, s e e Se r r e -T ate [ 3 2 ] , § l . COROLLAR Y - Let E and E I b e two e lliptic curve s whi ch are i sogenous ( over K) . If one of them ha s goo d r e duction at a place v, the same i s true fo r the othe r one .
IV- 6
ABE LIAN 1 -ADIC REPRESENTATIONS
(Re call that E and E ' ar e said to be i so genous if the r e exi s t s a no n - trivial morphi s m E � E ' . )
T hi s follow s fr om the theorem, since the 1 - adi c r e p r e senta
tions a s s o c iated with E and E ' a r e i s o mo rphi c . R emark Fo r a dir e c t proof of thi s cor olla ry , s e e Koi zumi - Shimura [11] . Exe r ci s e Let S b e the finite s et o f pla c e s whe r e E doe s not have good r e duction.
If
v e E K - S , we denote by t the numbe r of k -point s v v of the r e duced curve E . v ( a) Let 1 be a prime number and let m be a po sitive -
integer . Show that the following prop e r tie s ar e equi valent : m for all v e E K - S , p I:. 1 . ( i) t v !! 0 mod 1 v ( ii) The set of v e E - S such that t ;: 0 mod 1 m ha s K v density one ( d. chap. I, 2 . 2 ) . m (iii) For doll s e Irn( P l ) ' one ha s de t(l- s ) a 0 mod 1 ( The e quivalenc e of ( i i) and (iii) follow s from C e botar e v' s •
density the orem. The implications ( i) e a sy . )
�
( ii) and ( iii)
�
( i) are
( b) W e take now m = 1 . Show that the prope r tie s ( i ) , (ii) , ( iii ) ar e equival ent to : (iv) The r e exi st s an e lliptic cur ve E ' o ve r K such that: E'
�
( a. )
Eithe r E' is i somo rphic to E, or the r e exi st an i s o ge ny
E of degree 1 . ( 13 ) The gr oup E ' (K) contains an element of order 1 . ( The implication ( i v) � ( iii) i s e a s y . Fo r t h e proof of the
conve r s e , us e Exe r . 2 of chap .
I, L L )
....,..
[ fo r m > 2 , s e e K a t z [ 6 4 J . J
ELLIPTIC C UR YES 1. 4 .
I V- 7
.,
Safa r e vi c., ' s theo r eIT1 It i s the following ( d . [ 2 3 J ) : -
Let S be a finite set of pla ce s of K . The s e t of i so IT1orphi sIT1 cla s s e s o f elliptic curve s ove r K , with good r e duction at all plac e s not in S , i s finite . T HEOREM
Sinc e i so genou s curve s have the saIT1e bad r e duction s et ( d . 1 . 3 ) , thi s iIT1plie s :
COROLLAR Y - Let E b e an elliptic cur ve ove r K.
Then, up to
i s oIT1orphi s IT1 , the r e a r e o nl y a finite nUIT1be r o f e lli p tic curve s which are K - i sogenou s to
E.
T o prove the theoreIT1, w e u s e the following crite r ion for good r e duction: LE MMA
-
of
place s of K c ontaining the divi sor s of 2 and 3 , and s uch that the ring O s of S - intege r s i s Let S be a finite set
p r inci pal. Then, an e lli p tic curve E defined over K ha s good r e duction out side S if and onl y if it s equation can be put in the Z 3 W eier stra s s for IT1 y = 4x - g z x - g 3 with gi 6 Os and * 2 3 � = g - 2 7 g e O s (the group o f unit s of O S ) . 3 2 Proof. The sufficiency i s trivial . To pr ove ne c e s sity, w e wr ite the curve E in the forIT1 Y e
2
= 4x
3
- g' x - g' 2 3
( *)
K . Le t v be a place of K not in S. Then, since the r e i s good reduction at v, and since the divi sor s of 2 and 3 do not be long
with g',
1
I V- 8
to
A B E LIAN l - A DI C RE PRES E N T A T IONS
the curve E c a n be written in the fo rm
S,
with g .
�
a unit in v thi s ring . U s ing the p r op e r tie s of the W eie r stra s s fo rm, the r e i s an .." 4 6 12. element u e K "O such that g 2. , v = u v g 2.' , g 3 , v = uvg 3' , �v = u �' ; v moreove r , a s we can take g . = g � fo r almo s t all v , w e s e e that 1, v 1 we can a s sume that u = 1 for almo s t all v � S . Sinc e the r ing O s v t,: i s princ ipal , the r e i s an e lement u Ei K with v(u) = v( u ) fo r all v -2. v f: S . Then, if we repla c e x by u x and y by u 3 y in ( �, , the 1,
v
in the lo cal ring at v and the di s c r imina nt
)
curve
E
take s the for m
6 12. 4 with g 2. = u g z ' g = u g 3 and � = u �I . Sinc e , by cons truction, 3 g e aS and � e O� , the lemma i s e stabli she d . i Proof o f the theorem . After po s sibly adding a finite numbe r of pla c e s o f K t o S , we may a s s ume that S contains all the divi s o r s of 2. and i s p r in cipal . 1£ E i s an elliptic cur ve de S fined ove r K ha ving good r e duction out s ide S , the above lemma 3 , and that the ring a
tell s us that w e can wr ite
E
in the fo rm
. * 3 2. with g i e a and � = g - 2. 7 g 3 e aS · But , s inc e we a r e fr e e to S 2. to< 12. to< 12. * i s a finite and since 0s l ( aS ) multiply � by any u E ( a S )
g r oup , we s e e that the re i s a finite s e t X C O � such that any elliptic
E L L I P T I C C UR YES
IV-9
c u r ve o f the a bo v e t yp e c a n b e w r i tt e n i n the fo r m ( ':' ) with g and
� e X.
B ut , fo r a g i v e n �,
i
the e quat i o n
r e p r e s e nt s an affi n e e lliptic cur ve .
e
Os
U sing a the o r em of Siegel ( g e n
e r ali z e d b y Mahl er and Lang , d . Lang [14] , chap . VII) , o n e s e e s that
thi s equation ha s only a finite number of solutions in O S ' Thi s fini she s the proof o f the the o r e m . R e ma r k The r e ar e many way s in which one can de duc e S afa r e vi c ' s the o r em fr om Siegel ' s .
The one we followe d ha s be en shown to us by
Tate .
§ 2 . THE GALOIS MODULES A T TAC HED T O E In thi s se ction, E denote s an elliptic curve ove r K. We a r e inte r e s ted i n the structur e of the Galois modul e s E de fine d in 1 .
3.
1n
, T l ' V1
2 . 1 . The i r r e duc ibility theor em Re call fi r st that the r ing End ( E ) of K - endomorphi s m s of E K i s eit he r Z o r of r ank 2 ove r Z . In the fir st ca s e , we say that E ha s " no compl ex multipli cation ove r K . " If the sam e i s true fo r any finite exten sion of K, we s ay that E ha s " no complex multiplica tion. " THEOREM - A s sume that E ha s no complex multipli cation ove r K .
A B E LIAN 1 - A DI C R E PR ES E N T A T IO NS
I V-IO
T he n :
( a ) V 1 i s i r r educ i bl e fo r all p r im e s 1 ; ( b) E i s i r r e du c i bl e fo r almo st all p r i m e s 1 . 1 W e ne e d the fo l lowing e l ementar y r e suIt: LEMMA - Let
E
be an e lliptic curve defined ov.e r K with
End ( E ) = Z . Then , if E I � E , E " � E � K - i s o genie s with K non- i somorphic cyclic ke rne l s , the curve s E I and E " a r e non i somorphi c o ver K. Proof. Let n l and n" be r e s p e c tive ly the o r der s of the ke r ne l s of EI
�
E and E "
�
over K, and let EI
E. Suppo se that E I and E " ar e i s omorphic
-->-
E " be an i s omo rphi s m . If E
transpo s e of the i s o g e n y EI nl , and hence the i so g eny
�
�
E I i s the
E , it ha s a c yclic ke r nel o f order
E �
E, obtained by compo sition o f
E � E I , E I � E " , E " � E , ha s fo r ke rnel a n extens ion of Z / n" Z by Z / nl Z. But, sinc e End ( E ) = Z , thi s i s ogeny mus t be K multiplication by an integer a , and its ke rnel mu st the r e fore be of the form Z / a Z X Z / a Z . He nce nl and n" divide a . Since 2 a = n l n " , we o btain a = nl = n " , a contra di c tion. Proof of the the o r em. ( a) It s uffi c e s to show that, if End ( E ) = Z , ther e i s no one K dime nsional Q sub spac e of V stable unde r Gal( K / K) . Suppo s e 1 1. the r e w e r e one ; i t s inte r s e ction X with T 1 would be a s ubmodule -
with X and T / X fr e e Z -modul e s of rank 1 . For n � 0 , 1 l n consider the image X(n) o f X in E = T / l T . Thi s i s a n l n submodul e of E whi ch i s cyclic of order 1 and sta ble by n 1 Gal( K / K ) . Henc e it co rr e spond s to a finite K - algebrai c subg roup of of T
E L LIPTIC C U R YES
I V - ll
E and one can d e fi n e the q uo ti e nt c ur ve E ( n) = E / X( n) . T he ke r n e l o f t h e i s o g e ny E � E ( n ) i s c y c l i c o f o r de r I n . T h e abo ve l e mma then s ho w s that the c u r ve s E ( n ) , n � 0 , v
.,
a r e p a i r wi s e non- i s omo r p hi c,
c o ntr adi c t i ng the c o r o l la r y to Sa fa r e vi c ' s th eo r e m ( 1 . 4 ) .
( b) If E i s no t i r r e duc i bl e , the r e e xi s t s a Gal o i s submodule l of E whi ch is o ne - dime n s ional o ve r F l . I n the same way a s X l a b o ve , thi s d e fi n e s. a n i s o g en y E � E / X who s e ke r nel i s c y c li c of 1 o r de r 1 . The above lemma show s that the cur ve s which co r r e s p ond to d i ff e r e nt
value s o f "
1
a r e no n - i s o m o r phi c ,
.,
and
one
again
applie s
the c o r o ll a r y to Safa r evi c 1 s the o r e m .
Rem a r k One can prove par t ( a ) o f the a bo ve the o r em by a quite diffe r ent method ( d . [ 2 5 ] , § 3 . 4 ) ; instea d of the S afarevic ' s theorem, o ne u s e s the proper tie s of the decomposition and ine rtia s ubg roup s of
Im(p ) ' 1
d.
Appendix.
2 . 2 . Determination o f the Lie algebra of G 1 Let G = Im(p 1 ) denote the ima g e of G al( K / K) in Aut( T 1 ) , 1. and l e t " C End( V ) be the Li e a l g e b r a of G " 1 1 .f:L1 THEOREM - If E ha s no complex multipli cation & 1
=
(d.
2 . 1) , then
End( V ) , i. e . G is open in Aut( T 1 ) · 1 1
Proof. The i r r e ducibility the or em of 2 . 1 s how s that, fo r any open subgroup U of G , V1 is an i. r r e ducibl e U - modul e . He nc e , V 1 1 i s an i r r edu c i bl e .& - mo dule . B y Schur ' s lemma , it follow s that the 1 commuting algebra &1 of & in End( V 1 ) i s a field ; since 1 dim V = 2 , thi s fi eld i s eithe r Q o r a quadratic extension of Q 1 " 1 1 then &1 i s equal to e ithe r End( V 1 ) , o r the subalg ebra If " I = Q l' .f:L1
I V - 12
A B E LIA N 1 - A DI C R E PR ES EN T A TIONS
S l ( V ) of End( V ) c o n s i sting of the endo m o r phi s m s with tr a c e 0 ; 1. 1. 2 but , in the s e c on d c a s e , the a c ti o n of .& o n A V would be trivial, 1. 1.
2
and thi s woul d c o nt r adi ct the fa c t tha t the Galo i s modu.l e s AV V ( /-I ) a r e i s om o r p hi c ( chap . I , 1 . 2 ) . 1. impo s s i bl e .
Henc e '&
1
=
1
a nd
s l ( V1 ) i s
Sup po s e now that '&1 i s a q uad r a t i c ext e n sion o f Q ' 1
Then
is a one - dimen s ional '&l - ve cto r spa c e and the commuting 1. a l g e br a of '&1 in E nd( V ) i s '&1 it s elf. Henc e '& i s containe d in 1 V
1 '& 1 ' and is abelian ('&1 i s a "non- split Cartan algebr a " of End( V 1 » ' Afte r r e pla cing K by a fini t e e xt e ns i o n ( thi s doe s not affe ct '& ' 1 d . c hap .
I , L l) , we may the n s uppo s e that G
itself i s abe lian . The
i s then s e mi - simple , abelian and rational. 1 i s , mo r e ove r , lo cally algebraic . T o s e e thi s , we fir st r emark that ,
1. - adic r epre s entation V It
1.
a t a plac e v dividing 1 ,
w e have v(j ) � 0 sinc e otherwi s e the de
composition group o f v in G 1 would be non - abe lian by Tate ' s the o r y ( d . Appendix, A . 1 . 3 ) ; henc e , aft e r a finite extension of K, we can a s s um e that E ha s g o o d r e du ctio n at all plac e s v dividing 1 ( d. 1 . 2 ) .
Let E ( l ) be the 1 - divi sible group attached to E at v :::: V1 ( E ( 1 » and thi s ( d . Tate [ 3 9] , 2 . 1 , example (a » . W e have V 1 modul e i s known to be of Hodge - Tate typ e (loc . cit. , § 4 ) . U s ing anothe r re s ult of Tate ( chap .
I ll ,
1. 2 ) , thi s implie s that the r ep r e
s e ntatio n V 1 i s lo cally alge braic , a s c laime d above . ( Thi s could al so be s e e n by using, in stead of the the o r y of Ho dge - Tate module s , the l o c a l r e sults of the Appendix, A2 . )
W e may now apply to V1 the r e s ults of chap . III, 2 . 3 . Henc e , the re i s , fo r each prime 1 ' , a r ational , abelian, s emi s imple l ' - adi c r epr e s entation W l ' compatible with Vl ' But Vl '
i s compatible with V ' and V 1 , i s s emi- s imple . Hence V , i s 1 1. i s om o rphi c t o W l ' ( d . c hap . I , 2 . 3 ) . But we know ( c h a p . Ill , 2 . 3 )
I V - 13
E L LI P T I C C U R YES
that we ma y c ho o s e l '
s u c h tha t W
l'
i s the d i r e c t s um of o ne -
dim e n s i o nal s ub s pa c e s s t a b l e unde r Ga l ( K / K ) . ir r e duc i bi l ity of V , 1 .& = E n d ( V ) , q . e . d . 1 1
T hi s c o nt r adi c t s the
H e nc e , we mu st have .& ' = Q and 1 1
Remark End( E X K) K i s the c o r r e sponding ima gina ry quadratic fi eld, one show s ea sily that If E ha s complex multiplication, and L
=
Q
�
"&1 i s the Car tan s ubalgebra of End( V 1 ) defined by L l split s if and o nly if 1 de compo se s in L .
=
Q QP L . It 1
Exe r ci s e s ( In the se exe rci s e s , we a s sume E ha s no complex multipli cation. Let S b e the set of place s v e
.E
whe r e E ha s bad
- S , we denote by F v the Frobenius endomor K phi sm o f the r educ e d curve E ; if 1 f. p , w e ide ntify F to the v v v c o r r e sponding automo r phi sm of T . ) 1 1 ) Let H { X , Y ) be a polynomial in two inde terminate s X, Y r e ductio n. If v
e
.E
K
with coefficient s in a field of chara cte r i s tic z e r o . Let V
be the H s e t of tho s e V € .E - S for whi ch H( T r ( F ) , N v) = O . If H i s not K the zero polynomial , s ho w that V ha s density O . (Show that the s e t H of g e. GL( 2 , Z ) with H( T r ( g ) , det(g ) ) = 0 ha s Haar mea sur e zero . ) 1 2 ) The eigenvalue s of F may be identified with complex v numbe r s of the fo r m -1 +1. 1'" 2 - TV
(Nv) e
cf. chap . I , Appendix A . 2 . Show that the s et of v for which cp v i s 2 2 a given angle cp ha s den s ity z e r o . ( Show that T r ( F v ) = 4(Nv) co s cp
and then u s e the p r e c e ding exer ci s e . )
A B E LIAN 1 - ADIC RE PR ESENTAT IONS
I V- 14
3 ) Let L = Q( F ) b e the field gene rated by F v . By the v v p r e c e ding ex e r c i se , L i s qua dr ati c imaginary exc ept fo r a set of v v of den s ity O . (a) Let 1 be a fixed pr ime . Let C b e a semi-simple commutative a lge b ra of rank 2 . Let X c be the s et of element s 5 e Aut (V 1 ) that the su ba l ge b r a Q [ 5 1 of E nd( V ) generated by s is i somo r 1.
1.
Show that X c i s open in Aut( V J. ) ' and show that it has a p hi c to C . non - empty inter se ction with e very open subgroup o f Aut( V 1 ) , in par ticular , with G J. . ( b) Show that F v e X c if and only if the fi eld L v i s qua dr atic and L � Q is i s o morphic to C . v 1. ( c ) Let J. , . . . , 1 be di stinct prime numbe r s , and choo s e fo r n l each an algebra C . o f the type conside r e d in ( a) . Show that the set 1 of v fo r whi ch F e X c . fo r i = 1 , . . . , n ha s dens ity > O . v 1 ( Us e the fact that the image of Gal(K/ K) in any finite pr oduct of the Aut( V1 ) is ope n ; thi s is an e a s y con sequence of the the o r em proved above . ) ( d) Deduce that, fo r any finite s e t P of prime num be r s , the r e exist an infinity of v s uc h that L v i s ramifi e d at a l l 1. e P . In pa rti cular , the r e ar e an infinite number of di s tinct fie lds L v . 2 . 3 . The i sogeny the o rem THEOREM - Let E and E ' be elliptic cur ve s over K, let
J.
be a
pr ime numbe r and let V ( E ) and V ( E I ) be the co r r e s ponding 1 J. J. - adi c r epr e s entations of K . Suppo s e that the Galoi s modul e s V ( E ) and V ( E I ) a r e i s omorp hic and that the modular invariant j J. 1 of E ( d. 1 . 1 ) is not an integer of K. T hen E and E' are K i s ogenous .
E L LI P T I C C UR VES
I V - IS
W e ne ed the following r e sul t : PROPOSITION - Le t E and E' be e lliptic curve s over following c onditions ar e e quival ent:
K.
The
(a) ThE' Galo i s modul e s V1 (E) and V1 ( E ' ) are i s omo rphi c
fo r all 1 .
( b) The Galo i s module s V 1 (E) and V1 ( E ' ) ar e i s omo rphi c
fo r one 1 .
( c ) If Fv and F v' a r e the Frobe nius e s of the r e duced curve s E v -and E'v , we ha ve T r ( F v) = T r ( F v' ) fo r all v whe r e the r e i s good reduction. ( d) Fo r a set of place s o f K of density one we have T r ( F ) = T r ( F v' ) . v Clearly ( a ) implie s ( b) , and ( c ) implie s ( d) . The implication
( c ) follow s from the fa ct that T r ( F ) i s known when V1 i s v known. T o pr ove ( d) � ( a) one r e marks fir st that the repr e s enta tions o f Gal( K / K ) in V/ E) and V ( E' ) have the s ame tra c e , by (b)
==:I>
1
.,
Cebotar e v ' s den s ity the o r em ( chap . I , 2 . 2 ) . Mor eover ,
V1 ( E) (and
al so V 1 ( E ' ) ) i s s e mi - simple . Thi s i s c lear if E ha s no complex multipli cation over K since V i E ) is then i r r e ducible ( 2 . 1 ) ; if E
ha s compl ex multiplication, it follow s from the Remark in 2 . 2 . Since
V i E) and V/ E ' ) are s e mi - s imple and have the same trac e , they
ar e i s omo rphic . Remark s 1)
If E and E ' have go od re duc tion at v, let t -
be the number o f k -points of E ( r e sp . E ' ) . v v v fo rmula s ( cf. 1 . 3 ) :
( r e sp . t ' ) v v We ha ve the
A B E LIAN l - A DI C R E P R E S E N T A T IO NS
IV-16
t
v t v'
==
1 - Tr(F
v } + Nv
- T r(F' }
=
v
+
Nv
Henc e conditi on ( c ) ( r e sp . condition ( d ) ) i s equivalent to saying that tv
=
v' s
t v' fo r all v whe r e the r e i s good r e duction ( r e s p . fo r a s e t of o f d e n s ity o ne ) .
2 } If E and E ' a r e K - i s o genou s , it i s clear that co nditions
(a) , ( b ) , ( c ) , ( d) are s ati sfie d . Proof o f the the o r e m . In view o f R emar k 2 ) above , i t suffi ce s to show t h a t t h e e q u iv a l e n t c o n d it i o n s ( a ) , ( b ) , ( c ) , ( d ) i m p ly t h a t t h e e l l i p t i c
curve s E and E' are i s ogenous when the modular invar i ant j of E
i s no t an inte g e r of K . L e t v be a pla c e o f K such that and let p be the char a cte r i s tic If j '
=
that v(j ' ) � 0 .
j ( E ' ) , we
fir s t
the r e sidue field k . v s how that v( j ' ) i s al s o < of
v(
O.
j) < 0,
Suppo s e
T he n, afte r po s s i bly r eplac ing K b y a finite
exte n sion, we may a s s um e that E ' ha s good r e duction at v. v
Then, if L
(d. 1 . 3 ) ;
f.
p , the Galoi s -module
but V / E) i s ramifie d at
v:
V (E' ) L
th i s follow s either from
the c rite rion of N e r on - Ogg - S af a r e v i c ( 1 . 3 ) or
fr
of the ine r tia gr oup given in the Appendix, A. 1. fa ct that V ( E ) and V ( E ' ) a r e i somo rphi c .
L
i s unr amified at
om the dete rmination
3.
Thi s contradi c t s the
L
Let now q and q ' be the elements of K v whi ch corr e s pond to j and j ' in Tate ' s the o r y ( d. Appendix A . l . I ) , and let E and q E q , be the co r r e sponding elliptic cur ve s ( lo c . cit) . Since E and E q have the same modula r inva riant j , the r e i s a finite extens ion K ' of K wher e they be c om e i somo rphi c , and we can do the same v fo r E' and E q " H e nc e , the T at e modul e s T p ( E q ) and T p ( E q , )
b e come i s omo rphic ove r
K' .
B ut , i n thi s c a s e
the i s o g e ny
I V- 17
ELLIPTIC CUR YES
the o r em i s tr ue ( d . Appe ndix A . l . 4 ) , i . e . the curve s E q and E q l , hence also E and E I , a r e K I - i s ogenous . Howeve r , if two elliptic cur ve s are i s ogeno u s ove r some exte n sion of the ground field, they a r e i so genous ove r a finite extension of the g r ound fi eld. We may thus cho o s e a finite ext e n s ion L o f K and an L - i s o g eny f
:
E� L
�
E I X L . We will s how that f i s automatically defined K s ove r K. For thi s , it s uffi c e s to s how that f = f fo r all s e Gal( K / K) , or , e quivalently , that V( f) : V ( E ) � V ( E I ) p P _ commute s with the a ction of Galoi s . Howe ver , if G = Gal(K / L) i s L the open subgroup o f G = Gal(K / K ) whi ch c o r r e sponds to L, then
V( f) commute s with the a ction of G . It is then enough to show that L Hom ( V, VI ) = Hom { V, VI ) . But V and VI are i somo r phi c a s G G L
G-module s . Hence we ha ve to show that End thi s i s clearly true ; in fa ct, G and G
( V) = End { V) . But G G L ar e open in Aut{ V) by the
L theorem in s e ction 4 , and he nce thei r commuting algebra i s r educed to the homothetie s in each c a s e , i . e. End Thi s complete s the proof of the the or e m .
G
( V) L
=
End { V ) G
=
Q . P
Remark It i s very likely that the theorem is true without the hypothe sis that j i s not inte gral. T hi s could be proved (by Tate ' s method [3 8]) if the following gene ralizatio n o f S afarevi c! ' s theorem were tr ue : given a finite sub set S o f E K , the abelian varietie s ove r K , of dime nsion 2, with polarization of degr ee one , and good reduction outside S , are in finite number ( up to i somo r phi sm) . b e e n p r o v e d b y F a lti n g s , s e e [ 5 4 ] , [ 5 6 ] , [ 8 2 ] . ]
� [ th is h as
IV-1S
A B E LIA N £ - A DIC R E PRESENTAT I ONS
§ 3 . VAR IA T ION OF G £ AND G £ WI T H £ 3 . 1 . P r eliminar i e s W e ke ep the no tations o f the p r e c e ding parag r aphs . Fo r each p r ime num be r 1 , we denote by p £ the homomo rphi sm
defined by the action of Gal ( K / K ) on T £ ' The p l ' s define a homomo rphi sm p : Gal( K / K )
�
n Aut ( T 1 ) 1
,
whe r e the p r o duct i s take n o ve r the s e t of all prime numbe r s . and G 1 = Im( p 1 ) C A ut ( T 1 ) , so th that G 1 i s the image o f G unde r the 1 p r oj ection map . Let G 1. be the imag e of G in Aut( E ) = Aut(T 1 i T 1 ) :::- GL( 2 , F 1 ) . 1 1 1 Let G
=
Im(p)
e n Aut( T 1 ) 1
( 1) The image o f G E.Y de t : n Aut( T ) ---+ n 1 �: * ( 2 ) For . Z 1 almo = st all , det( and de t(G ) G ) . 1 = F 1. 1 1 LEMMA
-
z;
i s open.
__
-
1 . 2 ) that det( P 1 ) : Gal(K / K ) ---+ Z l i s the character X giving the actio n o f Gal(K / K) on 1 n -th r o ot s 1 of unity . Hence det(G) C n Z ; i s the Galo i s gr oup Gal(K / K) , c whe r e K = 0 K i s the extens ion of K gene r ated by all r oo t s o f c c unity . Since one know s that Gal(O 1 0 ) = n Z * ( d . fo r instanc e [1 3 ] , 1 c chap . IV) it follows that det( G) i s the open subg r oup of n Z ; c o r r e sponding to the fi eld K n Q , henc e ( 1 ) . A s se r tion ( 2 ) follow s c W e know
( d.
chap . I ,
,�
�
�
ELLIPTIC CUR YES from
(1)
I V- 19
and the defi nition of the p r oduct topol o gy .
A s sum e now that E ha s n o complex multi plication . W e know ( d . 2 . 2 ) that each G l i s open in Aut( T 1 ) . T hi s do e s no t a priori imply that G its elf i s ope n . Howe ve r : PROPOSITION - The following prope r ti e s a r e e quivalent: ( i ) G i s open in n Aut( T 1 ) . 1 ( i i) � 1 == Aut( T 1 ) fo r almo s t all 1 . ( iii) � 1 == Aut ( E 1 ) fo r almo s t all 1 . (iv) G 1 contains S L( E 1 ) fo r almo st all 1 . plication ( iv)
==:»
==:»
==:»
==:»
(iv) a r e trivial . Im ( i ) fo llow s from the following gr oup - theo r etical
The impli cations ( i)
( ii )
( iii)
r e sult, who se proof will be g iven in s e c tion 3 . 4 below : MAIN LEMMA - Le t G be a clo s e d s ubgroup of n GL( 2 , Z ) and let 1 ..... G 1 and G 1 denote it s image s in GL( 2 , Z l ) and GL( 2 , F 1 ) a s above . A s sume : (a ) G 1 i s open in GL( 2 , Z l ) fo r all 1 . i s open . ( b) The image of G � de t : n G L( 2 , Z 1 ) � n ( c ) G contains S L( 2 , F 1 ) fo r almo st all 1 . 1 Then G i s open in n GL( 2 , Z l ) '
z;
R emark Fo r each integ e r n � 1 , l e t ..... of o r de r dividing n, and let G be n :::: / / nZ). K ) � Aut(E ) GL( 2 , Z Gal( K n ( i ) above i s equival ent to
E
b e the g r oup o f point s of E(K) n the image o f the cano nical map One s e e s ea sily that prope r ty
( i ' ) The index of G n in Aut( E n ) i s bo unded.
IV- 2 0
3. 2.
A B E LIA N L -A DI C R E P R E S E N T A T I O NS
The ca s e o f a non - i nte g ral
j
T HEOREM - A s s um e that the modula r inva riant j o f E i s not an inte g e r of K . Then E e nj o y s the equi valent properti e s ( i ) , ( ii ) , (iii) , (iv) o f 3 . 1 . Sinc e
( 1 ) .
V
p
'
i. e. , (1)
E
q
==='I> ( 3 ) .
q
A
= q,
B
a s Gal( K / K ) -modul e s . ( E ' ) a r e i somo rphic q
and E
q
A
are
e ve r y me r omorphic function F I G invariant A by q i s i n va r iant und e r mul tip l i c a t i o n by q ;
hence the function field of E A
s u c h that
B ut
unde r m ul tipli c a t io n
q
1
It suffi c e s to show that E
i s ogenous ove r K .
E
A, B �
and E
A
q T r i via l .
q
i s contained in the function field of
a r e i s ogenou s .
I V- 3 5
ELLIPTIC C UR YES C hoo s e an i s omor phi sm
t> containe d in sp . W e have a priori thre e po s si bilitie s : •
( a) i = g_ = End( V ) . p t> t> ( b) i = g . i s a no n s plit Cartan subalgebra o f End(V ) . p t> t> ( c) � i s a Car tan subalgebra and sp = End( Vp ) . Howe ve r , � i s an ideal of � . He nce , ( c ) i s impo s s ibl e , and thi s
prove s the theorem . Remarks 1)
By a the o r em of Tate ( [3 9 ] , § 4 , cor .
1
to th o 4 ) , the alge bra
sp i s a Ca rtan s ubalg e br a of E nd( Vp ) if and only if E(p) ha s
" fo r mal complex multiplication, " i . e . if and only if the ring of endo
morphi sms of E(p) , o ve r a suita ble extens ion o f K , i s of rank 2 over Z . The r e exi s t elliptic c ur ve s without complex multiplication p (in the algebraic s e n s e ) who s e p - c omple tio n E ( p) have formal complex multipli cation.
Z ) Suppo s e that sp is a Cartan s ubalge bra o f End( V ) , and p l e t H = � n Aut( Vp ) be the c a r r e sponding Ca r tan subg roup of Aut( V ) . If N is the nor malize r o f H in Aut( V ) , then one know s p p that N / H i s cyclic of order 2 . Sinc e G e N , i t follows that G p p i s commutative if and only if G p C H. T he ca s e Gp C H corre spond s to the ca s e whe r e the for mal co mplex multipli cation of E (p) i s
I V-40
A B E LIA N l -A DlC R E PR E S E N T A T lONS
d e fi n e d o v e r K,
a n d the c a s e
G
rr p I-
H c o r r e s p-o n d s to the ca s e whe r e
thi s fo r ma l multipl i c a t i o n i s d e f in e d o ve r a qua d r a t i c e xte n s i o n of K . 3)
S up p o s e that G
fi e l d k i s fin it e .
L et
m ul t i p l i c a t i o n ( i . e .
o f E nd ( V ) ) .
p Ga l ( K / K ) o n
�
p
F
i s c o m mu ta t i ve , a nd t ha t t h e r e s i due
be
the
it se lf,
p
vi
e w e d a s a n a s s o c i a t i v e s u ba l g e b r a
d e n o t e s t h e g r o up o f uni t s o f F , F i s g i ve n by a ho mo m o r p hi s m
If U
V
qu a d r a t i c fie l d o f fo r m a l c o m p l ex
q>
G a l ( K I K ) --7 U
t h e a c ti o n o f
F
B y l o c a l c la s s fi e l d the o r y , w e ma y i d e n tify the ine r tia g r oup o f
ab Ga l ( K / K ) with t h e g r oup
U
K
o f uni t s of K .
Henc e the r e s t r i ction
CP I of
( b ' ) . Conver sely, if D is stable unde r �' i t s transfo rm s by G a r e in finite numbe r ; p a standa rd me an value ar gum ent then show s that the seque nc e ( * ) split s , hence (b ' ) ='l>(b) . The impli cation ( b) =='l> ( a ) ( the only non
trivial o ne ) follow s fro m the cor ollary to theo r em 2 of A . 2 . 3 . C on ve r s ely , if E ha s complex multiplication by an imaginar y quadratic field F , the group Gal( K / K ) a c t s on Vp thr ough F � Q ( s e e p chap . I I , 2 . 8 ) and thi s action i s thu s s em i - simple . Co n s e que ntly, the
I V- 44
A B E LIA N l -A DIC R E PR E S E N T A T IO NS
exact s e quenc e (a' )
=:l>
( 1,, )
split s ; thi s s hows tha t ( a ) =:l>
( b ' ) . Since (a)
=:l>
( b) , hence al s o that
( a l ) i s trivia l , the the o r e m is pr ove d .
If E ha s no c omplex mul tiplication, � is the Bor e l subalge b of End( V ) fo r me d b -y tho s e u e E nd( V ) - bra X P P s uch that u( X) C X ; the ine r tia al gebra � i s the subalgebra !.X o f C OR O L LAR y
1
-
b formed b y tho s e u X Le t Xx and
e
E nd( V ) s u c h that u ( V ) C P
p
x.
Xy
be the characte r s o f Gal( K / K ) defined by the one - dimensional module s X and y . Since k i s finite , Xy i s
o f infinite o rder . I f X i s the cha r a c t e r de fine d by the ac tion of Gal( K/ K ) on V ( IJ ) , the i somorphi s m s p X
�
y
- A2 V - V __
__
P
P
( IJ )
-1 s how that X X y = X · Hence the r e striction of X x and X X y to X X the ine r tia subgr oup of Gal(K/ K) a r e of infinite orde r . T hi s show s
fi r st that � i s either � o r a C artan subalgebra of � ; s inc e X X the s e c ond ca s e would imply ( b ' ) , i t i s impo s s ibl e , henc e � = �X .
Similarly, one s e e s fi r st that i
i s c ontained in r ' the n that its X action on X i s non tri vial ; sinc e it i s an ideal in � = � ' the s e X p r operti e s imply i = r ' X -p -p
Remark
T he above r e sult is g ive n in [ 2 5 ] . p. 2 4 5 , Th . 1 , but mi s stated: the algebra !. has be en wrongly defined a s fo rme d of X tho s e u such that u( X) = 0 (in stead of u( V ) C X) . p C OROLLAR Y 2 - If E ha s c omplex multiplication,
�
is a split
Cartan s ubalg ebra of End( Vp l . If D is a sup plementary sub s p ace
I V-45
E L LI P T I C C UR YE S
t o X s table unde r Gal( K / K) , the n X and D a r e the characte r i s ti c s ub spa c e s o f � and the inertia a lgebra � is the s ubalgebra of End( V p ) fo rm e d b y tho se u e End( V ) s uch that u( D) p The p r oof i s analo gou s to the one of C o r . s imple r ) .
1
=
0, u(X) C X.
( and in fac t
B IB LIOGRAPH Y
[1]
[2 ] [3 ]
E . ARTIN - Coll e c te d Pap e r s (e dited b y S . Lang and J . T ate ) , Addi s on - W e sley, 19 6 5 .
E . ARTIN and J . TATE - Cla s s field the o ry . Ha rvard, 1961 . M. ARTIN e t A. GROTHENDIECK - Cohomo10gie e ta1e d e s s ch e m a s . S e m . G e om . a1 g . ,
sur Yvette. [4 ] [5] [6 ] (7]
[8 ] [9]
[1 0 ]
W.
1. H.
E. S . , 196 3 / 6 4, Bure s
BURNSIDE - The Theo ry of Group s (Sec ond Edit. ) . Camb r idg e Uill v . P r e s s . 1911 .
J . CASSELS - Diophantine equations with special r efe re nc e to elliptic curve s . J . London Math. Soc 41. 19 6 6 . p . 193 - 2 9 1 . .. J . CASSELS and A. FROHLICH - Algeb raic Numbe r Theory . Ac ademic Pr e s s . 196 7 . N. CEBOTARE V - Die B e s timmung de r Dichtigkeit eine r Menge von P rimzah1en. we1che zu e ine r geg ebenen Substitutionskla s s e g ehl:>ren. Math. Annalen. 9 5 . 19 2 5 . p . 1 51 - 228 . C . CHEVALLE Y - Deux th e or�m e s d a r i thm e tiqu e . J . Math. Soc . Japan. 3. 19 51, p. 3 6 - 4 4 . M. DEURING - Die Typen d e r Multiplikato renring e ellipti sche r Funktione nk�rp e r . Abh. Math . Sem. Hamburg. 14. 1941, p. 197 -27 2 . J . IGUSA - Fibre sy stems of Jac obian va rietie s . 1 . Ame r . • •
'
J.
Ill.
o f Ma th s . , 7 8 , 1 9 5 6 , p . 1 7 1 - 1 9 9 ; II, i d . , p . 7 4 5 - 7 6 0 ; id . ,
81,
19 5 9 .
p.
453 -476 .
B -2
[ 11 ] [1 2 ] [1 3 ]
AB ELIA N l -ADI C R E PRES ENTATIONS
S . KOI Z U MI and G . SHIMU RA - On Sp e c ia l i z a t i o n s of Abe lian Var ietie s . Sc . Pape r s C ol I . G e n . E d . , U niv . T okyo . 9 . 19 5 9 . p . 18 7 - 211 .
o S . LANG - Abe lian va r i e tie s . Inte r s c . T ra c t s n 7 . New York. 19 5 7 . S . LANG - Alg e braic numbe r s . Addi s o n - W e sley. Ne w Yo rk.
[ 14 ]
1964 . o S . LANG - Diophantine Geometry . Inte r s c . T ract s n 11 .
[15 ]
S . LANG - Intr oduc tion to tran s c endental numbe r s . Addi s on -
[ 16 ] [1 7 ] [ 18 ]
New York. 1 96 2 . W e sley. New Y o rk, 1 9 6 6 . J . LUBIN - One parame ter formal Lie g r oup s ove r p -adic inte g e r ring s . Ann. of Maths . • 8 0. 19 64 . p. 46 4 -484 . J . LU BIN and J . TAT E - F o rmal c omplex multipl ication in lo cal field s . Ann. of Math s
.
•
8 1 , 1 9 6 5 . p . 3 8 0 - 38 7 .
G . D . MOST OW and T . TAMAGAWA - On the c ompac tne s s of a r ithmetic ally define d homo geneous spac e s . Ann. of Math s .
•
7 6 , 19 6 2 . p . 446 - 4 6 3 .
[1 9 ]
D. MUMF OR D - Geometric Inva r iant Theory . Erg ebni s s e
[ 2 0j
A. P . OGG - Abe lian curv e s of s mall c onduc tor . J ourn . fUr
[21]
T . ONO - A rithme tic of alg e b raic tori . Ann . of Math s
[22] [23]
der Ma th •
•
Bd. 3 4 . Sp r ing e r - Ve rlag . 196 5 .
die reine und ang . Math .
•
2 26 . 1 9 6 7 . p. 2 04 - 21 5 . .
•
74.
19 61 . p . 1 01 - 1 3 9 . ,
-
G . POL YA und G. S Z EGO - Aufgaben und Lehr sa.tze aus de r te Analy s i s . Band 1 . Sp r ing e r -V e rlag . 2 Aufl . . 19 5 4 .
1.
�AFAREVI � - Alg ebraic Number Field s . Proc . Int .
Cong r e s s . Stockholm . 1 9 6 2 , p . 16 3 -1 7 6 (A . M . S . T rans l . , S e r . 2 , vol . 31, p . 2 5 - 3 9 ) .
B-3
B I B L IOGRAP H Y
[24]
J . - P o SERRE - Sur l e s g r oup e s de c ongruenc e de s va riete s abelienne s . I z v . Aka d . Nauk . S . S . S . R . , 28 , 1964 , p. 3 -20.
[25]
J.
-Po
SERR E - G r oupe s d e L i e l -adique s attac he s aux
c ourbe s e lliptique s . ColI. Cle rmont -F e r rand, C . N . R . S . , 1964 , p . 239 -2 56 . SERRE - G r o up e s p - d iv i s i bl e s ( d ' apr e s S em . B ourbaki, 1 9 6 6 / 6 7 , exp o s e 318 .
Tate ) .
[26]
J. -Po
[27]
J . - P o SERRE - Sur l e s g roup e s de Galois attac he s aux
J.
g roup e s p - div i sibl e s . P r o c e e d . Conf . on Local F iel d s , Sp ring e r -Ve rlag, 1 9 6 7 , p . 113 -13l . [28]
J . - P o SERRE - L i e alg e b r a s and Lie g roup s . B enjamin, New Y ork, 196 5 .
[ 2 9]
J . - Po SERR E - C o rp s Locaux. He rmann, Paris , 196 2 .
[ 3 0]
J . - P o SERRE - Dependanc e d' exponentiell e s p -adique s . S e m . De l an g e - P i s o t - Poi t o u , 7 e anne e, 196 5/66 , e x p o s e
15.
[31]
J . - P o SERRE - R e sume de s c ou r s 1 96 5 /6 6 . Annuai r e du College de F ranc e , 1 9 6 6 -6 7 , p .
[ 3 2] [33]
49 -58 .
J . - P o SERRE and J . TATE - Good r e duc ti on of abelian v a r i etie s , A n n . o f M a t h . 8 8 ( 1 9 6 8 ) , p . 4 9 2 -5 1 7
G.
SHIMURA - A r ec ipr o c ity law in nons o1vab1 e extensions .
J ournal fUr die r e ine und ang . Ma th . , 2 21 , 1 9 6 6 , p . 2 0 9 220. [ 3 4]
G.
SHI MURA a n d Y .
T A NI Y AMA - C om p l e x
m ult i p l i c a t i o n of
abelian va rietie s and it s applic ation s to numbe r the ory . Publ . Math. So c . Japan, 6 , 1 9 6 1 .
[ 3 5]
Y . TANIYAMA
-
L fun c tions
of
fun c t i o n s of a b e l i a n v a r i e t i e s .
numbe r fields and z e ta J ou r n .
M a th .
Soc .
Japan,
B -4
[ 3 6]
A B E LIAN l -A DI C R E PR E S E N T A TIONS
J.
9, 19 5 7 , p. 3 3 0 - 3 6 6 . TATE Alg ebraic cycl e s and pol e s of zeta func tion s . _
P r o c . Purdue U niv . C o nf . , 1 9 6 3 , p . 9 3 -110, New York , [3 7]
J.
1965 . TAT E - On the conj e c tu r e of B i r c h and Swinne rton - Dy e r and a g eometric anal o g . S e m . B ourbaki, 1 9 6 5/6 6 , expo s e 306.
[38]
J . TAT E
- Endomo rphi s m s of Ab elian Varietie s ov e r finite
fields . Inven. math. , 2, 1 9 6 6 , p . 13 4 - 144 . [ 3 9]
J . TAT E
- p - divi s ible g roup s . Proc . C onf . on Lo cal Field s ,
Spr ing e r -Ve rlag, 1 9 6 7, p . 1 58- 1 83 . [40]
A.
W ElL
- Variete s abe1ienne s et c ourbe s al gebrique s .
He rmann, Par i s , 1948 . [41 ]
A.
W ElL
- On a c e rtain typ e of c ha rac te r s of the idMe -cla s s
g roup of an alg ebraic numbe r fiel d . P r oc . Int . Symp . T okyo -Nikko , 1 9 5 5 , [4 2 ]
A . W ElL
[4 3 ]
A.
p.
1 -7 .
- On the the ory of c omplex multiplication . Pr oc . Int . Symp . T okyo - Nikko , 1 9 5 5 , p . 9 - 2 2 . W ElL
-
AdM e s and alg e b r a ic g r oup s (Note s by M .
Demazur e and T . Ono ) . P r inc eton, Ins t . Adv. Study , 1961 . [44]
A.
W ElL
-
[ 4 5]
A.
W ElL
- U eb e r die B e stimmung Dirichl et s cher Reihen
B a s ic Numbe r T he o ry . Sp ring e r -Ve rlag , 1 9 6 7 .
dur ch Funktionalgleichunge n . Math . Annalen, 1 6 8 , 1 9 6 7 , p . 149 -15 6 . [46 ]
H.
W E YL
- Ub e r die Gl e ichve rte ilung von Zahlen mod . Ein s .
Ma th . Annal en, 7 7 , 1 914 , p . 3 1 3 - 3 5 2 .
SUPPLEMENTARY B IB LIOGRAPHY
[47] , S. BLOCH and K. KATO-p-adic etale cohom olog y, Pub!. Math. I.H.E . S . 63 (1986), p. 107-1 52. [48] F. BOGOMOLOV-S ur l 'algebricite des representations l-ad iques, C . R. Acad . Sci. Paris 290 (1980), p. 70 1 -703 . [49] H. CARAYOL-Sur les representations l-adiques associees aux formes modu laires de Hi l bert, Ann. Sci. E.N. S 19 ( 1 986), p. 409-468. .
[50] P. DELIGNE-Fonnes modulaires et representations l-adiques, Se m ina i rc Bourbaki 1 968/69, expose 355, Lecture Notes in M ath . 1 79 , p. 1 39- 1 86, Springer-Verlag, 1 971 .
[5 1 ] P. DELIGNE-Valeurs de fonctions L et periodes d ' in te gra le s , Proc. S y m p. Pure Math. 33 , A.M.S . ( 1 979), vol. 2, p. 3 1 3-346. [52] P. DELIGNE-Hodge cycles on abelian varieties, Lecture Notes in Ma th . 900, p. 9 - 1 00 , Sprin g er-Verla g , 1 982. [53] P. DELIGNE-Motifs et groupes de Taniyama, Lecture Notes in Math . 900, p. 26 1 -279, Springer-Verlag, 1 982. [54] G. FALTINGS-Endlichkeitssatze fUr abelsche Varietaten fiber Zah l korpcm, Invent Math. 73 ( 1 983), p . 349-366; Erratum , ib i d . 75 ( 1984), p. 38 1 .
[55] G. FALTINGS-p-adic Hodge t heory, Journal A.M.S. 1 (1 988), p. 255-299 . [56] G. FALTINGS, G. WOSTHOLZ e t aI-Rational Points, Vieweg, 1 984 .
[57] J M . FONTAINE-Groupes p-divisibles sur Ies corp s locaux, As te risque 4 748, S .M.F., 1 977. .-
[58] J.-M. FONTAINE-Modules galoisiens, modules filtres et anneaux de BarsoLli Tate, Asterisque 65 (1 979), p. 3-80. [59] J.-M. FONTAINE-Sur certains types de rep resentat ion s p-adiques du groupe de Galois d ' un corps local, construction d ' u n anneau de B arsot t i -Tate A n n . of Math. 1 1 5 ( 1 982), p . 529-577. ,
S u p p l e m e n tary B i b l i o g r a p h y
B-6
[60] 1.-M. FONTAINE-Formes differentiel les e t mod u les d e Tate d es varictcs abeliennes sur les corps locaux , Invent. M ath . 65 ( 1 982) , p. 379-409 . [6 1 ] 1.-M . FONTAINE-Representations p ad i q ues , Proc. Int. Congress 1 9 8 3 , vol . I , p. 475-486. -
[62] 1.-M. FONTAINE and W. :MESS ING-p-adic periods and p-a dic etale coho mology, Contemp. Math. 67 ( 1987), p. 1 79-207. [63] G . HENNIART-Representations /-adiques abeliennes, S emin ai re de Th cori e des Nombres 1 980/8 1 , Birkhau ser- Verlag 1 982, p. 1 07- 1 26. [64] N. KATZ-Galois properties of torsion points on abelian varieties, I n v en t . Math. 62 ( 1 98 1 ), p. 48 1 -502. [65] . R. P. LANGLANDS-Modular forms and /-adic representations, Lecture Notes in Math. 349, p. 36 1 -500, Springer-Verlag, 1 973 . [66] R. P. LANGLANDS-Automorphic representations, Shimura varieti es, and motives. Ein Mru-chen, Proc. Symp. Pure Math. 33, A.M.S . ( 1 979), vol . 2, p. 205-246. [67] D. MUMFORD-Families of abelian varieties, Proc. Symp. Pure Math . IX, A.M.S. 1 966, p. 347-35 1 . ' [68] M. OHTA-On /-adic rep rese n ta ti on s attached to automorphic forms, lap. 1 . Math. 8 ( 1 982) p . 1 -47. ' [69] K. RIBET-On /-adic representations attached to m od ul ar forms, Invent. M a Lil . 28 ( 1 975), p. 245-275 ; II, Glasgow Math. 1 . 27 ( 1 985), p. 1 85- 1 94. (70] K. RIBET-Galois action on division points of abelian v arieti es w i th many rea l multiplications, Amer. 1. Math. 98 ( 1 976), p. 75 1 -804. [7 1 ] K . RIBET-Galois representations attached to eigenforms with Neben typus, Lecture Notes in Math. 60 1 , p. 1 8-52, Springer-Verlag, 1977. [72] S. SEN-Lie algebras of Galois groups arising from Hodge-Tate m odules , A n n . of Math. 97 ( 1 973), p. 1 60- 1 70. (73] loP. SERRE-Une interpretation des congruences relatives a la fonction t de Ramanujan, Seminaire D.P.P. 1 967/68, n0 1 4 (=Oe.80). (74] 1.-P. SERRE-Facteurs locaux des fonctions zeta des varietes al gebri q u e s (definitions et conjectures), Sem inaire D.P.P. 1 969nO, n0 1 9 (=Oe. 8 7 ) .
[75] 1 . -P. SERRE-Sur les groupes de congruence des varietes abel iennes II, Izv. Akad. Nauk S . S . S .R. 3 5 ( 1 97 1), p . 73 1 -735 (=Oe.89) .
B-7
Supplementary Bibliography
[76] J.-P. SERRE-Proprietes galoisiennes des points d ' ordre fini des courbes elliptiques, Invent. Math. 15 ( 1 972), p. 259-33 1 (=Oe.94). [77] J.-P. SERRE-Congruences et formes modulaires (d'apres H.P.F. S w i n nerto n D y er) , Seminaire Bourbaki 1 97 1n2, n04 1 6 (=Oe.95). [78] J.-P. SERRE-Representations l-adiques, Kyoto Symp. on Algebraic Num ber Theory, 1 977, p. 1 77- 1 93 (=0e. 1 1 2).
[79] J.-P. SERRE-Groupes algebriques associes au x modules de Hodge-Ta te , Asterisque 65 ( 1 979), p . 1 55- 188 (=0e. 1 1 9). [80] J.-P. SERRE-Resume des cours de 1985-86, Annuaire du C o l l ege de France, 1 986, p. 95- 1 00. [8 1 ] H. P. F. SWINNE RTON-D YER-On l-adic rep rese n tati on s and c o n g r u en c e s for coefficients of modular forms, Lecture Notes in Math . 350, p. 1 -5 5 , S prin ger- Verlag, 1973; II. ibid. 60 1 , p . 63-90, Springer-Verlag. 1 977. [82] L. SZPIRO-Seminaire sur les pinceaux arithmetiques: la conjec ture de Mordell, Asterisque 1 27, S .M.F., 1 985. [83] M. WALDSCHMIDT Transcendance et exponentielles en plusieurs vari a b le s Invent Math. 63 (198 1), p. 97- 1 27. -
,
[84] A. WILES-On ordinary }..-adic representations associated to modular forms, preprint, Princeton, 1 987. [85] J.-P. WINTENBERGER-Groupes algebriques associes a certaines represenla tions p-adiques, Amer. J. Math. 108 ( 1986), p. 1425- 1466. [86] Y. G. ZARHIN-Abelian varieties, l-adic representations and SLl, MaLh . US S R lzv. 1 4 ( 1 9 80), p . 275-288.
[87] Y. G. ZARHIN-Abelian varieties, I-adic representations and Lie algebras. Rank independence on I, Invent Math. 55 ( 1 979), p. 165- 1 76. [88] Y. G. ZARHIN-Weights of simple Lie algebras in the c oho m ol og y of alge brai c varieties, Math. USSR Izv. 24 ( 1 985) , p. 245-28 1 .
INDEX
Admis s ib l e ( cha ra c t e r ) : A lmo s t loca l ly a lgebra i c Ani s o t ro pic ( to rus )
:
I I I .A . 2 . :
111 . 3 . 3 .
I I .A . 1 .
Arithmet i c ( subgroup ) :
I I .A . 1 .
As s oc i a t ed ( a l gebra i c morph i s m
•
a lgebra i c repres enta tion ) :
•
with a loca l ly
•
111 . 1 . 1 ,
111. 2. 1 .
Aut ( V ) : No ta tions . C, C
:
m
"
11 . 2 . 1 .
Cebo ta rev ' s theo rem
1 . 2. 2 .
Character g rou p ( o f a torus )
ll1 .A . 2
c ( ep ) : C
..
11 . 2. 1 .
•
K : 111 . 1 . 2 .
=
c
: l I I . A . 6 . Exer . 1 . K/ E Compa t ib le ( repres ent a t ions ) : Comp l ex mu l t i p li ca t ion :
1 . 2.3, 1 . 2. 4.
11. 2.8,
IV . 2 . 1 .
Conducto r ( o f a loca l ly a lgebra i c repres enta t ion ) : C oo , c l1l D :
:
11 . 3 . 1 .
ll . 2 . 1 .
Decompo s i t ion group : De f ined over
1. 2. 1 .
k ( repres enta tion
Dens ity ( o f a s et o f p la c es ) :
E
et
: 11 . 2 . 2 . :
II . 2 . 3 .
E l l i pt i c curve E
IP
:
IV . 1 . 3 .
IV . 1 .
1.
•
•
•
) :
1 . 2. 2.
11 . 2 . 4.
111. 2 . 2 .
I nd ex - 2
Em : I I . 2 . 1
•
: IV . A . 1 . 1 . q E q uid is tribu t ion : E
'"
I .A . 1 .
E : IV . 1 . 2 . v Ex c e p t i o na l s e t ( o f a s tr i c t ly compa t ib l e s y s t em ) � "...J � '
�t
: I11 .A . 2 .
: 11 . 2 . 5 .
F ' f : I!.2.3 v v Fro b enius e l ement : •
1. 2. 1 .
Fro benius endomo rph i s m :
IV . A . 3 .
rE
Gt
Gt
IV . 1 . 2 .
IV . 2 . 2 .
...
�t GLy
11. 2.8,
IV . 3 . 1 . ..
IV . 2 . 2 ,
IV . App
•
: No ta t i ons .
, Om :
II. 1. 1.
Good reduct ion ( o f an e l liptic curve ) :
Gro s s encha rakter o f type ( A ) : o He i ght : 1V . A . 2 . 2 . Hod ge- Ta t e d ecompo s i t ion Hod ge- Tate modu le :
1,
1
m
:
Id e le : :
111 . 1 . 2.
111 . 1 . 2 .
11 . 2 . 1 . II . 2 . 1 .
Id e l e c las s es
1t
:
IV . 1 . 2 .
11. 2. 7.
11. 2. 1 .
1V . A pp .
Inert ia group :
1.2. 1 .
Inte gra l ( repres enta t ion ) :
1. 2.3.
Is o geny , i s o genous curves
1V . 1 . 3 .
j : IV. 1 . 1
•
K, K s : No ta t io ns .
t-ad ic repres enta t ion ( o f a fi eld )
1.1 . 1 .
1.2.3.
Ind ex - 3
1.2.3.
A - ad ic repres enta t io n ( o f a fie ld ) La t t i c e :
1. 1 . 1 -
L- func t ion :
1. 2.5.
Loca l ly a l gebra ic ( repres entat ion ) :
111. 1 . 1 , 111 . 2 . 1 ,
111 . 2 . 4, 111 . 3 . 3 .
Modu lar invariant ( o f a n e l l i pt ic curve ) a
Modu lus ( o f
•
Ra tiona l ( repres enta t ion )
•
•
•
•
) :
1V. 1 . 3 .
: 1. 2 . 3 , 1 . 2 . 4. 1V . 1 . 2 .
: 11 . 2 . 4.
S a farevi c ( theo rem o f
•
•
•
) : 1V. 1 . 4 .
: 11. 2 . 2.
Stric t ly compa t i b l e
( sys t em o f repres entations )
1. 2. 4. Supp( m ) : 1 1 . 2 . 1 . Tate ' s e l l i pt ic curv es : 1V . A.1 1 1 . Ta t e ' s theorem : 1 1 1 . 1 . 2 , 1 1 1 . A . 7 .
g
: 1 1 . 2 . 4.
ql
T .r, ( \.l) : 1 . 1 . 2 . T : 11.2 . 2 . m Torus : 1 1 . 1 . 1 . Transve c t ion :
T
Um
111. 2 . 2 .
: 11 . 1 . 3 .
•
Reduct ion ( o f an e l l i p t i c curve ) :
Sm
IV . 1 . 1 .
loca l ly a lgebra ic repres enta t i o n ) :
Mu l t i p l i ca t ive type ( grou p o f ) S Neron- Ogg- a fa revi c ( c r i t erion o f
Re � ( H )
:
=
1V . 3 . 2 . 11. 1 . 1 .
� / Q ( Gm/ K ) :
, U
: 1 1. 2 . 1 . Uni formly d i s t ribu ted ( s equenc e ) : I . A . 1 . Unrami f i ed ( repres enta tion ) : 1 . 2 . 1 . V .r, ( 1-1) : I 1 2 . Weiers tra s s form ( o f an e l l i pt i c curve ) : 1V . 1 . 1 . v ,m
•
•
1.2.3,
1nd ex - 4
'XE
:
1II.A. 4.
'XL
:
X( T ) , X ( T ) m
11. 3. 1 .
Y,
yO , Y - , y+
:
I: K
1. 1 2. •
1. 2. 1 .
11. 3 . 1 , II.A . 2 .