The Beiliqson
for
Algebraic
Conjecture
Number Fie1ds
Jürgen
Neukirch
pre facg
In
thi s expos ition
on the Artin
the galois
L-series
gfoup
\^/e consider
l,(Mrs)
of
a representation
B . Gros s
M
of
f = Gal- (O/O)
, which may be viewed. as the
general conjecture
Beilinson's
on mo-
zero dimensional case of
tivic
L-ser j-es . The con j ecture
tation
of
a con j ecture
gives
a K-theoretic
interpre-
of
the transcendental
nature of the values of
L(Mrs)
at the integral
places
s = n € n . Included are those places
where the L-series vanishes.
fn this case the "value" is meant
to be the
first
pansion and is
non-vanishing
denoted by
coefficient
L(Mrn)*
in
the Taylor
ex-
.
For example, let
F be a finite
algebraic
number field
and
let
X = Spec F . Then X gives rise-to
H(X)
a representation
r , namely Ho(x I 6,e) - ,Hom(F'Q) . The associ-ated. L-seof
L GifX ), s )
ri-es
is in this case the Dedekind zeEa f unction
6"(s)
of
F . We consider
the
following
two groups,
denote as cohomology groups:
" 1 t x , e ( n ) ) = K 2 r , -(1F ) o 0 ,
"[tXn,n(n)) =t6Ter{ n-1* ]*
which $/e
-f
ry
Here
cx runs
module
(2rL)"
under
n - 1'lR
Hom(FrC)
through
conjugation
complex
. By a theorem of
Phism
rg
lef t
right
n-th
of
map
is
Hom(FrC)
and
!!re have a canonical
isomor-
1
lft'+Hö(XR, lR(n)).
E
are equipped with
"Q-structures",
canonical
1
"X(1-n)
linear
W^ . It
\u
of
on
ue = "A (X , Q ( nl )
and the
Q-subspace
n-1e]*
*e
the e-subspace
. The
[ 1ll- irTTi)
with
regulator
the
Borel
fixed
means the
l+
wi-th the
side
side
t
acting
1
, HA(x,Q(n))
Both lR-vectorspaces
the
and
F is
of
,O
with
determined
defined
respect
to
to be the determinant
a Q-basis
up to a rational
of
ano
Ue
number, i . e.
. x ( 1 ' - n ) e r nY g * .
rie
Gross conjecture
this
for
the representatj-on
case has been proven by Borel)
then
(which in
nlx)
says that
n )
for
1,
c*(1-n)= 6F(1-n)* mod P* o
For an arbitrary
is
f ormulated
Artin
L-series
L(Mrs)
the conjecture
that
x = spec (F )
quite
the same vüay, j ust
has to be replaced by an "Artin motive " , which produces
representation
M . The main purpose of this exposition
in
to give
a presentation
of
jecture
in
Dirichlet
'
is
The exposition
rather
serve
ft
the case of
independent
as an elementary
comprises
Deligne
and it
Artin
cation
is
the
Beilinsonrs
divided
of
L-series.
the
two parts.
into
introduction
of
link
of
these
The main result
of
the Gross con-
of
The first
part
theory
and may
Beilinson
into
the
cohomology and the higher
describes
of
is,
L-series.
the . generäl
explanation
proof
the
the general
absolute
regulators
concepts
this
part
set
up.
cohomology,
the
of
with
is
Artin
motives,
the values
of
the verifi-
the Gross conjecture for Diri-chlet
L-sef ies. Thj-s
j-s obtained by elementary arguments up to a theoverification
rem on the explicit
description
of the regulator
map tD ,
the proof of_which is subject to the second part.
r have tried
to write the first
part in such a !,lay that it may serve as a
basis
for
a student's
seminar.
ttl
The second part
proof
of
the
regulator
theorem
the
cult
general
incorrect,
Schneider.
due to
them.
cl-aim
(7 .O.2)
the
version,
this
So the
part
(2.4) ) , so that
The proof
remains
helpful
U. Jannsen,
discussions.
of
t3I
emergence
Beilinson's
by M. Rapoport
able
to
gap in
co-authors
G. Tamme and K.
the
understand
"crucial
l e m m a"
the
proof.
to
am grateful
'f
or many
Wingberg
f
and
j-s essentially
(see the
a serious
theory.
and partially
proof
the
w€ \,'y7erenot
paper
the
the
n
has been rewritten
of
The
by means of
for
whj-ch was incomplete
Besides of tlre abcnrementioned
Ch . Deninger,
for
higher
the
function.
point
generality
fuIl
Beilinson's
there
whi-ch describes
by S. Bloch,
before
Beilinson's
of
polylogarithm
the
presentation
Unfortunatelyr
in
above,
was an initial
conjecture.
Based on a f irst
P.
proof
This
and makes use of
also
mentioned
has been proven
dilogarithm.
a presentation
d,evoted to
map tD by means of
case n = 2
of
is
Part
Regulators
S1 .
and Val-ues of Artin
Requlators
which \^Ie shall-
The theory
I
for
Algebraic
develop
is
L-series
Number Fields
based on a canonical-
homo-
morphism
r:
given
which
is
when
n = 1 , then
for
each
K 2 r r _ 1( A ) - + I R ( n - 1 )
n
K_ (c) = c*,
1
6:ß -) lR
In hhe general
following
1) .
and
,
facts
is
z r+ log
case the definition
three
r
of
the homomorphism
lzl
the map
r
relies
on the
( s e e t 2 OI ) .
We have a canonical
homomorphism
K-(C)
n ^ ( B G L * ( A ) ) + H ^ ( B G L + ( A ) )= H - ( G L ( A ) , a ) ,
q
q
y
q
the Hurewicz-map.
2).
V ' I eh a v e a c a n o n i c a l
3).
pairj-nq
Hq(cL(c),R(n-1)) x "n(cL(e),%)
',t)
In the
of
continuous
cohomofogy
Hä
GL(A)
with coef f icients
in
lR (n- 1)
"Borel regulator
structed
elements"
1 )
These elements
yield
:R(n-1) .
the
topological
we have canonically
group
con-
an identification
H ä( c L( 0 ) , I R ) = A i l ( v 1, r 3 , v 5 , . . . )
of
the conttnuous
generated
degree
cohomology with
the
by the cohomology classes
2n-1
(see llzl
and t6l)
.
free
exterj-or algebra
u2n-1 =
b2rr-1 of
*(.z, 'T t L )
which
are
invariant
con jugation
under
Gt (A)
on
the
involution
induced
by complex
lR (n- 1) . For the def inition
and
of
u2n-
\ and b 2n_1 vle re f er the reader to Rapoport
1 ,
S 1 . we denote the image of
b2rr_., under the canoni-cal
lzo 1 ,
map
.u. 2
c n - 1 \, n
r *. 7,r, , \ ü, )l R ( n - 1 ) ) - + H 2 t - t ( " " ( C ) , n 1 ( n - 1 ) )
also
by
b2,.-.,
composite
and obtaj-n
our
homomorphism
tbzrr-1
is
called
Now let
be the ring
x :
"Borel
the
F
of
regulator
be a f inite
integers
Spec(F) , xn
Any complex
o*:
no\^r as the
of
'-)
K 2 r , - 1 ( a ) + H 2 r r - 1 ( c r - ,( a ) , a ) - g = J
It
r
imbedding
map".
algebraic
in
F
lR (n-1 ) '
number f ield,
0,
l.et
and let
= Spec(OF) , x(CI) = Hom(F,C) .
cx,: F + C
K 2 r r - 1 ( F ) + K 2 r r - 1( A )
j-nduces a map
by funcLoriality
and we obtain
a
homomorphism
K 2 r r - 1( F ) + K 2 r r - 1( a ) x ( a )
which is
functorial
in
, a r ' ( . . . r c r{ . ( a ), . . . ) c l € x ( a ) ,
F.lVe
consider
the composite homo-
mor,phism
K 2 r r - 1( F ) - + K 2 n - 1 ( a ) x ( a ) g
Ttre complex conjugation
acts
on
x(c)
on
lR (n-1)
on the middle
r
is
this
dule
by
, and thus
compatible with
t
l+
r \^r€ obtain
, on
K 2 n _ 1( a )
and the right
Indicating
a canonical
tD r K2r,-1(r)
and, äs composite with
action.
lR (n-1)x(a)
group.
the
homomorphlsm
+ tn (n-1)x(c) l*
,
(0F) + K2rr_1(F) ,
K2rr_.1
and
fixed
mo-
This
homomorphism is
two fundamental
(1.1) Theorem
map
rD
cal-led
theorems
about
(Dirichlet's
together
with
the n-th
regulator
map.
We have
these maps:
n = 1 , t h e
(c)
% - nX
induces
unj-t theorem):
the diagonal
map
For
an isomorphism
( K 1( 0 r ) o n ) o l R : r [ n x ( a ) r +
(1 .2) theorem
(Borel):
For
n
i somorphism
K2n-1 (r)
For the proof
of
2)
E lR3 tp(n-l)x(c)l*
theorem
(1 .2) we refer
the
results
language
reader
to
tS l
and [6].
we now reformulate
Beilinson.
these
the
of
We set
nf,(xo,e(n)):=
K 2 r , - t ( o ro) o
For the
in
reader
who is
familj-ar
" l C X , e ( n ) ) : = K 2 r , - (1F ) o O .
with
the
general
Beilinson
theory
we Semark, that this designation
is justified
by the
f act that the Adams operator
,!k act s on Kr- _, ( 0f ) O 0 as
zn-l
_
multiplj-cation
by kn because of
r o Uk = Ld,
and (1 .2) .
From this and from
R 2 n ( 0 F )@ 0 = o f o l l o w s
- O for i- + I . lVe set on the other
hand
also
"i txn,A (n) )
1
H ; ( x R , l R ( n ) ) 3 - H o ( x ( a ), c / r R ( n ) ) +
- H o ( x ( a ) , R ( n - 1 ) ) += t n . ( n - 1) x ( a ) l * ,
regarding
gulator
that
C = lR (n-1)
e lR (n)
. lVe then
obtain
a re-
map
1
1
,0, HÄ(un,e(n)) + ^;(xR,lR (n))
Hetween the "absolute
2\
cohomology"
Hl
and the
K . ( 0 , ,) E 0 = K i ( F ) E 0 f o r
i
1 . This
is a consequence of the rocalization
sequence and of the fact
that the K-groups of the finite
r e s i d u e c l a s s f i e L d s of
0,',
groups.
are finite
Vte remark that
"Deligne
cohomology"
appendix
to
the
$2 of
regulator
generally
rn
shown that
Grothendieck
and Borelrs
Theorem:
-
(HÄ(.xz,Q(1))e Q) I
map
, but
varieties
it
the
clearly
sense.
1
extends
Namely,
let
to
over
induces
number fields.
isomorphisms
for n = 1r
forn>
for
arbitrary
finite
rg
denote
of
unions
functorial
in
a
the groups
one of
, HÄ(x,(D(n)) , Hö(xo,tR(n)).
X =
zero-dimensional
is
1
1.
schemes
disjoint
map
H(X)
1
HÄ(x2,0(n))
,O
above situation
over
to
0 , i.e.
Spec (F) . The regulator
twofold
motives
nä(xR,R(l))
X
schemes
quite
by Beilinson
jH;(xR,lR(n))
We have formulated
r^rith
map coinciiles
1
;
n
(Hi(x,Q(n))8R
Spec (F)
this
the
theorem we obtain
The regulator
1
(see also
t2ol
has been defined
arbitrary
From Dirichletrs
(1.3)
is
[2])it
map that
for
.
nl
Amorphism
P : X + Y
of
O-dimenslonal
varieties
homomorphisms
where
p*"=
H(p) -..In
usual
K-theory.
which
then
t!,ro
" = "1
the case
the map
p*
is
in-
transfer
tt
by the
induces
Q
p*
(x)
H
i-t H (Y)
pt
duced by the
of
over
In the
' K2rr-1 (x) + Krrrcase
" = ,l
t
(Y)
the map
p*
is
induced
homomorphi sm
associates
Px : R (n)x(a)
-r lR (n)Y(c)
to
f:
a function
(p*f) (y) =
t
,
X(C) -r lR (n)
_ ,' ( v )
the. function
f (x)
x€P
on
Y(C)
. If;
in
then
p*
and
p*
Quite
generally
tor
maD.
particular,
are mutuall-y
the maps
p*
p:
X + X
inverse
and
p*
is
an automorphism,
automorphisms
commute with
of
the
H(X) .
regula-
The central
description
that
F
result
of
is
the
the
logarithm
with
this
regulator
exposition
map
Q (U*)
field
X = Spec (F)
and
in
rD
of
description
which j-s defined
function
for
the
. This
an explicit
n
N-th
is
roots
obtained
for
atl
of unity
by the poly-
complex
and
s
l"l
@
k
L= ( z )r . I rF
It
extends
in
both variables,
(cf
is
to
a function
For
[18]).
for
which
all
this
defined,
and
z
in
and holomorphic
the region
a- [ 1 ,-)
e € UN-{1}cF and n
Ln(6) -
Viewing
s
is
'
( . . . r l r r ( C r e r) . . . ) c x , : F + c
as an element
(1 .4) Theorem:
For each
of
(a/R, (n) ) x
(C)
, rnrehave
n
1
tr,t uN-{1} + HA(X,Q(n))
such that
e € UN-{1} ,
for
tD(err(e)) = Ln(6) .
hle shall
see in
complete and explicit
X = Spec(A (U*) ) .
of
general it,y of
tation
for
part
cohomology and regulator
of
it
part
in
the values
is
II , S 1 , that this theorem yields
a
description
of the regulator
map for
theorem (1.4) is difficult
and uses the full
Beilinson ' s theory of absolute cohomology,
The proof
Deligne
part
II
of
of Dirichlet
devoted,
will
this
map. We shall
exposition.
L-series,
be obtained
give
a presen-
The consequences
however,
by elementary
to which this
methods.
z
for
52. Regulators
Let
k
Artin
Motives
be a finite
algebraic
number fie1d,
F an algebraic
c1osureandr=Ga1(k|k)theabso1utega1oisgroupofk.Let
X
be a zero-dimensional
varj.ety over k , i.e.
a finite
disjoint
union of schemes Spec(F), where
F is a finite
field
ex_
tension.
Then X gives rise to a representation
of f , namely
-
H(x) = lio(i,0)
rür
= O"ot^'
,
* = X xr. E and n^(*)
d.enotes the set of connected como '
- Ä .').
ponents of
X
To this representation
belongs an Artin
Lseries, which is nothing but the zeta funCtion of
f..
tt is
our purpose to study arbitrary
Artin
L-series
and relate
them
to regulators:
For this reason.we view the situation
of $1-as
attached to the representatj.on
H(X)
and refine
it now by replacing - H(X) - by a -representation
which is associated
to an
arbitrary
Artin motive.
By this we mean the folLowing.
where
Let
E be a finite
algebraic
number fie1d.
--ij.n
--'E
--'is
-a pair
-.
over
k $rith coef f icients
,
where
X
An Artin
- :-
motive
= (X,p) ,
is
a zero-d.imensional
l--homoryqrphisrl _ - -
variety
over
k
and
p
is
a
p: H(x) O E -) H(x) O E
with
in
2
p- = p
of
the E-vector
? T ^( X )
space
H(x)
o
o E = E
. we set
particular
E X = ( X ,i d )
To each Artin
J? t)
motive
M = (Xrp)
we associate
Each connected.
component of X consists
=
X
Sp'ec(F), thenwe obtaln
a bijection
X([) = HoTt(F,[) ]
,
which
assocj-ates
ponent
that
of
f
the
acts
to
a [-rational
assoc j-ated point
on X([)
from
the
point
of
left
the
of
the E-vector
one point
space
on1y.
ff
no(f),
of
X the
scheme
and on no(I)
X.
connected
Note
from
com-
however,
the,right.
n,-.,(X )
= P E
H(M) := P(u(x)sE)
which
have
is a representation
j-n particular
f
of
with
coefficients
tt^
H ( E X ) = H ( X )o n
A m.orphism between
two Artin
=
E
I
in
E . l f e
rür
\A,f
v
motives
with
coef f icj-ents
in
F ,
'f :
is
an E-linear
(X,p) -
f-homomorphism
H(Ey) -r H (Ex)
e:
p o g = g og
with
coefficients
(Y,q)
in
.
fn
E
\n/ay the
this
form
Artin
an abelian
motives
k
over
with
category
M= ru
Stnl
K
Note ,
fn
that
this
we have
wav the
let
and.
A
M
K
the
oppos ite
d.irections
.
functor
v:')
from
go into
f
catesorv
of
V:
K
x P EX ,
,
zero-dimensional
varietles
over
giv j-ng the motives
to the category
i s covariant , thus
M
"
"
character of
spaces . This convention coinci-d.es with
Beilinson t3l and Jannsen t141. It
of t g I . We remark, that
that
k
the
of
i s h o w e v e r o p p o s i t e t o that
H o * M . ( E X , E Y ) H o * n , ( H( E y ) , H ( E x ) ) 3 c n o ( x x y ) s E ,
f
.=
At
where
cHo(xry) =
is
the
free
abelian
components
of
cycles
XxY. In
on
c u o ( X x Y )I
by associating
XxY ,
E =
zno(XxY)
group
over
the
i.e.
the
group
connected
(zero-codinensional)
of
f : no (X*V1 = no {X) r ro (?) - Z'
to a function
E
\
,
,
_ nIo { I, )
+
( g : n ( ! ) + E ) F + ( 9 : n o ( x ) - E ),
s i n c e H( E X )= " ' o
canonically
of
w€ have
fact,
n^ (XxY) r
(z o''
)l'en e Homr,f("(Ey),H(nx))
n . .(, v )
we have
no (xxY)
set
the
map
I
6lxl=
E
(x)
has the ".r'or,1";
{Jl,y)g(y)
tl]"r=
;* , .; € x,
Hom"(H (nX) ,E)
Therefore
the
dual
to
each Artin
Artin
moti-ve
motive
= H (EX)
M =
(Xrp)
, there
is
associated
M= ( x , ü ),
ü
being
the dual- map of
M P H (M)
The functor
H(EX) + H(Ex)
p:
ls
contravariant
and yields
an equi-
va lence
H : M ;
between
cients
the
in
RepE(f)
over
E
of
and the
finite
Artj-n
of
opposite
dimensional
motives
motive
determined
mology
not
by the
presentation
We are
is
simply
selection
over
category
k
of
with
the
identify
f :
of
GalC[ltcl
'-'o
F( e P n ( t ' ) i
with
M
a representation
of
coeffi-
category
representations
E . one shourd. however not
an Artin
is
category
' - ' o
R' e P e ( i )
of
f
I but
a direct
summand of
to
Artin
it
a re-
H(X) O E
now going
groups
"l
to
associate
and
H;
every
as follows.
motive
coho-
Ler
H: V3 + Vec^
K
U
be any contravariant
sional
varieties
functor
over
whj-ch send.s sums into
f : X -+ Y functorially
k
into
products
two
H(v)
where
f*
= fl(f)
and
on the
f*"f*
the
category
category
V;
of
and assoclates
of
O-d.imen-
Q-vector
to
spaces,
a morphism
homomorphlsms
f * .
-H
- *( x )
= 1
if
f
is
an isomorphism.
Let
f f( E x ) : = H ( x ) o n
To every morphism
f
r^/e'assoeiate
as follows.
.
€ Homr(EX,Ey) = cHo(xxy)@E
two homomorphisms
f*
( EX )
H (Ey ) <--+H
f*
The group
(x'Y)
= TT
cHo (Xxy)@E = Eno
E
Z€no(XxY)
I
acts
on
t f( n ( x x Y )) =
H{ f F z )
TT
Z€no(XxY)
componentwise
by scalar
endomorphism
fu
of
multiplicatiorr
then defined as the composites
p
fU
- +r f @ 1
l f( n v )
H ( u ( x x Y ) fru
=
!n.ot
yiel-ds
f
r so that
H(E (XxY) ) . The maps f {< and
f *
an
are
of
pr,
*@1
fl(e(xxY) ,r r- + f{(EX)
Fry,ti-
and f =graph(p)O1€
If
isamorphismin
e:X-)Y
{
C H o( X x Y ) O E , t h e n i t i s e a s y t o
see that
f * = g * E 1 r f * = g * @ 1
we may noh/ extend
the category
M
the
functor
of Artin
H
motives
from the. category
to
v;
M = (X,p) , by setting
t {( M ) : = P * H ( n x 1
We apply this
to'the
functors
H(x) :
1
H 1( X , e ( n ) )
H(X) =
änd
Hi(Xn ,lR (n) ) :
(2.1 ) Definition:
with
M = (X,p)
Let
coefficientö
in
E
be an Artin
p
Then
is
motive over k
an idempotent
in
the
ring
E n d , , ( E X) , a n d v / e s e t
lvl
't
1
H i ( t { , Q ( n ) ) = P * H A ( E X , Q ( n))
1 .
1
H ; 1 l . i i ' R , . Q ( . n. =) )p * r i i ( e x o ' n ( n ) ) .
The fj-rst
group is
an E-vector
space and the
rR E E-mod.u1e. The E-vectorspaces
in
the same \^/ay starting
As an immediate
first
part
tives
over
(2.2)
Theorem:
from the
nf,Wo,
functor
consequence of
of Beilinson's
Q (n ) ) are def ined
tf (x) - a\txo,
e (n) ) .
( 1 . 3 ) we obtain
theorem
conjectures
for
second a free
Artin
arbitrary
rf
M = (X,p)
is
an Artin
in
E , then the regulator
duces isomorn-ntsms
motive over
map
tO
for
k with
x
t D ' ( H l ( M n , e ( 1 ) e) H ( M ) r ) s n3 n / r u r * , R ( 1 ) ) f o r n =
:
mo-
k:
coefficients
tD
the
1
HO(M, Q(n))
E lR
N
- ' H1 i ( M R ' l R ( n ) )
for
n
in-
1
10
From the regulator
motive
V
M
as follows.
an A-module,
a free
maps
then
E-submodule
ff
t
E
we obtain
0
is
a ring,
of
fi-nite
A
rank
of
an E-algebra
(or E-lattice
an E - s t r u c t u r e
Vn
regul-ators
) of
V
the
and
is
such that
V = V E " O A .
u E
Now to
any isomorphism
: V -' Vl
f
A-modules
of
and
VrV'
which
\^Ie associate
V*
k'
.L.t
the
are
endowed with
regulator,
the
the
choice
determinant
of
lVe apply
this
isomorphisms
with
an E-basis
of
a canonical
E-structure,
HÄW%,Q(1)) oH(M)'
right
matrix
Vn
the maps
1
the
the
of
i-s the
element
assocj-ated
and
Vi
to
f
after
.
theorem (2.2) , which are
free- rR @ E-modules . eoth sides äre e q u i p p e d
to
1
and
of
which
Vf
€ g*/9,*
R(f)
gi-ven by
E-structures
side
t0
of
the
left
side
with
1
for n=1 and Hi(M,e(n))
for
n
with
H o ( M R , e ( n - 1 ) ) : - p * ( H o ( X l R , 0h - 1 ) ) @ E ) ,
regarding
that
\
1
' f 'r = H
;(XnrBln)).
!o lxn ra !1 1)l = t o ( n - 1 ) x ( a ) r c+ t n ( n - 1 ) x ( a
(2.3)
tD
Definition:
with
respect
We denote the regulator
of
these two E-structures
by
to
"rrl(1-n)e (ne
and call
it
the
n-th
regulator
S3. L-series
We are now going
to
the isomorphism
E)*/s*
of the motive M
of
explain
Artin
Motives
the second and third
part of
j
e
c
t
u
r
e
s
Beilinson's
con
, concerning the values of L-seri-es ,
here for the case of algebraic
number fields.
The third
conjecture
is identical
with a conjecture,
previously
posed by
11
(which,
B. Gross l12l
(see [25])).
s=O , is
for
We consider an Artin
ef f ic j-ents in the f inite
a representat j-on
in
turn
Stark's
yI = (X, p)
motive
algebraic
conjecture
over
number f ield
k
with
yields
M
E;
co-
p : I + Aut"(H(M))
of
f = Galt[lf<)
on the finite
space H(M) . For every
o € Hom(ErC)
galois
the absolute
mensional- E-vector
group
diwe ob-
a complex representation
tain
p o : f - >A u t n ( n ( r t r ) o ) ,
H ( M ) o = H ( l , t )o c ,
Ero
to
which
\^/e associate
t{
(r-;;i ,=; rurf) -"; (H(M)o) )
nder
runs over H; finite
prirnu." ";
k and e^,- j-s
E tfrat
'decompositj-on
in a
group f* c f - over
: L ( M or s )
&
element
mapped
onto
Frobenius
group
To the
by
the
inertiä
the
L-series
=
) =
-
Here
the
Arti-n
in
Tp
o
motive
automorphism
A' )
6
.
M
,
g
I
in
4
v/e associate
t^e/Ita., I p
'
0
the
an
is
being
L-series
of
M
-
settl-n g
L ( M r s ) = ( . . . t L ( t t t o r s )r . . . )
o€Hom(ErC)
Thts
is
a meromorphic
To be precise,
every
function
on
o € Hom(E, C)
with
C
values
induces
in
C g E.
a homomorpfri8m
o * : C E E + G , z E a r rz. (oa) .
We obtain
in
this
"lt.-")-
C @ E =l|Tc
way a decomposition
€ c o E
*L-serl".
is
defined
and the
by
;;tlM,;i-=ilMo,'r .
Our aim is to study the valu e s o f t ( M , s ) a t t h e i n t e g r a l
points
s =m : O . The order
d (M) of L(Mo,s) at s = m is
mindependent of
o and i-s given by
n)
w e s t r e s s that
\^7e have
('f
.
putr
-1
og and not
o f L ( M o , s ) . S o L ( M " , s ) i s not the u s u a l
o
H( M ) b u t o f the dual- of H ( M ) o .
g
t
Artin
in
the
L-series
definition
of
12
q.g
t
(M)
dir5H
E
8ldm (M) =
dimrH (M)
u
E
älE
r
dimuH (I',1)
is
well
't+
E d i m , ( H ( M) / H t * l t t
u
8l--
dimrH (M)
u
functional
factors
equation
contribute
On the
tion
known
of
the
(see
1251,
CH O, 56)
of
L(MorS),
which
the
group
cohomology
M =
(3.1) Proposition:
Let
efficients
Then,
E .
(
1 (MR
a
L+
runs
(t
only
s h o w s that
H] tu*
(Xrp)
motive
be an Artin
Let
S = Spec(k).
co-
with
ly
canonical
s ) 8lI t 'dtO
I compr e x
the
f-
descrip-
explicit
rM
ö ) E ]R
rM
H( M) o e S H ( M ) / H( M )
over
the
(n) ) :
,n
k
for
n odd,
)@R
for
n even,
and
TM c f =
o
E-.
rear
in finite
primes
G a 1t t c t t < ) i s a decomposi r i on grou p over
Proof:
from the
a n d follows
following
the
L
where
) for m<o odd
l r O H( M)
,lR (n) )
mcO even
order.
hand r w€ have
in
for
rea I
to
other
m=O
tt
8lcomplex
This
for
of
k
I
X + S
The morphism
yields
a map
x(a) '+s(a)
and $re let
fix
X ( C)
s
an extension
be the f ibre
ät [ '+ c of
s € S(A) = Hom(k,c). We
over
k - ' C . Then
s:
d
s
induces
a
bi j ection
x ( F , 13 1 ( a ) s
The idempotent
and
p
ind,uces
fr'("*):= Ex(c)s
have a commutative
endomorphisms
as described in
p* l,l
ä*,
(ä*t)(x)
-
f (3"x)
Ex([)
l
(c)
Ex = --5
.
J,n*
nx
p*
of
f l ( E X ): = e x
5 2 . By functoriality
diagram
Ex(a)s
where
X H sox .
,
(k)
(k)
we
I
i
13
Let
now
be an infinite
tü
be an imbedding
de f ining
prime
of
"Ut k - r C
and let
lE . Then
" l t t l * , r R ( n )) = p * ( t n ' ( n - 1 ) x ( a l) * I E ) = p * [ E x ( n ) " * .( n - 1 )] + =
x (' c' ") 8. 0 8x (' -a' s) 8- ) o n ,
_
O p * [ n x ( a ) s , E r R( n - 1 ) ] + e @ p * [ ( n
{l -ülrea I
(n-1)]+
complex
Let
be real
I
diagram
tive
X(C)^
E
s = =t
and let
1 , , r . o , /4t -
s
E lR(n-1)
E*al/Qni)"
and consid.er
n - 1 ' r e x ( [-)
l
]R
I
rR.
e.f
r/
E
I
t
p.I
X(C).
the commuta-
E lR(n-1)
ä ** o@
11
// (t 2
- ni)"
E
Since complex conjugation
; (O)
on
of
-
n - 1' ,
rx([)
s
corresponds
to
the action
the non-triviat
element
,g
on x ([)
under the
eg in
+
map äo : x([)
x(c)s
r \ a r €= 3 " t h a t t h e c o m p l e x c o n j u g a t i o n
on the left
side of the d.iagram corresponds to the action of
(, - 1. .)I-I-- ' l * X o n t h e r i g h t
sid.e. Regarding that
are compätible with these actions , and that
as lra-modules we obtain
o
the tgo maps p*
(k ) =
p*gx
H (M)
x (' G
- ' )s.
.n-1
n odd
[ "- ,- *' -l k- '" * . ,
@ rR(n-1)]+=(p*EX(k))(-1)^^ @rR=1
rr0
([,r)/H (M)6 E]R,n even
LH
where the exponent (-1)n-1 denotes the eigenspace of
gp with
p*[n
ej-genvalue
Let now
be the pair
(-1)n-1
I
of
o
.
be a comprex prime and let
s = =X, 6, k + c
con j ugate imbeddings def ining g . donsid.er the
homomorphism
x(aD
) ^o
( n
given
E
x(a);)
o r R( n - 1 ) e : d
E x ( [ ) @r R
by
s*(f + s) =
+ (-1)t-1ä*g)
]tS*r
The fixed
given
module of
complex conjugation
by the- elements
f(*)=f(;)
the restriction
module yields
1
-
on the left
sid.e is
(c)
s and
f € Ex
+ f) , where
iq
S i n c e s * ( + ( f+ f l l = 3 * t = f
of
the map
an isomorphism
s* @ 1/(Zni1n-1
o 3 , w€ see that
to this fixed
14
x ( c )D^ o E x ( a ) = )
Applying
t ( n
E lR (n-1) l+ = Ex(k) E lR .
as above the maps
p*
x ( c )D ^
p*[(E
CIE
fnserting
these
1
H0 (MR ' lR (n) )
x ( c )- ):
([)
I F. (n-1) l*:(p*nX
lr*,
results
into
\nte obtain
of
the L-function
theorem
(2.2)
jectures
for
we obtain
arbitrary
(3.2) Theorem:
the
the
Taylor
series
of
the proposition.
of
o
motives:
n
we have
-
ord__n _L(Mrs)
s=l-n
l_n
composi tion
the results
on the pole
L(Mrs)
at
s = 1-n , and applying
the second part of Beilinson's
con-
Artin
the
= H( M ) o ] R
( 3 . 1 ) with
For every
l{e now consider
above d.irect
assertion
the
Comparing proposition
order
we obtain
first
d i m " H l( M n , e ( n ) )
non-vanishing
coefficient
o
expansion
L(M,s) = ao(s
at
the
point
s = 1- n
(c E E)* = IJo* ,
. This
where
L(Mr1-n)*
coefficient
=
lim
s+1-n
"value"
1
(s-(1-n)) n-r
this
we no$r haver äs a special
t s general conjecture
:
value
Beilinson
of
by
'urlarDl
it
the
an element
o € H o m ( E r O ) . We d.enote it
and call
briefIy
is
of
at
L(M,s)
case
of
(M)
s = 1-n . For
the last
part
of
15
(Gross):
5)
(a o E)*/E*
,
9onjecture
For every
tl[(1-n)
where
IU
is
For the
6 (k's)
the
dual
trivial
, this
'= L(14,1-n)* mod g*
motive
motive,
conjecture
n
of
M .
i. e . for
the
Dedekind zeta
has been proven by A. Borel
function
(see [5]) .
t(M, 1-n) + o by theorem
,IR (n) ) = o , then
(3.2) and the conjecture is a special case of Deligne's conjecture
t g ] on the "critical"
values of motivic L-series.
Here it says that
hthen Hltu*
t(M,1-n) e s* c
and this
these
is
in
algebraic
fact
true
(A O E)* ,
by the results
of
Siegel
values
resurts.
$re have further
going
For exampre, let
G = Gal(Q(u[) lOl
group of
the
the, canonical
with
.e,-th cyclotomic
isomorphism
f ierd..
and let
lzZJ.
con j ectures
For
and
be the galois
e : G ; (z/ g) * be
Let
M(i)
u"
the Artin
motive
character
0i : G -+ (n/[)*"*
e ( Up , - l
c eg , i even.
(i )
with
n = i mod(g,-1) we then
, 1-n)
obtain an interpretation
as multiplicative
Euler characteristj-cs in .Q,-adic cohomology (cf t 1 I )
For the values
L (M
l r , t l n t i ) , 1 - nl n) =
T-I
j
# Hi1i'4(il
S. Lichtenbaum
has proposed. conjectures
s ) . hle
remark,
that
map
P
C E
a
( . . .
E =rJC'.
r ] d ,1 . .
the imbeddins
that
E lgg
)o€Ftom(8,c, after
pred.ict
C E E
the identi
formulas
becomes
fication
the
16
of
this
type
a most general
in
Of particular
goes back to
conj,ecture
vial
interest
ESpec(k )
motive
function
H. Stark
t t 0I )
case s = O , for
the
l, (M,s )
k . At
s = 1
which
IvI is
. !{hen
t25J
then
of
6(krs)
is
(see
frame
the
the
tri-
Dedekind zeta
is
the
it
has a si-mp1e pole with
residue
zt
6 ( k , 1 ) *=
where
h
in
and
k
is
the
R
class
s = O via
the
rect
of
the
functi-onal
trivial
H(M)
the
has rational
an arbitrary
at
proof
the points
cases=O
of
Dirichlet
of
this
roots
of
unity
is
the classical
formula from s = 1 to
we get
= R mod q* = n(rp) .
len
and s = O
class
our conjecture
number formula.
values,
Artin
The main purpose
son's
. This
then
the
(see t25l) . There is
been proven by Tate
for
k
equation
motive
consequence of
of
Transforming
C ( k , -o ) *=
So for
hR
e
number, e the number of
the regulator
number formula.
class
1 ( 2 n , 1 2t
Stark
If
is
the
a di-
character
conjecture
no proof,
has
however,
motive.
of
this
exposition
is
to present
Beilin-
the Gross conjecture
for the Dirichlet
L-series
s = 1-n
O . The method does not apply to the
f o r w h i c h h r e have fortunately
a classical
proof
(see [ 25 ] ) .
S4. Dirichlet
L-series
character,
and let
f
.Let |: n/Nn + C be a Dirichlet
b e t h e c o n d u c t o r , so that
X comes from a primitive
character 'X ' mod f . The link between the Dirichlet
L-series
- : l
1 7' :
L(x,s=
) ; ry
k=1 kj-s given
regulators
and the
, Re(s)
polylogarithm
by the
o
o
k
L = ( z )- . E .
=
k=1 k"
which
hre have introduced
in
extends
for
(see tt8l
s + 1
L" (e2nix'
(Or1) to a continous
from
on R/Z
function
L e m m a 7 ) a n d \ ^ r es e t
Z ( X , s )= :
a€z
For this
(4 .1)
,
The f unction
S1 .
z= (x) -
function
function
Lemma:
/N
X ( a ) s. Lt r" ( f t )
h/e have the following
CH. I.
52)
\ ^ I eh a v e
s € A- { 1}
For all
(see ttS1,
N s - 1 2 1 xs, ) = T - Tt 1 - x ( p )p = - t ) t = - 1L ( X ' , s ) .
plN
plt
Proof : By a straightforward
that
for
pohrer series
one proves,
computation
m
L
=(z)
- ms-1 :
L =( w ) .
v
t*="
For the
L=,
function
{= (x) = ms-t
(- i . e .
L^ (x)
s
have the
factors
is
this
implies
- m-1
(*)
N
-r Q
R/n
a
It
N = pf
is
same prime
through
direct
a distributj-on
the
,,]o
of w
- 'e
- -i Jg-h
-t
.
s-1 ) . If
f actors , then
function
consequence of
X':
(*) .
will
,
Lr(St
nohl suffice
to prove
(prf) = 1 . lVe have
f
and
Xz% /N -+C
the f unction
-) c
z/f
and our formula
the
( x ,= ) = N s - t
Ns-12
I
formula
a€Z /N
= Ns-1 r
for
the
case
( a ) .s( (- f=t )
r
(x)zs (ft)
x
b€z/f
x:tr/Ä
18
Now X(x) -X'(b)
x=bmodf
for
a = pc for some c
X(a) = O if
:
Ns-12(x, ")
1
x ' ( b )N =
E
(*)
Applying
the
second the
Ns-1 :
D r\
/N
a€% /N
x=b (f)
m = p
for
to
the
whereas
. Hence
Z^ (*)
r
x€E
b€.n / f
(xrp) =1,
and
x ' ( a ) L ^D ( r\3 ) .
a=pc
first
change of variables
sum and making in
a = pc
c € n/f
where
v/e obtain
' - 1 I - =( ? )
x ' ( b )f
e !
N s - 1 x - ( xs ,) = E
b€n
=
/f
(t
(4.2)
as
c€%/f
x'(p)p=-t)t=-1 x-(X'rs)
The Dirichlet
logarithm
x ' ( p )p s - 1 r s - 1 E . l x '( c ue, " ()f
L-series
E
g,.e.d..
,
m a y n o T , vb e e x p r e s s e d
by
the
poly-
follows.
Proposition:
For
a1t
s € A-{1}
\^/ehave
r , ' ( f , s ) = g ( X , s ) . L ( X , s ),
where
e ( X ,=s") # t ä l = - r m #
,
ptrt
t(X)
Ueing the Gauss sum
t(X)
Proof :
Sj-nce both
suffice
to consider
X
is
sides
-
f-1
. ,: X(a)"''ITLa/r .
a=O
are holomorphic
in
the case Re(s)
i.e.
f = N . Then, for
primitive,
s , it
will
k
r
X(a)"2niaklN = t(x)x(t<).
a€% /N
(k,N) -
1 , this follows from X(a) = X(akl XtLl , and if
( k r N ) + 1 , t h e n b o t h s i d e s a r e z e r o . N o \ a /d i v i d i n g
both s ides
and summing over
by ks
k
If
since
the
E
^ (. 2 n i a l N ) = t ( x ) L ( x , s ) .
X ( a ) LD
.a€n/N
- zs(ft), this is the desired. formula
L^("2ria/N)
S
case
that
X
is
primitive.
In
the
general
case
for
\ ^ r 7 eh a v e
19
r ,( f , s ) = T - T ( 1- i ' ( p )p - s )L ( x ' , s )
p lN
ptrf
1
(\ 1 - i ' ( p )p - s ) . L ( x ,' s )
+
T
T
t(xl pl*
p/.f
The proposition
Bernoulli
The generalized
the
nov/ f rom (4 . 1 ) .
follows
B
are
IIrX
defined
by
expansion
f
E
a='l
They are c o n n e c t e d
B n € o
ze
^ t ä t
where
with
n
E
(cf
the
negative
(4.3)
(i)
ordinary
l-'
integers
fheorem:
For
n
z
rlrX n l
D -
n=O
Bernoulli
numbers
f
E. X (a) tner,(T)
,
a=l
is
l-
F o r t h e values
[13]).
@
Z
(?) e..* n - i
i=O
mial
z
eTz-1
by
er, (x) =
a
- - :
BD ' X = rI
the
numbers
the n-th
of
Bernou]Ii
the L-series
polyno-
L (X, s)
at
we noTn/obtain:
n
x ( - 1 ) + ( - 1 ) n - 1 , then
Bt,
x
L ( X, 1 - n ) - + o
tf
n
(ii)
If
x(-1) :
L'(Xr1-n) -
(-1)t-1
a(!
(2ril t-1
where
a(X) - !=U|
Proof : It
is
well
known ( cf
, then
t(Xr1-n)
= O
and
L(x,n) + o ,
- x ' ( p )p - n
Nn-t 11 1
p l N 1 - x ' ( p )p t - 1
plf
t 1 3l ) tfrat
L(X,1-n) -
B
n'X
n
for
every n
x ( - 1 ) + ( - 1 ) n - 1 . s o u = = r r f i Ä *x ( - 1 ) = ( - 1 ) t : 1 . T h e f u n c r i o n a l
equation for the Dirichlet
L-series may be written
L ( x , s ) = + t ? * [rl =
2io
L(x-'1-s)
r(s)cos|(s-d)
20
6 = O
with
p.
-1
(cf [13],
according X(-1) = 1 or
X(-1) = (-1)t-1, n-6 must be odd and there-
or
1
1o4). Since
fore
cos TT
s = 1-n
has a simple pole at
f(s)
and
r,(frn)
the
in
1-l-ö = (-1) (1-n-6)/2
2
\
.
,
.1-n-ö
\-'1.
(n-1)
nor^/ each
at
s =
functional
in
equation
a Laurent
'
!
residue
a s i s w e l l - known. Expanding
+ Or*
n t O-1
,
with
series
'
term
1- n ,
\^7eSee that
,,T'
2 t r, 1 - n
L ' ( X , 1 - n-) t +
(n-1) !
, - \ n-1 .1-n-ö L(xrn)
( - 1 ,
2i-"
= t ( x ) (n-1) !
2
Inserting
tain
in
number
f iel-d
to
an Artin
X
the
(4.2)
of proposition
for
s = n
we obcl
formula.
lrle no\^r consider
values
fn-1r,(f,n) + o .
(2nD t-1
the result
the desired
L
a character
group
multipl-icative
E
and extend
motive
it
over
by
A
+ g{c
(n/N)*
X:
of
a finite
O
to
with
algebraic
n / N.
values
takes
that
lVe as soc iate
E
in
as
fol]ords.
and
F = Q(UN) , UN the group of n-th roots of unity,
G o n x = S p e c( r )
The right action of
c = Gal (r lOl
Let
let
induces
a, left
action
on _
H(x) - Ho(I,e) - ,no(x)
thus
and H(EX) - H(x)sE E
n (X)
o '
-E;
I
an identification
EIc] = End",f(H(Ex)).
Vj-a the
canonical
as a character
of
isomorphism
G , and $/e consider
€.,=
A
r x-1(r)'
"X
,
is
an
an Artin
defines
n[c]
. the
X
element
I
no (X) + E}
( e . . f ) ( x )-
P x =1 r ee -X.
in
we may view
t€G
H(EX) = { t:
which acts on
N) *
(n/
G:
-1
r x '(t)f
T€G
in
idempotent
by
(xt)
o
"tdu, f (H (EX) )
and therefore
motive
M = (X,pX)
over
txl
A
with
and call
coefficientsin
it a piric$e'!
E.We
m-otive.
denote
this
motive
by
21
we assume F c 0 . we then have a homomorphism r =
GaltölO) n c and a distinguished point
; € spec(F R Ol (Lhe
q/
kernel of F I 0 + 0 , a8b r+ ab) by which hre get a 6ijection
63ne(x), r *;P
t:no(X)+e
, of right
G-sets. Associating to a function
I f (xt
r
the element
morphism
H(EX)= E
This
is
an isomorphism
on
E[G]
the
endomorphism
€ E[G]
no_'( x )
we obtain
lef t
H (EX)
of
an iso-
= Etcl
f -modules , if \^re let
n(o)
by
from the left.
of
as multiplication
pX
- 1' ) t
becomes after
the
o € f act
Therefore
identifica-
just multiplication
tion with
E[C]
of
E[c]
by the element
1
1
1
=
€ E[G] . The associated representation of
m.X
+G,:X'(t)t
I
is the 1-dimensional
subspace
erElGJ = ""X
of
E[G]
on whi-ch
presenlation
X
f
acts
associated
via
with
the
the character
Dirichlet
X . So the remotive
is
tXl
f + G + E *
We noh/ consider
the
r , ( t x ,l s )
of
the
=
L-seri-es
(...rL(Xors),
motive . On the
Oirichlet
I-(lxl ,s) =
E
_1
polynomial,
primitive
Nn-tfl % '(p)p'^
plN 1-1
PIt
€ Qlxl
is
n
ilo
f the
hand \^re set
z e" t * g l o x ( a ) e c I E ,
a(txl) = +)]
Here
other
o € H o m( E , C )
?€n{N
a _€
=+
i
" n" tr , r # t € E ,
x
(
a
)
r
t
t
f
a=1"!s''!
R
"nr[x]
B r ,( x ) =
' )
(ll er"n-i
conductor
of
the n-th
X , and.
X'
_h
I
Bernoulli
the associated
character.
E + CI | z E e r> zoe,
E
identification
The homomorphisms o*:
o € Hom(E7C), yield
the
C
C E E
= ll c,
o
€ E.
22
bywhichthesubspace
E= 1 EE:CEE
b e c o m e st h e i m a g e
of
E -r TIA , e F (... 16€r...). After this identification
rlvrehave with the notions
cf theorem ( 4 . 3 )
l , ( t x ] r s ) = ( . . . , L ( x o r s ) , . . . ) o € H o m ( E r c )|
BrtrIX]
= (""8t,
a(tx])
= ( . . . r a ( X o )r . . o . . ) o € H o m ( n r c )
lrle may therefore
(4.4) Theorem:
refonmulate
Let
")o€Hom(Erc) t
xo""
( 4 . 3 ) as fol lows
theorem
n
(i)
r ( t x l , 1 - n ) = - B n 'nI x l + o
(ii)
rf
x(-1) + (-1)t-1
if
X(-1) = (-1)n-1 , then
L( tXl ,1-n) - o
and
L ' ( [ x ] , 1 - n ) = a ( t x l I - !- l ( t x l , n ) + o
(2ni)^'
with
€ E c E @c .
a([X])
$5. Regulators
of
Dirichlet
Motives
!
Let again F = Q(U*) , where no\,v N >,1 . Let G =
and
G a l ( r l O ) , x = s p e c( F ) , € . , = E x - 1 ( t ) t € E l c l
l,(txl,s) =
^
t€G
E
l ,D. (tf\ t ) E x ( a ) m o dE ' F.
a€%/N
In
the
preceding
values of
let
rrre have explicitely
s = 1-n
L(tXlrs)
at
the L-series
motive
tXl = ( tXl,+a
now determine
computed the
paragraph
"X )
the regulators
. By means of
theorem (1 .4) we
.IX, (1-n)
(5.1) Theor:em: The n-th
regulator
.IX1(1-n)
n > 1 , of
motive
iÄ"given
the Dirichlet
of ttre Dirich-
tXl
e (IR@E)*/E*,
by
23
"[x1(1-n) -
(i)
1
if
X(-1)+(-1)n-1
(ii)"[x1(1-n)="fu0(tX1],')modE*,!fX(-1)=(-1)n-1.
Proof:
We first
of
the E-structure
explicitely
determine
1
1
H ; ( [ x J * ' n ( n )) = " i ( H ; ( X ß , ] R ( n))E E ) .
Since
G
acts
r € G
a c t o n
X
on
X(C)
and
from
the
right,
elements
the
" [ t x n , n ( n ) ) E E = t a l n ( n ) x ( a )E E ] + = t c l n ( n ) E E x ( a ) r +
(see 51, p. 4) . Directly
( see S2 ) \^/e see that the endomorphi sm
1*
by
nition
"i
given
is
from the defi-
from the left
1
1
. Hö(xn, ,lR (n) ) E E -+ ,ö(Xß, ,rR (n) ) E E
by
ei = E 1* @ X-l(t)
X
t€G
lrle choose a primitive
N-th root of unity
.
and \^Ie l-et
e
cr: F + C
imbedding
be the
sider
the bijection
induced
at, = "2ni/N
by
determined
G + X(C) ,
of G-sets
isomorphism of
o*: nx(c) 3 Etcf
left
EIG]
,
(f : x(c)
. we may then
r t> o orr
con-
and the
modules
-+E) r, r f (crt-1). .
T
Complex conjugaQion
cation
with
X (A)
on
c € G r c!
e +
becomes multiplication
by
becomes multiplication
by
and
if
In
a € (n/N)*
'particular
corresponds
then to multipli-
e-1 , the action
of
1*
on nx(a)
(a)
on nx
and the operator
eI
X
- 1' ( t ) t
lVe have
"X = E1
€ E[G]
r
e..E[Gl = Ee
X
X
"xr = ."x = X(a).*
corresponds to
e_c = ce
X
r € c
under
= X(-1)e.. .
X
X
(n/ N)* = G
24
By the
above
h / e obtain
identifications
" l t X , e( n ) ) o n 3 ,ltxrR , rR(n) ) o E
I
I
I
g*
t a l r R ( n )E E t c l l
$
+
I
*
J'
.1, X
diagram
a commutative
l1E'x
X
v
H l r r x, Ql ( n ) ) 3 ^ l r r x l o , n( n ) )
t c l n ( n ) @E " * l * .
S
complex conjugation
is multion A/lR' (n) E E .X
t
1
(-1 )
plication
by
by
X (-1 ) , since it is multiplication
(-1)n-1 on a/rR(n) = rR(n-1) and by X(-1) on E "X .
Therefore
tC/n (n) E E e .l+ = O and thus
X=
if
"IXr(1-n) 1 ,
X(-1) + (-1)n-1
The action
of
Assume X(-1)
= (-1)n-1
. Then the E-structure
t a l r R ( n -) E E e l + = C / R ( n ) O E " X = *
of
(n-1) O E .X
x-
(2ni)t-1e
1-dimensional E-vectorspace
O E "X with
basis
o-. = (2ri)t-1
@ "X . (vtorking with
a/R (n) irr place
X
\ ^ r eh a v e t o i n t e r p r e t
of IR (n-1)
2nj- as 2ri mod n.(n) ). In
is
the
order
the regulator
"[X1 (1-n)
we must choose
-dimensional
E-structure
n' X 1 of the 1
to determine
a basis element
r
HÄ([X],4(n)) of
HÄ([X],Q(n)) o IR . rf
o*t7(n*)
then
.
ment
= .rX
,
after
a € ( l R @E ) *
c r - , . , ( 1 - n ) = a m o d E { < . N o ! , /t h e o r e m ( 1 . 4 )
txl
1
e r , ( 6 ) e H Ä( x , Q ( n ) ) a n d w e s e t
this
choice
,
yields
the e1e-
1
n x = " i ( e r r ( e ) )€ H Ä ( [ x ] , Q ( n ) .)
(Note that
N
is
assumed to be
By (1 .4 ) we have
t1(err(tr))= Ln(C) = ( .. .,
From this
\^te obtain
the
"[x1 (1-n) -
>1 r so that
lrr(re),.. .)r
desired
( 2_n* ir )- "
e € UN-{1 }).
(c/R (n))x(c) .
: F - r G€
result
'
t( tx
1],r)
mod E:F,
once v / e h a v e s h o w n t h a t
lX + O. For this ii suffices
"
show that
_
.q,(tx-11,r) E L ^ ( e 2 ' n j - t ) o x - 1( " )
a€.%/N
rr
to
25
A/R (n) E E . B y ( 4 . 4 ) w e
considered as an element of
g(tX-1 l rt) + o as an element in
c I E . Therefore
have, that
1l,t)
e ]R (n-1) E E = c @ E
l,(tx
it sufficäs to show that
is
+O
Now for
each summand Lrr( "2nia/N)
L r ,( e 2 n i
o x-1 (.)
( - a )/ N )
w e have
E x-1 (-.)
L r ,( e 2 n L a / N ) o x - 1 ( . )
*
L r r ( e 2 n L a l N )o x - 1 ( . )
+ x-1 (-1)Lrr(.2ni(-a)/N)
e X
- 1'
(a) =
o x - 1( . ) .
( L r ,"(2 n i a l u ) e x - 1( - r ) m )
ta/N)
2 TLr a / N )
we have
e x-1 (-1 ) rr,("2r
For the term L :: Ln (e
T = X ( - 1 ) t = ( - t ) n-1 L and hence L c lR(n-1 ) = C . From this
E
follows [(tX-1] ,n) € lR (n-1) O E c C E E .
-1
Regarding that
[X-']
is
the dual motive
-V.
tll
of
[X],
v/e
(4 .4 ) the
above theorem and from proposition
(which is the third part
that the Gross conjecture
result,
motives,
holds true for Dirichlet
conjectures)
of Beilinson's
n = 1 and the case
if \^/e take into account, that the case
obtain
from the
X = t
has been established
(5.2)
For every Dirichlet
Theorem:
coefficients
by Dirichlet
in
E
and every
n
and Borel:
motive
t Xl
\^7ehave
. f 1 1 ( 1 - n )- t ( [ x ] , 1 - n *) m o d E * .
over
0
with