SPECIAL
VALUES
OF HECKE
ABELIAN
G ,. Harder
L-FUNCTIONS
AND
INTEGRALS
and
N. S c h a p p a c h e r
Max-Planck-Institut
f~r M a t h e m a t i k
Gottfried-Claren-Str.
5300 Bonn
In this article
nality
we attempt
conjecture
in the p a r t i c u l a r
ters
"motive"
relates
certain
tion,
proved
analytic
equation.
braic Hecke
values
to algebraic
Hecke
In this
little
Hecke
"geometry"
good
command
number
perused
the s o - c a l l e d
formula
This
in fact,
field,
is
due to D. Blasius
Deligne's
conjecture
to certain p e r i o d s
come up in a g e o m e t r i c
situation
is quite
are among
complex
of the c o r r e s p o n d i n g
of
situa-
In the
different:
special
The L-
those
for w h i c h Hecke
plane
and f u n c t i o n a l
motives
we n o w have of the motives
that many n o n - t r i v i a l
of c h a r a c t e r - i d e n t i t i e s .
in
[Seh],
and we shall
and Selberg:
goes back
conjecture
really
charac-
the c o n j e c t u r e
their L-functions.
to the whole
of C h o w l a
in the m o t i v i c
both r e s u l t s
the
about
characters
reveals
in fact but r e f l e c t i o n s
of D e l i g n e ' s
[DI])
has e m e r g e d
- see §3 below.
characters
formula,
case
results,
of its L - f u n c t i o n
however,
recently
is sy s t e m a t i c a l l y
relations
attached
see §5 below.
continuation
But the
fairly
The r e l a t i v e l y
are
here,
of a l g e b r a i c
ratio-
(see
A0").
an a l g e b r a i c
special
of D e l i g n e ' s
L-functions
Most of the time when motives
case e n v i s a g e d
only
over
we tend to know very
functions
the f o r m a l i s m
of motivic
of two c o m p l e m e n t a r y
respectively:
For any
the motive.
of type
by virtue
and G. Harder,
values
case of L - f u n c t i o n s
("Gr~Bencharaktere
now a theorem
3
to e x p l a i n
for special
26
to the year
with w h i c h we start
formalism,
should be v i e w e d
d i p t y q u e '~, as A. Weil has p o i n t e d
but
we hope
"comme
in
attached
to alge-
period r e l a t i o n s
This point of v i e w
illustrate
it here by
see § 6.
1897,
as does the
in § I. Tying
to make
up these
it apparent
les deux volets
[WIII],
instance
p. 463.
two
that
d'un m~me
18
Contents
§ I
A
§ 2
Algebraic
§ 3
Motives
§ 4
Periods
§ 5
The
§ 6
A formula
§ 1o
In
formula
A
of
of
Hurwitz
(I)
all
~ =
the
conjecture
the
[Hu]
proved
1
(3)
rl
21_
u
Both
formulas
this
in H u r w i t z '
L-functions
that
4v
~
4v
×
(rational
number),
, where
£(
dx
-
of
2.62205755,..
these
zeta-function
E'
a£ ~
Hecke
(a+bi)
analogy
Riemann
for
Lerch
~ = 1,2,3,...
(2)
Notice
characters
Hurwitz
E'
a,b6~
for
Hurwitz
rationality
formula
1897,
of
Hecke
1
2~
-
=
identities
at p o s i t i v e
(2~i)
2v
with
even
the
well-known
formula
for
integers:
x (rational
number).
a
are
special
case,
we
cases
look
of D e l i g n e ' s
at t h e
conjecture.
elliptic
curve
A
To
understand
given
by
the
equation
A
A
the
A
is d e f i n e d
field
admits
over
~
k = ~(i)c
complex
: y2
= 4x 3 _ 4x
, but
~
.
we o f t e n
. Over
this
multiplication
by
prefer
field
of
the
same
to
look
at
definition,
field
k :
it as d e f i n e d
we
can
see
over
that
19
k
>
End
(A)
®
--X
i
I
>
I--->
D e l i g n e ' s a c c o u n t of Hurwitz'
that both sides of
formula would start from the o b s e r v a t i o n
(I) express
H I (A) ® 4 9
The left hand side of
iy
c
information about the h o m o l o g y
H4~
(A 4~)
(I) carries data c o l l e c t e d at the finite places of
k , as does the right hand side for the infinite places.
In fact,
look at the d i f f e r e n t c o h o m o l o g y theories:
- Etale cohomology:
n ~ 1 , by
A[£ n]
Fix a rational prime number ~ , and denote,
the group of
a l g e b r a i c closure of
V£(A)
~
in
n
~ - t o r s l o n points
for
being the
= I<lim A[£n])®ZZ£ ~£
n
A×k~
with coefficients
~£
By functoriality,
a
A(~), ~
~ . Then
is the dual of the first £-adic c o h o m o l o g y of
in
in
the i s o m o r p h i s m
k ~ ~ ® ~ End A
makes
Vi(A)
into
k ® ~£-module, free of rank I. The natural continuous action
of
Gal
(W/k)
on
V£(A)
is
k ® ~ -linear,
and therefore given by a
c o n t i n u o u s character
~£: Gal
(~/k)ab
- - >
GL k ® ~ £ ( V Z(A)) =
(k®~z)* .
This character was e s s e n t i a l l y d e t e r m i n e d
- if from a rather d i f f e r e n t
p o i n t of v i e w - on July 6, 1814 by Gauss,
[Ga] . The e x p l i c i t analysis of
the G a l o i s - a c t i o n on t o r s i o n - p o i n t s of
ly "modern"
fashion)
is a prime element of
~ I
(mod (I+i) 3)
A
in 1850 by Eisenstein,
~ [i]
, and if
not dividing
F
6
Gal
was carried out
(in a stunning-
[El]. - In any case,
2Z
(~/k)ab
,
if
n o r m a l i z e d so that
is a g e o m e t r i c Frobenius
20
element
at
dividing
(~)
(~),
~
The
(i.e.,
any
(F)
characters
ter"
=
~£
~ defined
on
I
k
for all
by
makes
6 k* c
F
to c o n s i d e r
,s)
over
of H u r w i t z ' f o r m u l a
V~(A)®k®~
- Betti
> Gal
(~/k)ab
the
of
kab
an " a l g e b r a i c
of
k
that
Hecke
are p r i m e
characto
2:
k*
~
~4~
that
> (k®~g)*
can be d e f i n e d
k
is e m b e d d e d
I
on all
into
(Re(s) > I-2~)
~ (~) 4~
I
~
ideals
of
, so t h a t
it
,
s
all p r i m e
ideals
is s i m p l y
value, of the L - f u n c t i o n
and de R h a m c o h o m o l o g y .
A
Denote
by
(if not its c o m p l e x
B
HI(A) = H I ( A ( ~ ) , ~ )
surface
A(~)
, with
B
H I (A) ®~ •
Complex
P
finds
L-functions
= ~
(I)
one
of
~ [i]
. Then
4 .L(~ 4~ ,0). We h a v e
afforded
the
shown
b y the l - a d i c
left
hand
side
how
this
is
cohomologies
4~
curve
Riemann
ideals
>
. Remember
for any p r i m e
then
( k ® Q£)*
to g i v e
of
~ Z I, the c h a r a c t e r
ranges
a special
>
~
p
p
12
(mod P )
x 6 kab),
fit t o g e t h e r
2
L(~ 4~
where
integer
group
(~) ~__> - 4 v
sense
-I
all
the
I2~
Then
F-1(x)~ ~ x ~ ~
algebraic
conjugation
F
on
: HI(A) ~
=
A×~
Here
we
shall
multiplication)
the
first
the H o d g e
H -I'0
the
rational
decomposition
an e n d o m o r p h i s m
"Frobenius
fact
at ~ " ) .
that
defined
singular
@ H 0'-I
induces
(the
use
is a l r e a d y
the
over
homology
~.
of the
21
Call HB
this
onedimensional
Let
B
the f i x e d p a r t of
logy of
G-vector
= H D1R (A)V
H R(A)
A
over
HI(A)
under
be a b a s i s
, and let
F
of
space.
be the dual of the f i r s t a l g e b r a i c
~ , g i v e n w i t h the H o d g e
de R h a m c o h o m o -
filtration
DR
+
H I (A) ~ F
D {0}
where
F + ®~
~ H 0'-1
u n d e r the G A G A
I : H~(A)
I induces
an i s o m o r p h i s m
i+
Then,
:
+
HB(A)
®~
~I . i +(n)
= f 1 ~dx
~*
,
Q
®~
(1
of
>
H I R (A)
onedimensional
6 H I R ( A ) / F + , for
~
is a r e a l f u n d a m e n t a l
is the d e t e r m i n a n t
calculated
over
¢:
®(~(E
~-vector
spaces
.... > (HIR(A)/F +) ®~
in t e r m s of R - r a t i o n a l
t h a t of the m a p
defined
by
(2) . In fact,
p e r i o d of our curve,
and so, up to
of the i n t e g r a t i o n - p a i r i n g
(HB(A) ®(~(~) x (H0(A,~ I) ®(~(Z)
equals
isomorphism
I+
since
S > (~
b a s e s of b o t h
spaces.
H 0 ( A , ~ I) c H I (A)
DR
This d e t e r m i n a n t
is the dual of
H ~ R (A) /F +
Passing
to tenser p o w e r s
the p e r i o d s
In a sense,
~4~
occuring
setup:
character
have considered
k
s p a c e s a b o v e we find
(I).
a little
in d e r i v i n g
to o b t a i n a o n e d i m e n s i o n a l
~ ). In the c a l c u l a t i o n
H~R(A/k).
the p e r i o d
~
f r o m the
situation
of the p e r i o d ,
(i.e.,
embeddings
and two c o p i e s of
of the b a s e
field
k
into
B
k
via
the
too, we s h o u l d
= H~R(A)j ®~ k , e n d o w e d w i t h the f u r t h e r
via c o m p l e x m u l t i p l i c a t i o n ,
the two p o s s i b l e
vector
In the ~ t a l e c a s e we h a v e u s e d the a c t i o n of
complex multiplication
of
in
we h a v e c h e a t e d
cohomological
k-valued
of the o n e d i m e n s i o n a l
HI(A),
action
i n d e x e d by
• .... B u t
in the
22
presence
ficial,
of an e l l i p t i c
and
As a final
fairly
tion
the g e n e r a l
remark
easily.
such
Any
I = ~
then
It is t h e s e
lattice
k
, this
will
(I),
would
be t r e a t e d
it s h o u l d
F = I . (ZZ + ~ i )
we g e t
numbers
and
g2(F)
- g2lF)
have
in
too a r t i -
§ 4.
be n o t e d
gives
seemed
that
it is p r o v e d
a WeierstraB
~func-
= 4. T h e
~(z,F),
rational
the c o e f f i c i e n t s
that H u r w i t z
Hecke
E
3
= 4~(z,r)
essentially
§ 2. A l g e b r a i c
Let
formula
~
that
for
(1) are
over
procedure
about
~' (z,F)
and
curve
numbers
of the
studied
in his
left
unspecified
z-expansion
of
papers.
Characters
be t o t a l l y
imaginary
number
fields
(of f i n i t e
degree
over
~) , and w r i t e
Z = Hom
the
sets
on
Z×T
of c o m p l e x
(k,~)
and
embeddings
, transitively
T = Hom
of
on e a c h
k
(E,~)
and
E. The
individual
factor.
group
8 : k*
induced
>
: Rk/~
This
means
is g i v e n
(4)
that,
for all
• oS
: k*
~oB
(x)
E*
by a r a t i o n a l
(~m)
• 6T,
character
> RE/~
(~m)-
the c o m p o s i t e
>
~*
by
= ~E-~
O(x)
n(~,~)
,
Gal(~/~)
An algebraic
phism
is a h o m o m o r p h i s m
in
~(z,F) .
acts
homomor-
23
for certain
p6 G a l
Let
, such that
,f ~,, > k ~
i.e.,
those
x 6k*,
let
be the topological
id~les
x
the finite
to
1.
An algebraic
whose
also
xf
id~le
Hecke
components
denote
obtained
character
~ , is a c o n t i n u o u s
s u c h that,
f o r all
for a l l
is t h e
group
at the
by changing
~
of
k
>
8
a subgroup
that
has
to k i l l
the
infinite
c : complex
+ n(co,T)
conjugation
T6 T , w e g e t a c o m p l e x
of the
Hecke
k~
components
in
,and
of
x
E , o f (infinity-)
of f i n i t e
character
index
= n(o,T)
Y , then,
of the units
of
by
k.
on
~)
+ n(o,cT)
is i n d e p e n d e n t
valued
Gr~Sencharakter
id~le-class-group:
*
k]A,f
ToY
..........
~.
>
klA
*
klA/k.
the array
I. F o r
of
~,T.
It is
of
to a q u a s i c h a r a c t e r
Consider
in
integer
w = n(o,T)
For any
are
idele
k -
= B (x)
It f o l l o w s
the weight
places
of
E*
continuity,
(where
the
with values
o f an a l g e b r a i c
called
infinite
id~les
homomorphism
infinity-type
(5)
of f i n i t e
x £ k*,
(xf)
~
= n(a,T)
the corresponding principal
: k*
~,f
If
n(~o,~)
(~/~) .
k~
type
n(o,T)
integers
ToY
of L - f u n c t i o n s ,
~.
>
indexed
by
T:
Toy w h i c h
extends
24
L
where,
for
(~,s)
=
(L(To~,S))~6 T
,
W
Re(s)>
~ + I ,
I
#
the
product
being
value
~(~p)
wp
of
kp
The
point
factor
pole
at
there
either
s =
it
is
does
s = 0
on
, for
0
out
a disjoint
words,
prime
depend
is c a l l e d
. This
turns
all
not
side
Z ×T
In o t h e r
over
of
ideals
on
the
critical
the
s =
of
choice
for
functional
is r e a l l y
that
p
~
, if
of
is c r i t i c a l
for
which
the
uniformizing
equation
a property
0
of
k
for
any
of
L(To~,s)
the
T
parameter
, no
infinity-type
for
~
if
and
F-
has
a
B
only
of
if
decomposition
:
{ (o,T)
for
every
n(o,T)
TET
<0}
0 { (d,T)
, there
is
a
I n(co,7)
"CM-type"
< 0}
~(TOB)C
Z
such
that
• (~ToB)
= ~(ToB) ~
,
for
~6 G a l ( ~ / ~ )
(6)
• O6~(~8)
For
~ such that
~(~)
:
s =0
(Q(~}''T))T6T
(7)
In
Y
¢¢ n ( d , T )
is c r i t i c a l
6 ({*)T
=
~(~)
other
words,
>{
he
conjectured
~
n(co,<)
Deligne
(E®~{)*
L(~,0)
: E r~
<0
6 E c
that
, and
->0
defined
array
of
periods
that
> E®{
there
is
x 6 E
,
L(To~,0)
an
conjectured
= T(X) . ~ ( ~ , T )
such
that
for
all
25
The
definition
motive
of
~(~)
to an a l g e b r a i c
is d i s c u s s e d
Hecke
in § 4. It r e q u i r e s
attaching
a
character.
§ 3. M o t i v e s
3.1
In t h e e x a m p l e
characters
~4m
rect
of
factor
illustrates
Starting
right
what
by taking
fairly
well
cycles"
mily
Here
- see
be
between
to
of w h a t
of
II.
A little
with
motives
In
isomorphic
such motives
field,
defined
this
when
and
we have
N
the
Just
weak
using
theory
be:
various
a fairly
and
"absolute
motives
their
more precisely,
M
should
between
di-
This
its c o h o m o l o g y .
the d i f f e r e n c e
version:
a certain
theories.
a motive
a number
be c o n c e r n e d
I and
be
HI (A) , i.e.,
cohomology
parts
for our Hecke
L - funcgiving
amounts
a homo-
to g i v i n g
a fa-
of homomorphisms
HO
(M) - - >
HDR(M)
H£
compatible
Hodge
H°
-->
w i t h all
H£
(N)
the n a t u r a l
comparison
(Betti cohomology depends on the choice of
o : k - - > ~ yielding M}--> Mxo~ )
(N)
HDR(N)
(M)-->
decomposition,
w i t h the
His
shall
coincide.
two
over
certain
a "motive"
of
idea
constitutes
we
[DMOS],
shown
and periods
morphism
variety
choose
half way manageable
often
powers
the g e n e r a l
means
of m o t i v e .
therefore
tions
tensor
in the v a r i o u s
f r o m an a l g e b r a i c
"consistenly"
Hodge
§ 1, we c o n s t r u c t e d
H4~(A4m),
to c o n s i s t e n t l y
notions
can
of
Hodge
(for all
structures
filtration,
isomorphisms
£ )
on these
cohomology
Gal(k/k)-action,
between
HB
and
HDR
groups:
as w e l l
, HB
as
a n d the
•
3.2
L e t us
braic
Hecke
A/~
(This
state
defines
of D e u r i n g ,
[Sh I]...)
plication
what
But
of
this
A
and
precisely
~
the m o t i v e
is r e a l l y
a result
more
character
should
HI (A)
what
be[
a motive
over
~
whose
Gauss
observed
in 1814;
which
has been
further
is n o t w h a t
therefore
we are
the
attached
- In the e x a m p l e
Hecke
L-function
nowadays
this
generalized
looking
to a n a l g e -
o f § I, t h e c u r v e
for.
character
The
~
is
L(~,s).
follows
from
by S h i m u r a
complex
multi-
are n o t v i s i b l e
28
over
the
~
. That
@tale
happened
and
for
cohomology,
to be
k
a general
~
has
field
mology
of
rank
Gal
(see
3.3
A
the
over
field
k
in o u r
of v a l u e s
onedimensional
The
E ~ ~®~
treatment
of
~
of
(which
again
Galois-representations,
End/kA
of
In fact,
values
is t h e
embed
of
HI (A)
[E
weight
, for
k
in the
such
is an
of
. Thus
eoho-
E®~i-module
E
on
the
with
their
isomorphisms.
Betti
M
that
various
compatible
the
In o t h e r
rank-condition
cohomology
an a b e l i a n
The
fact
that
character
was
multiplication
abelian
such
a motive
H
(M)
space.
: Q].
with
-I
M
§ 2,
field
action
be
that
Hecke
complex
occuring
. The
End
saying
E-vector
is
in
M
comparison
into
by
base
of
course
various
stated
of
~
of
like
Z, Hz(M)
via
an a l g e b r a i c
theory
those
that
by
A/k
these
motives
one
of
Shimura
varieties
n(o,7)
variety
of
the
and
main
Taniyama.
CM-type
6 {-1,0}
with
are
, for all
.
given such
in
E
field
. Then
E
A
, we
an
can
generated
algebraic
assume
by
over
values
(i.e.,
of C a s s e l m a n ,
defined
Hecke
without
the
is a C M - f i e l d
a theorem
variety
the
2 dim A =
to
the
all
should
example
and
rise
the
(o,T) 6 Z × T
for
acting
M
should
be
over
~
realizations
that
with
E
characters
precisely
of
character
defined
the
(T/k)
also
typical
Hecke
all
a onedimensional
give
results
Hecke
such
and
3.1),
can
form
always
and
I with
HI(M )
should
on
realizations
words
and
used
a motive
acts
structures
k
and
) to obtain
to be
E
extra
The
considered
algebraic
theories,
various
on
we
thus
Given
the
is w h y
[Sh
k
loss
of
can
that:
~
of
k
of g e n e r a l i t y
~
quadratic
1],
such
character
be
on
the
over
to
. 2 dim A =
• there
that
finite
a totally
applied
get
[E
is an
with
E
id~les
real
of
subfield),
an a b e l i a n
: ~]
isomorphism
N
E
• HI (A)
> ~®~ End/kA
is
a motive
for
~
.
27
3.4.
When
~
above
still
forces
with
has
lweight
order
arbitray
the
to be a b l e
to assemble
as t e n s o r
HI(A)
in 3.3.
E
is h o w e v e r
a Hecke
{n(J,T) } =
8
to c o n t r o l
a motive
product
the n a s t y
. For example,
then
infinity-type
is e a s y
essentially
There
(%0)
the
if
k
{1,0}
for any given
k
can never
take
of the
quadratic
Hecke
form
(5)
lilBi ,
finite
expect
character
HI(A)
the f i e l d s
{-1,0}
in
6 =
with
naively
with class
{n(d,7) } =
all v a l u e s
form
one w o u l d
of c o n t r o l l i n g
with
condition
twisting
algebraic
of c o n s t i t u e n t s
problem
of
. Since
motivically
is i m a g i n a r y
character
homogeneity
to be of the
(Bi) I : I , n i ( d , 7 ) 6 {±1,0}
characters
like
weight
or
of v a l u e s
number
h > I ,
or
E = k
, but
its
h-th
p o w e r may.
Constructing
a motive
of a m o t i v e
the
for
for the
field of coefficients
3.5
This
"descent"
o f the
B u t we g a i n m u c h
formalism
due
Deligne
was
group
we
3.6
E
the m o t i v e
SO,
give
CM-abelian
over
varieties
onedimensional
ture
would
~
- along
admit
complex
to the
defined
Hecke
with
k, w h o s e
given
£-adic
over
subgroup
as c a n
finite
order)
characters
of
over
proper-
(isomorphic
irreducible
~
an E - a c t i o n
£-adic
its ~ - r a t i o n a l
T
to a
represen~
of
k,
the c o r r e s p o n whose
repre-
.
character
by ~
Langlands
beautiful
by lifting
k
IV.
of
of
Sk
Hecke
~
.
, of w h i c h
multiplication
other
whose
algebraic
to a g i v e n
back
equipped
the
by a character
[S£])
in
b y the
T
category
with many
±h
elegant
of t h o s e m o t i v e s
by Serre
E ®Q£-modules
imply that
the
subquotient
Sk
over
which
in w e i g h t
[DMOS]
group"
, a certain
algebraic
k
"Taniyama
twisting
k
power
to s h o w t h a t
c a n be d e a l t w i t h
§ 5, a n d D e l i g n e ,
, the
over
the m o t i v e s
for e v e r y
, a motive
to
if we use a v e r y
to s h o w t h a t
attached
of
"descended"
tensor
still has
of c o e f f i c i e n t s
group
k
, one
to the c a t e g o r y
after
precisely
representation
sentations
able
constructed
are g i v e n
"find"
ding
Taniyama
for every
scheme
tations
~
as an E - l i n e a r
±I
insight
[La]
is e q u i v a l e n t
varieties
the
t i e s - has,
field
more
over
(eventually
from abelian
. Since
scheme
subsequently
representations
be o b t a i n e d
c a n be
to L a n g l a n d s ,
a group
power
of w e i g h t
E
directly.
defined
h-th
the c h a r a c t e r
of
k
Galois
representations
. Furthermore,
realizations
with values
c a n be c o n s t r u c t e d
determine
in
from
are
the T a t e - c o n j e c a motive
up to
28
isomorphism
As we
not
are
too
tives
- even
dealing
surprising
that
can
necessarily
are
theorem
ically
we
Hodge
field
find
same
periods...
§ 4.
Periods
in t h e
of
the
first
look
at
for
an
are
well-known
carry
over
4.]
As
and
M
~
be
Then
is
these
for
Hecke
the
are
cohomology
cohomologies
to t h e
In
this
fact,
from
closely
We
are
cohomology
of a l g e b r a i c
character
does
to
seem
to be
on Deligne's
an
algebra-
of
will
from
abelian
have
the
a comparison
So,
case
use
(not
I. A n y w a y ,
motive.
in the
going
not
over
to a r i s e
character.
Hecke
they
of o u r
of m o -
k
cohomology
character
going
category
over
[DMOS],
the
cycles).
is p e r h a p s
it h i n g e s
variety
- see
groups
more
same
it
of
some
varieties,
let
us
a motive
facts
and
which
which
to m o t i v e s .
in § 2,
let
an a l g e b r a i c
a motive
for
same
in the
Still,
Hodge"
(algebraic
cycles,
varieties
on an a b e l i a n
§ I, p e r i o d s
Hecke
I.
constructed
the
Rham
prove:
motives.
cycle
to
of
de
algebraic
[Sch],
"motives"
Hodge
abelian
attached
of
motives
belong
and
actually
is a b s o l u t e l y
two
example
Betti
- see
of
absolute
all
motives
"every
that
As
can
category
closed
varieties
two
sense
for
from
larger
that
whenever
one
obtained
isomorphic
in a n y
strictest
motives
that
be
CM) , any
actually
known
in the
with
any
over
k
and
Hecke
k
space
o6Z
of
be
attached
embedding
an E - v e c t o r
E
totally
character
of
to
, the
~
imaginary
k
with
(in the
singular
dimension
1. T h e
Ho(M) ® ~
:
number
values
sense
rational
E-action
of
in
fields,
E
3.2
. Let
above) .
cohomology
respects
the
H
(M)
Hodge-
decomposition
H
(M) ® ~{
fined
at the
Starting
A/k
Hecke
is an
of
from
E ®~{
beginning
the
CM-type,
character
= {T - m o d u l e
of
special
and
(see
using
HP,q
P,q
of
rank
I.
(~
and
T
were
de-
§ 2.)
case
the
3.6) , o n e
where
M = HI (A)
uniqueness
finds
that,
of
with
the
for
an a b e l i a n
motive
any
attached
embedding
variety
to a
T 6 T
,
29
the
direct
factor
of
Ho(M) ®~
on which
E
acts
via
T
lies
in
Hn(o,x) , w-n(c,T)
(The
n(o,T)
are
given
by
the
infinity-type
of
~
: see
§ 2,
formula
(5) .)
4.2
Let
for
Hecke
• For
• ~
are
in p a s s i n g
and
~
over
~
have
Coming
the
back
is c r i t i c a l
T'
if
M(~)
of
k
and
with
M(~')
values
are motives
in
E
, then
the
.
06~ , H @ ( M ( ~ ) )
~'
that,
and
equivalent:
~ M(~')
some
4.3
note
characters
following
• M(~)
us
for
~ H
same
to o u r
~
(see
:
@
(M(T'))
, as r a t i o n a l
infinity-type
motive
§ 2,
M
B .
for
formula
Hodge-structures.
~
, suppose
(6)),
and
now
that
consider
the
s = 0
comparison
isomorphism
I
Note
that
k®~
~ {Z
. So,
I
~EZ
, let
e
For
On
the
and
HDR(M)
right
hand
H u(M) ® ~ {
is b y
definition
is a n
be
side,
> HDR(M) ®~
a k-vector
isomorphism
an E - b a s i s
choose
of
of
H
a basis
~
£0
(@,T)6ZxT
for
(M)
~
of
, and
and
that
- modules
put
HDR(M)
of
e :
rank
1.
( ~ ® I)~6 ~
over
k ®~E
,
decompose
L0 :
with
space,
k ® E ®{
~o,T
the
6
T-eigenspace
corresponding
(0,T)EZxT
that
of
r
0,7
H D R ( M ) ®kl {
decomposition
of
. Writing
I(e)
, we
find
I(e
for
) =
all
Z
T6T
I(e~) T
30
~
The
,~
= p(O,T)
, for
. I(eo) T
p(O,7)
6 {*
unit
C(~*) ZxT
(p(~,T))(~,T)E
gives
the
pends
only
4.4
Modulo
"matrix"
on
(8)
This
~
of
I
and,
p(o,T)
• p(co,T)
essentially
from
the
norm
character
context
identity
is e n o u g h
coefficients
one
up
to m u l t i p l i c a t i o n
by
(k®E)*
, de-
in ~
the
and
be
relation
to L e g e n d ~ e ' s
that
such
period
uniqueness
= ~w
discussed
know
has
N (2 z i ) w
(using
~
is
to
(k®E®~)*
.
a factor
in o u r
=
E×T
such
amounts
proved
it
some
of
relation,
motives
. _ The
motive
~(-I)
in
detail,
e.g.,
more
~(-1)
is
a motive
for
can
Hecke
characters)
attached
in
[DI],
defined
over
to
§
the
3.
~
For
(8),
, with
that
I
HB(~(-I))
with
trivial
tical
s
= ~
comparison
~
and
HDR(~(-I))
isomorphism.
, if c o n s i d e r e d
over
7
or
4.5
(see
cT
In
(7)
,
Incidentally,
a totally
With
(8) a n d 3.4, c a l c u l a t i n g
the
reduces
to i n t e g r a t i n g
holomorphic
= ~
~(-I)
imaginary
p ( o , T ) 's
(or
differentials
field
has
k
no
cri-
.
their inverses)
usually
on which
E
acts via
.
terms
above)
of
these
can
be
p(O,T)
defined
, Deligne's
period
componentwise
~(%~)C(E®~)*/E*
by
-I
(9)
For
~(~{,T)
the
definition
= D(~') T - o6~(~ToB)
of
the
"CM-types"
p(o,'~)
¢('~oB)
, see
§ 2,
formula
(6).
31
Note
that
in
the
product
(E ® 1)*
D(~)
one
=
- One
(D(~)T)T
computes
other
things,
D(~)
which
esp.
2.4.1
let
K
6 T
the
and
cE ~
(9)
can
be
has
born
to
of
by
5.7.2B
the
G a I ( ~ / K T)
of
the
permutes
kernel
of
the
the
[L
2
D (~)
T
NOW,
any
places
set.
: K
~
6 K*
T
] < 2
of
I P @
set
Call
: both
8(p)
induces
the
pT
sign
of
made
in d e f i n i n g
Let
us
some
properties
D(~)
then
d)
Let
is
depends
{¢(~o~)
If
(T°O)
a factor
factor
. A
arises
when
formula:
among
definition
of
computations
Start
with
one
- cf.
TET
[Ha],
, and
: ~(~o~) }
be
L T mK
. Let
s@n>
the
fixed
field
)
{-+I}
for
2
k
a permutation
some
of
O(pTo6)
this
= a(p)
choices
c)
of
follows.
and
the
m(~)
ordering
= K[(D(~)
~(To~)
(D(T) T ) < 6 T
b)
K~nneth
Z
to
D(~) T
with
set
infinite
T
array
a)
This
the
@('loB)
L
The
4.6
8.15.
by
> ~(~(ToB))
, and
D(~)
list
up
factor"
character
p 6 Gal(~/~)
k
[DI],
cohomological
field
~[
of
an
as
G a I ( ~ / K T)
Then
in
well-defined
"discriminant
Rk/Q M
is
{p 6 G a l ( ~ / ~ )
fact,
the
these
-
fixed
in
of
choose
out
Cor.
be
is,
found
cohomology
one
was
in
definition
are
permutation.
the
of
in b i j e c t i o n
Then
we
with
this
set
(D(~))P
T
independent,
its
of
only
to
a factor
also
[Sch].
in
(EOI)*
,9f
components.
D(%')
on
up
k,E
- ef.
, and
the
collection
of
"CM-types"
I • CT}
6(E®
is
I)* c ( E ~ ) *
a CM-field,
D(~) N 6 d i s c r ( k o T
F/k
be
a finite
with
maximal
, up
to
extension
totally
a factor
of
degree
real
in
(E®
n
subfield
I)*
. Then
ko,
32
c
up t o a f a c t o r
in
('[o 8)
(E ® I)*
TIT
. Here,
the r i g h t h a n d
side m e a n s
the
following:
Let
d(k*) 2 6 k * / ( k * ) 2
infinite
place
v
be
of
k
the relative
, choose
discriminant
a square
root
of
6
F/k
V
6Z , let
Icl
be t h e
infinite
place
of
k
. For
= /d-6k*
any
. For
V
determined
by
~
and
c~
,
w
and denote
the
by
continuous
changing
o(6toi) £ 6"
the w e l l - d e f i n e d
isomorphism
k j o I --~--~
> {
the r e p r e s e n t a t i v e
of
d
or
image
given
the
of
by
signs
~
of
61~16 klo I
. - Note
6
under
that
, a t some p l a c e s
V
v
, multiplies
the r i g h t h a n d
side
of o u r
formula
only
bya
factor
in
(E ® I)*
Assume
the
situation
a n d the p r o p e r t i e s
the b e h a v i o u r
of 4.6,d) . F r o m
4.6,a)
a n d d),
of the p e r i o d s
A(F/k,B)
the v e r y
one
under
finds
definition
the
extension
following
of the b a s e
(~ ONF/k)
D (~ONF/k)
(~n)
D (~n)
D(~°NF/k)
D(p) n
array
A(F/k,B) 6 (E~6)*
A(F/k,B)
present
Note
that,
if
in
[Ha],
somewhat
4.7
Let
in the
clumsier
us c l o s e
this
will
k
can be evaluated
still
formula
for
field:
D(~) n
=
§ 5 below.
p(c,~) ,
• -
(10)
The
of the
case
n = 2
section
For
Both
in the
the
under twisting.
factors
, although
the
employed
here.
the T a t e
(11)
2 ( ~ . ~ m) ~ (2 ~i) m ~(~)
twist,
second
second
with a few words
our p e r i o d s
•
D(~)
n-1
Ic I T 6 T
reappear
4.6,c).
than the one
~(6
D ( ~ n)
is a C M - f i e l d ,
by
77
of
theorem
factor
A
are already
formalism
there
o n the b e h a v i o u r
one
finds
of
of
is
of
33
If
~
is a c h a r a c t e r
one
passes
from
multiplying
properties
to k n o w
If
F
under
via
to
~
class
: Gal(kab/k)
details
in
words,
field
of
E*
X
and
is the
~
in
E*
and
eigen-
All we n e e d
of f i n i t e
on
of
order
X
on
to
Gal(k/F) , Gal(k/k) ,
, where
transfer
map),
then
~q(TONF/k)
:
us m e n t i o n
with
the r e s t r i c t i o n
~ : Xo V e r
2 ( ~ - ~ n)
Let
unchanged,
[Sch].
a character
~
considering
~(X" (To NF/k) )
(12)
in
values
lemma:
k, X
Gal(Fab/F)
with
numbers
c a n be f o u n d
, and
theory,
on
algebraic
invariance
extension
-->
K*
/k*
~,f
by leaving
D(T)
Q(~T)
. The
with values
order
by c e r t a i n
following
(in o t h e r
resp.,
Vet
finite
p(0,T)
is a f i n i t e
F~,f* /F*
k~,f*
~(T)
the
is the
of
£ (F/k,B)
~q(Tn)
in p a s s i n g
that
the p r o o f
of
(12)
also
shows
that
the
quotients
~(T)
may
always
§ 5.
The
The p r o o f
values
The
rationality
of D e l i g n e ' s
case where
there one
Let
values
But
it w a s
field
under
k
idea
who,
(see e n d
Hecke
L-functions
of
§ 2)
case
base
falls
into
is t r e a t e d
by a t h e o r e m
of the
for the c r i t i c a l
characters
is a C M - f i e l d
extension
describe
the m a i n
Damerell
for Hecke
conjecture
to the g e n e r a l
us b r i e f l y
sums.
of a l g e b r a i c
the b a s e
passes
Historically,
by G a u s s
conjecture
of L - f u n c t i o n s
of s p e c i a l
(I)
be e x p r e s s e d
about
two parts:
first.
From
the b e h a v i o u r
field.
the C M - c a s e :
for
in his
the C M - c a s e
thesis
[Da],
goes back
published
to
Eisenstein.
the f i r s t
34
comprehensive
account
of a l g e b r a i c i t y
Hecke L-functions
of
finer
theorems
rationality
(The c a s e
[GS] a n d
of
imaginary
imaginary
[GS'].)
~uadratic
quadratic
I n the F a l l
k
of
1974,
Damerell's
as an a p p l i c a t i o n .
later on developed
ralize
Damerell's
functions
needed
into
of a r b i t r a r y
a technical
Andr@
This
[WEK]
CM-fields:
assumption
Well
gave
among
of
course
them.
completely
of
things,
at t h e
IAS - w h i c h w a s
G. S h i m u r a
values
to g e n e -
of H e c k e
(At t h a t p o i n t ,
infinity-type
in
an e x p o s i t i o n
other
to c r i t i c a l
[Sh 3]
o n the
published
settled
- inspired
results
values
He a l s o a n n o u n c e d
never
later
including,
the b o o k
algebraicity
but
was
and Kronecker
for c r i t i c a l
fields.
in t h a t c a s e ,
w o r k of E i s e n s t e i n
theorem
results
he
L-
still
of the H e c k e
cha-
racter.)
To e x p l a i n
example
the
starting
in § I: t h e
an E i s e n s t e i n
point
L-value
series
of t h i s
there
method
appeared
of p r o o f ,
recall
of
our
I
~ ) as
relation
between
straightforward
- e.g.,
(up t o a f a c t o r
:
v
a,b6Z
relative
L-value
if,
to the
lattice
and Eisenstein
in § I, we w e r e
such
that
s = 0
pair
to study
are
symmetric
modular
k
lations
[WEK]).
theory
or,
series
among
But
can
them
with
L(~a,0)
in
for
(non h o l o m o r p h i c )
it r e m a i n s
L-function
combination
Well's
the p r o o f
which
is a v a l u e
series
real
of t h e i r
theory
properties
from explicit
treatment
for the Hilbert
o n an a l g e b r a i c
there
totally
in a n y
k
(viz.,
subfield
.
directly
models
that,
of E i s e n s t e i n
be d e r i v e d
e.g.,
differential
true
equation,
a # 0
to t r a n s -
of a C M - f i e l d
to the m a x i m a l
k
integers
, t h e n we w o u l d h a v e
to the f u n c t i o n a l
respect
the
the a l g e b r a i c i t y
in g e n e r a l
equivalently,
~a
as
quadratic,
(see,
of c a n o n i c a l
for
by c e r t a i n
lattices
is i m a g i n a r y
Eisenstein
the v a l u e s
as a l i n e a r
to
sometimes
is not q u i t e
o f an H e c k e
respect
forms
k ), r e l a t i v e
When
in
with
. Now,
4~
for s u c h o p e r a t o r s
values
c a n be w r i t t e n
Hilbert
of
series
But except
of c r i t i c a l
which
Z + Zi
series
is c r i t i c a l
f o r m the E i s e n s t e i n
operators.
(a+bi)
of the
polynomial
of D a m e r e l l ' s
algebraicity
modular
of H i l b e r t
group
depends
(as in
modular
re-
theorem
on a
[Sh 3])
forms.
35
This
latter
Katz
did not
the
approach
special
Katz'
main
stop
was
to
values
he
concern
used
look
had
was
by
Katz
at m o r e
determined
with
in
[KI],
precise
up
[K2].
Just
rationality
to a n a l g e b r a i c
integrality
properties
like
Shimura,
theorems
about
number.
and
p-adic
In fact,
inter-
polation.
When
Deligne
that,
up
turned
formulated
to a f a c t o r
o u t to
expresses
be
his
in
conjecture
~*
problem,
L-values
in t e r m s
structed
from
lattices
in
tion
k
hand,
the
field
k
, and
, with
multiplication
with
see
role
[DI],
his
8.19.
Deligne's
"reflex
Theorem
I:
(Note
field
definition
(Its
refinement
the
that
factors
most
managed
serious
made
to
~*
for
more
obstacle
solve
this
problem
on
Let
k
values
be
for
~
he
the
level
was
some
in the
by
motive
to
of
E
the
such
a
complex
). T h i s
rationality
was
dealt
state-
to p r o v e
down
over
other
dualization,
attempt
writing
of
, with
ad hoc
con-
the
coefficients
precise
Shimura
multiplica-
. On
character
related
an
This
reason.
E'
k
of
to c h e c k
varieties
over
and
need
complex
. The
of m o t i v e s
able
a CM-field,
in
E
- by
I].)
with
any
in
[Sch
Thus
have
a Hecke
closely
in
dualization
motive".
abelian
defined
(or s o m e
of
of
the
theorem.
field
field
field
to
~(~)
number
varieties
- up
L('~,o)
values
some
of
number
is t h a t
felt
following
therefore
some
as
conjecture
Blasius
in
abelian
the
periods
, which
over
he
Shimura's
for
E
k
remained
Deligne's
of
b_~
by Deligne
ments
Don
from
of
of
in q u e s t i o n
values
arises
k
defined
L-function
character
double
are
1977
, it p r e d i c t e d
a confusing
the
by
in
k
an
analogue
, resp.
E
:
k
,
to p r o v e
and
~
CM-field
a Hecke
E
. If
character
s = 0
of
is c r i t i c a l
, then
6E
~
algebraic
>
Hecke
E®¢
.
character
of
any
number
field
takes
in a C M - f i e l d . )
As B l a s i u s '
paper
[B]
is a b o u t
to be
available
we
shall
not
enter
into
36
describing
that,
course,
series
action
a very
Gal(~/~)
(i.e.,
of
in d e t a i l .
mentioned
on abelian
Suffice
above,
the b e h a v i o u r
Shimura's
description
shall
now describe
varieties
it to say
he n e e d s ,
of
o f the E i s e n s t e i n
reciprocity
- due t o T a t e
n o t be p u b l i s h e d
the
field,
algebraic
following
and
Hecke
finity-type
law
in C M - p o i n t s ) ,
and Deligne
of C M - t y p e :
F/k
of
see
> E*
A(F/k,8)
defined
in § 4.6,
Theorem
2:
conjecture
[Ha],
for H e c k e
GL 2
completely
before
some
time.
situation:
Let
a finite
k
extension
with
values
s = 0
- of the
[LCM],
be a t o t a l l y
the
of
finite
second
GL n
imaginary
field
Y
E
It
, and
n ~ 2 . Let
in a n u m b e r
for
the
L-functions.
to
of d e g r e e
is c r i t i c a l
be a c h a r a c t e r
Recall
:
in d e t a i l
from
character
in § 4.7 above.
bit more
§ 3,
8 . Assume
X : F~* /F*
a little
of D e l i g n e ' s
on a g e n e r a l i z a t i o n
Consider
like
analysis
Gal(~/~)
of the p r o o f
might
ber
careful
motive"
7.
We
relies
of his proof
"reflex
the e x p l i c i t
of
chapter
(II)
technique
f r o m the
under
and also
part
the
apart
~
, of
numbe an
in-
. Let
order,
and put
~ =Xlk~* '
array
(A(F/k, Yo8))76T
formula
(10).
L F(X- (~ONF/k) ,0)
6E
A (F/k, B)
m
>
E®~
Lk(~-~n,0)
R e m a r k s : (i)
A s the E u l e r
w
Re(s) > ~ + I , a n d
s = 0
the d e n o m i n a t o r
(ii)
Here
critical
in the
for
is c r i t i c a l
theorem
is h o w t h e o r e m s
values
product
is n o t
I and
of all H e c k e
L(T,s)
for
converges
for
Y , it is w e l l - k n o w n
that
zero.
2 imply
L-functions:
Deligne's
Given
conjecture
any totally
for all
imaginary
37
number
field
a number
(5)
of
F
, and
field
Eo
§ 2 forces
contained
in
F
any
, of
Hecke
Bo
to
factor
some
algebraic
a Hecke
character
= X -( ~ o N F / k )
choose
E mE
of
F
, the
through
the
, with
values
homogeneity
maximal
in
condition
CM-field
k
=
~o
NF/k
,
homomorphism
8
Choose
~
8o
:
8o
for
character
infinity-type
, for
big
: k*
~{
of
> E'o*
k
some finite
enough
to
with
infinity-type
order
contain
character
the
values
B
X
of
of
T
, write
F
as
, and
well
as
those
o
of
X
- Define
e : Xlk*
. Put
n =
[F
L(~0"T n,0)
~ ( w . y n)
But
we
know
extension:
the
see
behaviour
end
of
of
§ 4.
the
. By
theorem
I,
6 E
periods
Theorem
L(X-(
L(~,0)
: k]
Q
under
2 therefore
twisting
implies
and
base
that
~ o NF/k),0)
6E
> E@{
•
[2(X- ( T o N F / k ) )
Finally,
is
[D1],
This
may
gives
Hecke
that
real
now
under
be
replaced
finite
by
extension
Eo
of
because
the
Deligne's
field
of
conjecture
coefficients:
2.t0.
ginary
the
E
invariant
Deligne's
number
L-functions.
Deligne's
fields
case
of
conjecture
fields.
These
But
it
conjecture
follows
number
are
should
for
for
Hecke
the
only
be
Hecke
from
results
fields
which
of
are
L-functions
fields
said,
for
with
the
(=Dirichlet)
of
totally
honest
sake
of
completeness,
L-functions
Siegel's
(cf.
neither
totally
[DI],
real
ima-
regard
6.7)
nor
of
totally
and,
in
totally
38
imaginary,
no H e c k e
The r e m a i n d e r
theorem
(=Dirichlet)
of this
section
L-function
is d e v o t e d
has any c r i t i c a l
to s k e t c h i n g
value.
the p r o o f
of
2. Let us set up some n o t a t i o n .
We c o n s i d e r
the f o l l o w i n g
algebraic
Go/k
:
GLn/k
To/k
=
standard maximal
Bo/k
=
standard Borel
and the two m a x i m a l
<>ik :
groups
over
k :
-
parabolic
torus
s u b g r o u p of u p p e r t r i a n g u l a r
matrices,
subgroups
.~
(Op)
p 6 G L n _ I , t 6 GL I }
q 6 GLn_ 1 , t 6 GL I }
Dropping
to
the s u b s c r i p t
zero will m e a n
taking
~ . So,
G/(~
:
Rk/,~ (G O/k)
and so on.
We i n t r o d u c e
7P
the two c h a r a c t e r s
: g =
<0
tn
}
>
det(g)
and
tn
the r e s t r i c t i o n
of s c a l a r s
39
which
we v i e w
as c h a r a c t e r s
The
representations
are
the
~-th
sentation
Coming
Go/k
(resp.
of
back
of
Go/k
to the
on the t o r u s
b-th)
on
kn
with
situation
: P(~ih,f)
highest
symmetric
(resp.
extending
weight
power
(resp.
(resp.
standard
Qo
~TQ
)•
)
repre-
(kn) ~ )
2, d e f i n e
= Po(klA,f)
Po
~Tp
of the
its dual
of t h e o r e m
to
a homomorphism
> E*
by
> ~I (tf) ~ (det(gf))
We r e q u i r e that the c e n t r a l
that
~I
is d e t e r m i n e d
character
has
view
•
=
be our
~
. This
means
w(tf)_ = ~I (tf)_ ~ (tf)
n_
as an " a l g e b r a i c
Hecke
character"
on
P/~
, and
it
an i n f i n i t y - t y p e
type
Hence
we get
(~) = 7 6 H o m ( P / ~
an a r r a y
of
type(To~)
Recall
types,
, RE/~({m))
indexed
by
T 6 T
, with
components
= 'toT 6 HOrn(P,{ m)
that
H o m ( P , G m}
and
~
by
"" "t
We may
of
that
the type
8
of
=
@
gCZ
H o m ( P o ,~m }
is g i v e n
by
the
,
integers
n(a,<)
- see
§ 2.
40
It is then
easy
to c h e c k
TOy
for e v e r y
Given
T
• 6T
:
that
(n(o,T) .yp)o6E
,
.
, define
an array
i(~)
of d o m i n a n t
weights
(l(o,<))oE Z
=
by the rule
{
This
affords
with
nal
M(A (T))
highest
system
in
with
[Ha],
: Gx~
K
centre
The m o d u l e s
if
n(o,T)
=
~
(GLn/k)
oGZ
> GL(M(A(T)))
of r e p r e s e n t a t i o n s
we
are
study
~
the
M(i(7))
the
in the
sense
under
Gal(~/~)
cohomology
modules:
= G (~) ~ G ((~IA) /
u(n)Z~
o~bG(~)
and we c o n s i d e r
conjugate
in these
SK
the
n(o,~)yp
> 0
,
® M(I(O,T))
, M(I(O,T))
b e i n g the r e p r e s e n t a t i o n
o6Z
weight
k(O,T) . The s y s t e m
{M(A(T))~6 T
is a ~ - r a t i o -
coefficients
where
n(o,T) < 0
=
representations
As
if
a representation
p
where
(-n(@,~)-1)yQ
= G
H'(~,M(A(T)))
2.4
- i.e.,
subgroups
the
of
GLn(0)
the q u o t i e n t s
K o'Kf
, and where
maximal
Kf
coefficient
~ - G(~,f)-
[Ha],
of c o n g r u e n c e
Form
is a s t a n d a r d
provide
of
compact
is o p e n
systems
M(I(T))
module
: = lira
Kf
H'(SK,
subgroup,
compact
M(A(T)))
in
times
G(~
on
,f) •
SK
,
41
The
embedding
homotopy
of
SK
a stratification,
of p a r a b o l i c
SB SK
into
equivalence.
The
with
strata
subgroups
, corresponds
coefficient
its B o r e l - S e r r e
boundary
of
~S K
corresponding
G/~
. The
can be e x t e n d e d
H" ( ~ B ~ , M(A(T)))
:
class
conjugacy
of l o w e s t
H" ( ~ B S K
is a
and
has
classes
dimension,
of B o r e l - s u b g r o u p s .
to the b o u n d a r y ,
lim
SK
compactification
to the
stratum
to the c o n j u g a c y
system
compactification
of this
the
The
limit
, M(A(T)))
- - >
Kf
is a g a i n
a
G(~,f)
SK
induces
a
as
i
>
SK
in
of modules,
The
diagram
<
G(~lh,f)-module
rB
Just
~
-module.
~B S K
homomorphism
: H" (~, M(A(T)))
[Ha]
,II,
induced
the
from
> H" ( $ B ~ , M(A(~)))
right
hand
side
an a l g e b r a i c
n : B(QIA,f)
>
turns
Hecke
out
to be a d i r e c t
sum
character
~*
T(~IA, f)
on
B(~,f)
termined
particular,
(for
•
, up to
by K o s t a n t ' s
G(~,f
it is e a s i l y
as above,
and
}
. The
theorem,
checked
types
of
[Ko] ; cf.
that
the
these
[Ha],
II,
following
T 6 T ) is c o n t a i n e d
in the
characters
are
for
. In
n = 2
induced
de-
module
cohomology
of
~B ~
:
42
= Ind
V
G (QIA, f)
B (QIA, f)
\
h
is
C
h(bf_gf)
=lh:G(~,f)
-->
for all
, and
)
= (TOO) (bf). h(if)
bf 6 B(~ih,f)
,
and
_gf £ G(~IA, f)
(Here,
"C "
means
pact
subgroup
in
More
precisely,
right
>
T
1
d o : ~ [k : ~]
with r e s p e c t
under
a suitably
small open com-
.)
we have
VTo 0
where
invariance
G(~,f)
H(n-3}do
, and the
to the two o b v i o u s
(~B ~ , M(A(T)))
system of maps
Q-structures
{iT]<6 T
is Q - r a t i o n a l
on the systems
on both
sides.
Consider
the n o n - t r i v i a l
submodule
J~o~ = I n d G ( Q ~ ' f )
P(~,f)
Obviously,
of
{JToO}T£T
a Q-rational
Eis7
for all
TCT
structed
first
. Thus,
other
words,
that
{EisT}TC T
argument,
oo
>
system of
first
"section"
of
essential
G(Q~,f)
-submodules
step of the p r o o f
is
rB ,
H(n-1)d° (S, ~{A(T))}
,
rB o Eis
~
= Id on
J
. This section is con7
T°O
by m e a n s of r e s i d u a l E i s e n s t e i n series or, in
non c u s p i d a l
like
. The
: JTo~
over
V
c
is a Q - r a t i o n a l
H ( n - 1 ) d ° ( $ B ~ , M(A(7)))
to c o n s t r u c t
To~
Eisenstein
is d e f i n e d
in [Ha],
III.
over
~
But here
series
attached
to
P/Q
. To prove
one has to use a m u l t i p l i c i t y
this
is more
complicated.
One
one
43
has
to use
boundary
has
the
spectral
in t e r m s
to be r e l a t e d
sults
non
cuspidal
we have
which
to a u t o m o r p h i c
of J a c q u e t - S h a l i k a
discrete
Once
sequence
of the c o h o m o l o g y
computes
of the
forms,
the
strata.
and one
on m u l t i p l i c i t y
one,
cohomology
Then
has
of the
the c o h o m o l o g y
to a p p e a l
and of J a c q u e t
to reon the
spectrum.
the m o d u l e s
we can p r o c e e d
Eis z
(J o~)
c H(n-1)do
more
or less
in the
(S, M(A(T)
same
way
as
)
in
[Ha],
V: We c o n s t r u c t
an e m b e d d i n g
iH
H
being
struct
the
torus
homology
: F*
>
with
H(Q)
classes
GLn(k)
= ill(F*)
(compact
ZIi H, 7 o X, g) C H
depending
on a p o i n t
whose
restriction
As
[Ha],
in
to
(n-1)d o
and
_g6 G(~]A)
* /F*
X : F]A
>
k ~*
V, we get an
Using
modular
this
torus
we can con-
symbols)
(~, M(A(~)))
on a finite
order
character
E*
should
be
intertwining
~
.
operator
G ( ~ m , f)
Int(Z(iH,X))
: JTo ~
> Ind
H(~m, f )
by
evaluating
ning
E i s T ( J T o Q)
on
Z(iH,X,~)
. There
is a n o t h e r
operator
int l°c
: J
To~
>
Ind
G ((~]A,f)
H (~IA, f)
T oX
,
intertwi-
44
constructed
tors
are
as a p r o d u c t
U-rational,
of l o c a l
and
for
intertwining
some
x C E*
we
Both
opera-
for all
~6T
L F ( T O (X" (~ONF/k) ) ,0)
= T(X) A (F/k,ToS)
Int(Z(iH,TOX))
operators.
f i n d that,
,
intlOc
L k ( T ° (~.~n) ,0)
This
implies
theorem
2. - T h e
factor
x 6 E*
can
actually
be g i v e n
more
explicitly.
§ 6.
The
A formula
fact
that
produces
of L e r c h
a Hecke
a period
tions
of a motive
first
example
p(u,T)
These
monomial
leave
in t h i s
various
by D e l i g n e ,
we
f o r the
They
here
[Sch].
volving
. Anderson's
K = ~(/i-~-)
Assume
for
simplest
Instead,
simplicity
[1974 d]:
K
[~(~D ) : K]
dividing
D
that
Hecke
Write
n :
motives
in
-
the
well
quadratic
D > 4 . Recall
is c o n t a i n e d
for w h i c h
by m e a n s
of
K
~ ( ~ D ) , the
~(D)2 " F o r
seen
Shimura
of m o t i v e s
above
as
for J a c o b i - s u m
character
We h a v e
a
[Sh2]
(up to an a l g e b r a i c
field
the
~
But
case
sense
ideal
in-
characters.
construction
f i e l d of
,
referring
of d i s c r i m i n a n t
, in the
a prime
principle.
on a typical
Hecke
number).
over
some others,
let us c o n c e n t r a t e
be an i m a g i n a r y
Jacobi-sum
those
using
application,as
isomorphism
construc-
(8) o f § 4. T h e p e r i o d s
relations
reproven,
up to
geometric
c a n be g i v e n .
comprise
c a n be r e f i n e d
to
its m o t i v e
different
in f o r m u l a
period
were
two
character
formula
this
the r e a d e r
Let
same
monomial
relations
[D2].
aside
G
determines
whenever
of t h i s p r i n c i p l e
occuring
has proved
character
relation
of
D-th
P
of
-D
of the
[WIII],
roots
~(~D )
of
] .
not
, put
G(P)
: --
[
XD, P (x) "A~ (x)
,
x 6Z [~D]/P
with
"ZD, ~ (x) ---x (I~P-I)/D
cod P
, and
l(x)
:
exp(2~i
(cod
•
P )"
the
D th- power
tr( ~ [~D]/p)/]F p
(x))
residue
symbol
45
Then
extend
the
function
of
J(p)
multiplicatively
ties
of
Gauss
sums
Stickelberger
mula
for
then
its
K
to
and
one
all
prime
=
~
of
that
J
an e x p l i c i t
finds
G(P)
ideals
show
that,
K
J
h
is the
class
by genus
some
of
J
>
is
number
in
. Elementary
K
. By
analytic
algebraic
proper-
a theorem
class
Hecke
of
number
for-
character,
-(n-h)/2
X
of
K
t
. (Note
In o t h e r
J
which
finds
K~
deduced,
was
that
words,
if
n
and
J
h
have
is a H e c k e
the
cha-
.%,h
finite
character,
a motive
function
on
e.g.,
given
by
defined
representations
the
of
, if v i e w e d
be
for
of
D
of
a Hecke
also
l-adic
Z
D
order
and
some
Hecke
character
-I
in f a c t
dividing
Anderson
whose
:
K*
> Z
theory.)
~
of w e i g h t
it c a n
motive
set
character
K
places
But
the
is an
j. IN- (n+h)/2
That
p ~D
then
(13)
for
with
by
parity,
racter,
to
values
of
in+h)/2
same
K
infinity-type
X ~
where
of
prime
takes
if
p
,
version
: K*
is g i v e n
ideals
are
i.e.,
id~les,
is w e l l - b e h a v e d
was
proved
from
the
Greg
Anderson,
over
given
K
following
, with
by
J
by Weil
at
the
(loc.
construction
cit) .
of
[AI],[A2].
coefficients
, in
H n-2
of
in
the
K
,
zero-
a
46
~(~D )
> K
x
viewed
as a p r o j e c t i v e
of the K - v e c t o r
variety
space
that
construction
large
more
fields:
general
At any
M(J)
rate,
thanks
tained
from
induction:
[DMOS],
abelian
[DMOS],
J
varieties.
p.
217).
the
The p e r i o d
tually
same
as those
calculations
79 - 96. For
(This
Thus,
we h a v e
lies
by
by the
last
fact
(13),
[Sch].
at our d i s p o s a l
the p e r i o d s
a motive
of m o t i v e s
is p r o v e d
of
characters
of A n d e r s o n ' s
or to
in the c a t e g o r y
fact
action
Hecke
details
preprints,
ob-
by S h i o d a -
of the m o t i v e
: M(J) @ K K ( ( n + h ) / 2 )
of any m o t i v e
constructed
on F e r m a t - h y p e r s u r f a c e s
to B e t a - i n t e g r a l s .
space
well-known
out
to J a c o b i - s u m
to his
work,
, which
M ( J I N - (n+h)/2)
will be
. ~h
pp.
we r e f e r
by the
(carved
attached
to A n d e r s o n ' s
character
motivated
motives
groups)
see
, the p r o j e c t i v e
that
D
.+xD
0}cipn-1
x1+..
n =
contain
construction,
for the
]PK (Q(~D))
is of c o u r s e
automorphism
of c y c l o t o m i c
{
=
Fermat-hypersurfaces
their
in
~(bD ) . N o t e
Z X K ~ ( D D)
Anderson's
~----> tr~ (~D) / K(xm)
For
M(J)
7T
F
always
one e s s e n t i a l l y
a
{<~>)-
for the c h a r a c t e r
reduce
gets
even-
the p r o d u c t
I
X (a)=-1
Here,
K
X(P)
, and
-m)
:
(~-
the p r o d u c t
X(a)
= -I
. <~>
lies
between
0
is the D i r i c h l e t
is t a k e n
over
those
is the r e p r e s e n t a t i v e
and
I
character
a6
of the q u a d r a t i c
(Z/D Z )*
of the
class
field
for w h i c h
a
~ mod Z
which
47
A motive
for
5.~h
multiplication
values
in
elliptic
call
K*
K
(viewed
derived
terms
A
[GS],
values
in
E
§ 4.6
to
K
. Calling
shown
to be
) : cf.
[GS],
§ 4. U s i n g
any
, and
E
a motive
c a n be
AJ/H
takes
~ONH/K
can be
of t h i s m o t i v e
complex
. Choose
for
K
with
~ONH/K
field of
is a m o t i v e
of the c o n j u g a t e s
out
and
of
GaI(H/K)
the
for
formulas
computed
in
of o u r e l l i p t i c
the
4.7),
K*
: CI(K)
twists
one
by the n o r m
finally
generates
Multiplying
a n d the
obtains
the
finite
following
order
character
relation,
up to
:
6 C£ (K)
y
curves
that
, for
Straightening
where
class
of s c a l a r s
§ 9, the p e r i o d s
%
from elliptic
HI(A)
~ , HI(B) ® E h
d 6
a factor
the H i l b e r t
its r e s t r i c t i o n
of
as t a k i n g
in
up
for s i m p l i c i t y
such that
of the p e r i o d s
curve
(cf.
H
A/H
B = RH/KA
be b u i l t
. - Assume
, for
curve
f i e l d of v a l u e s
~h
can
by
~
the a b e l i a n
(14) w i t h
extension
its c o m p l e x
a
r (<~>)
X (a) =-I
of
Conjugate,
K
belonging
to
~
.
we g e t
(15
d 6 C£ (K)
for
some
of
z
z
with
a n d the
analytically
~
z 4 6 ~*
. Except
, this
is the e x p o n e n t i a l
by Lerch
(15),
up to a f a c t o r
which
in t u r n
Hodge
cycles
proving
in
in
inspired
on a b e l i a n
uniqueness
a6 (~/m~) *
[Le],
~*
for the d i f f e r e n t
p.303.
, was
Deligne's
varieties
of the m o t i v e
The
given
proof
first
geometric
by G r o s s
of the
- which
interpretation
o f an i d e n t i t y
in
[Gr],
theorem
again
proof
about
Hecke
of
a paper
is e s s e n t i a l
for an a l g e b r a i c
proved
absolute
in
character.
48
REFERENCES
[All
G. A n d e r s o n , The m o t i v i c
characters; preprint.
[A2]
G. A n d e r s o n , C y c l o t o m y
group, preprint.
[B]
Don Blasius,
[Da]
R.M. D a m e r e l l , L - f u n c t i o n s of e l l i p t i c c u r v e s w i t h c o m p l e x
m u l t i p l i c a t i o n . I, A c t a A r i t h m . 17 (1970), 2 8 7 301; II,
A c t a A r i t h m e t i c a 19 (1971), 311 -----317.
[ml]
P. D e l i g n e ,
Proc. Symp.
[D2]
P. D e l i g n e (texte r @ d i g 6 p a r J.L. B r y l i n s k i ) , C y c l e s de H o d g e
a b s o l u s e t p 6 r i o d e s des i n t 6 g r a l e s d e s v a r i @ t @ s a b 6 1 i e n n e s ;
Soc.
Math.
On the
and an
critical
V a l e u r s de
Pure Math.
France,
interpretation
extension
values
fonctions
33 (1979),
M~moire
n~2
of J a c o b i
sum Hecke
of the T a n i y a m a
of H e c k e
L-series;
preprint
L
et p @ r i o d e s d ' i n t 6 g r a l e s ;
p a r t 2; 313 - 346.
(2 @ m e
s~r.)
1980,
p.
23 - 33.
[DMOS]
P. D e l i g n e ,
and Shimura
[Ei]
G. E i s e n s t e i n , U b e r die I r r e d u c t i b i l i t ~ t u n d e i n i g e a n d e r e
E i g e n s c h a f t e n d e r G l e i c h u n g , y o n w e l c h e r die T h e i l u n g der
g a n z e n L e m n i s c a t e a b h i n g t , - a n d the s e q u e l s to this p a p e r
M a t h . W e r k e II, 536 - 619.
J. M i l n e , A. O g u s , K. Shih, H o d g e C y c l e s , M o t i v e s
V a r i e t i e s ; S p r i n g e r Lect. N o t e s M a t h . 900 (1982).
-;
[GS]
C. G o l d s t e i n , N. S c h a p p a c h e r , S @ r i e s d ' E i s e n s t e i n et f o n c t i o n s
L
de c o u r b e s e l l i p t i q u e s ~ m u l t i p l i c a t i o n c o m p l e x e ; J.r. ang.
Math. 327 (1981), 184 - 218.
[GS']
C. G o l d s t e i n , N. S c h a p p a c h e r , C o n j e c t u r e de D e l i g n e e t Fh y p o t h @ s e de L i c h t e n b a u m sur les c o r p s q u a d r a t i q u e s i m a g i n a i r e s . C R A S P a r i s , t. 296 (25 A v r i l 1983), S@r. I, 6 1 5 - 6 1 8 .
[Gr]
B.H. G r o s s , O n the p e r i o d s o f a b e l i a n i n t e g r a l s a n d a form u l a of C h o w l a a n d S e l b e r g ; I n v e n t i o n e s M a t h . 45 (1978),
193 - 211.
[Ha]
G. H a r d e r , E i s e n s t e i n c o h o m o l o g y
case
GL 2 ; p r e p r i n t B o n n 1984.
[Hu]
A. H u r w i t z , U b e r die E n t w i c k l u n g s k o e f f i z i e n t e n
der l e m n i s k a t i s c h e n F u n k t i o n e n ; f i r s t c o m m u n i c a t i o n in: N a c h r . k. Ges.
W i s s . G ~ t t i n g e n , M a t h . Phys. KI. 1897, 273 - 276 = M a t h .
W e r k e II, n ° LXVI, 338 - 341. P u b l i s h e d in e x t e n s o : Math. Ann.
51 (1899), 196 - 226 = M a t h . W e r k e II, n ° L X V I I , 342 - 373.
[KI]
N. K a t z , p - a d i c i n t e r p o l a t i o n o f r e a l a n a l y t i c
series; Ann. M a t h . 104 (1976), 459 - 571.
[K2]
N. K a t z , p - a d i c L - f u n c t i o n s
49 (1978); 199 - 297.
of a r i t h m e t i c
for CM-fields;
groups
- The
Eisenstein
Inventiones
math.
49
[Ko]
B. K o s t a n t , Lie a l g e b r a c o h o m o l o g y a n d the g e n e r a l i z e d
B o r e l - W e i l t h e o r e m ; Ann. M a t h . 74 (1961), 329 - 387.
[LCM]
S. Lang,
1983.
[La]
R.P. L a n g l a n d s , A u t o m o r p h i c R e p r e s e n t a t i o n s , S h i m u r a
V a r i e t i e s , a n d M o t i v e s . Ein M i r c h e n ; Proc. Symp. P u r e
3 3 (1979), p a r t 2; 205 - 246.
Complex
Multiplication;
Springer:
Grundlehren
255,
Math.
[Le ]
M. Lerch, S u r q u e l q u e s f o r m u l e s r e l a t i v e s
c l a s s e s ; B u l l . Sc. M a t h & m . (2) 21 (1897),
290 - 304.
[Schl]
N. S c h a p p a c h e r , P r o p r i & t 6 s de r a t i o n a l i t 6 de v a l e u r s
s p 6 c i a l e s de f o n c t i o n s
L
a t t a c h 6 e s a u x c o r p s CM; in:
S 6 m i n a i r e de t h ~ o r i e de n o m b r e s , P a r i s 1 9 8 1 - 8 2 , B i r k h ~ u s e r
(PM 38), 1 9 8 3 ;
267 - 282.
[Sch]
N. S c h a p p a c h e r ,
preparation.
[S£]
J-P. S e r r e , A b e l i a n £ - a d i c
c u r v e s ; B e n j a m i n 1968.
[Shl ]
G. S h i m u r a , On the z e t a - f u n c t i o n of a n a b e l i a n v a r i e t y w i t h
c o m p l e x m u l t i p l i c a t i o n ; Ann. M a t h . 94 (1971), 504 - 533.
[Sh2]
G. S h i m u r a , A u t o m o r p h i c f o r m s a n d the p e r i o d s
v a r i e t i e s ; J. M a t h . Soc. J a p a n 31 (1979), 561
[Sh3]
G. S h i m u r a , On some a r i t h m e t i c p r o p e r t i e s of m o d u l a r
of one a n d s e v e r a l v a r i a b l e s ; Ann. M a t h . 102 (1975),
[WIII]
A. W e i l ,
Springer
[WE~]
A. Weil, E l l i p t i c f u n c t i o n s
K r o n e c k e r . S p r i n g e r 1976.
Oeuvres
1980.
O n the p e r i o d s
of H e c k e
characters;
representations
Scientifiques
- Collected
according
au n o m b r e d e s
prem. partie,
in
and elliptic
of a b e l i a n
- 592.
Papers,
to E i s e n s t e i n
forms
491-515.
vol.
and
III.