The Bloch-Kato conjecture on special values of $L$-functions. A survey of known results

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

G

UIDO

K

INGS

The Bloch-Kato conjecture on special values of

L-functions. A survey of known results

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 179-198

<http://www.numdam.org/item?id=JTNB_2003__15_1_179_0>

© Université Bordeaux 1, 2003, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

The Bloch-Kato

_conjecture

on

special

values of

L-functions.

A

_survey

of

known results

par GUIDO KINGS

RÉSUMÉ. Cet article

_présente

un survol des cas connus de la

con-jecture

de Bloch-Kato. Nous ne cherchons _pas à _passer en revue

tous les cas connus de la

conjecture

de

Beilinson,

et nous laissons

de côté la

_conjecture

de Birch et

_{Swinnerton-Dyer.}

L’article com-mence _par une

_description

de la

_conjecture

_{générale. À}

la

fin,

nous

indiquons

brièvement les démonstrations des cas connus.

ABSTRACT. This _paper contains an overview of the known cases

of the Bloch-Kato

_conjecture.

It does not _attempt to overview the known cases of the Beilinson

_conjecture

and also excludes the

Birch and

_{Swinnerton-Dyer}

point.

The _paper starts with a brief review of the formulation of the

_general

_conjecture.

The final _part

gives

a brief sketch of the

_proofs

in the known cases.

Introduction

This is an extended version of a talk at the "Journ6es

_{axithm6tiques"}

at

Lille in

_July

2001. Our purpose is to _survey the known results of the Bloch-Kato

_conjecture

on

special

values of L-functions. We restrict ourselves here

only

to the cases where

actually something

is known about the L-value

itself and not

_only

_up to rational numbers

_{(Beilinson conjectures).}

For the Beilinson

_conjectures

there are other _surveys

_(see

[34],

_although

there is

some _{progress since} then

[28]).

The known cases concern

_only

certain number fields

(or

more

gener-ally

Artin

_motives),

certain

_elliptic

curves with

complex

multiplication

and

adjoint

motives of modular forms. Moreover all cases where the exact

formula of the

_{Birch-Swinnerton-Dyer conjecture}

is known

_{give examples}

of the Bloch-Kato

_conjecture.

Since we do not

_attempt

an overview of the

Birch-Swinnerton-Dyer

conjecture

and there are _many

_special

cases checked

individually,

we will say

nothing

about this.

The structure of the _{paper is} as follows: In the first section we review

briefly

the formulation of the

_conjecture

for Chow motives. As our

_K-theory

knowledge

is

_limited,

we also formulate a weak version of the

_conjecture.

The second section reviews the known cases and has

_naturally

two

_parts:

the first treats the case of Artin motives

_(mainly

abelian)

and the second

treats

_elliptic

curves with

_complex

_{multiplication}

(mainly

over an

_imaginary

quadratic

field).

The third section

_{explains finally}

the main

_ingredients

of the

_proofs.

Here two

_things

are decisive: on the one hand

techniques

from

Iwasawa

_theory

and the main

_conjecture

and on the other hand motivic

polylogarithms

and their realizations.

I would like to thank the referee for _very useful comments.

1. Review of the Bloch-Kato

_conjecture

In this section we recall

briefly

the formulation of the Bloch-Kato

con-jecture,

referring

for more details to the articles of Fontaine

_[13],

Fontaine and Perrin-Riou

_[14]

and Kato

_[23].

We follow

_mainly

the excellent survey

[13].

To understand the connection with the Iwasawa main

_conjecture,

_[23]

is

_{indispensable.}

1.1. Motives and their realizations. Fix two number fields K and E

and consider the

_category

of Chow motives

_{M K (E)}

over K with coefficients

in E as defined in

[21] §

4. To each

_object

M =

{X, q, r)

_E where

is a

_{smooth, projective variety, q}

an

_idempotent,

and r E

_Z,

are

associated several realizations:

E the de Rham realization

_equipped

with its

_Hodge

filtration .

the Betti realization each

sum-mand is

_equipped

with a

_{pure E}

Q9

_R-Hodge

structure over R on

MB 0o

R.

Q9Q E

the

_p-adic

étale realization with its

_Gal(K/K)

_action,

where _p is a

prime

number.

Remark. Note that one

_expects

that there is a

decomposition

in

_A4K(E),

such that the realization of

_{l~~’+2’’(M)(r)}

is

etc. In

_particular

it should be

_possible

to restrict attention to _pure motives

hw+2r(M)(r).

This is known for curves

[32]

and for abelian varieties

[31].

As we do not know it in

_general,

we have to use the above realizations.

For a motive M =

(X, q, r)

_we define its n-th Tate twist _to be

M(n)

:=

(X, q, r

+

_n)

and let

M~

:=

(X, qt, dimX -

r)

be the dual of M if X has

E the

_subgroup

_of

_{the Chow group of X}

_consisting

of

_cycles

homologically equivalent

to 0. We define the

_following

_objects

for M =

(X, q, r):

t

- ,-,- , -

_

this is motivic

_cohomology

- --,- I

-Recall that one has Chern class _maps defined

by

Soul6

where

_H¿ont

_(K,

_{... )}

denotes the continuous Galois

_cohomology.

_Together

_ .., .. - I "

-with the Abel Jacobi _map

we

_get

for all _p. Consider for all finite

_primes

p of K _{the group}

_Hf(Kp,

_Mp)

as in

[13]

3.2. This is a

_subgroup

of

HI

(Kp, Mp)

and we denote the

_quotient

_by

Mp).

Then let

and define

Conjecture

The E vector _spaces

_{H°(K, M)}

and are

finite

dimensional

b)

The _{map rp} induces an

_isomorphism

H)(K, Mp)

c)

H°(K,

M)

Q9Q

H°(K,

Mp) (Tate conjecture)

On the other hand there should be the

_following

relation of

_{H) (K, M)}

and to the Betti and de Rham realization: Consider the

com-parison

isomorphism

this induces the

period

map

where

_M£

:- are the invariants under

_complex

Conjecture

1.1.2. There exists a

long

exact sequence

The _maps are

_given

_by

the Beilinson

_regulators

_roo as in

_[13]

6.9.

1.2. The

_conjecture

of Bloch and Kato. Consider the motive M =

(X, q, r).

Denote

_by

detE

the

_highest

exterior _power of an E vector space

and

_by

_detEl

its dual. .

Definition 1.2.1. Assume that the

_conjecture

1.1.1

_{a) is}

true and

_define

the

_fundamental

line

_following

Fontaine and Perrin-Riou: Let

The fundamental line zs then

where are the invariants under

_{complex conjugation.}

Now we want to

_investigate

the

_special

values of L-functions. Let us

define an L-function for M: first consider a

prime p

of K not

_lying

over _p.

Put

_Dp(Mp)

:= where

_7p

is the inertia

subgroup

at _p.

_{For pip}

we let

Define an Euler

_product

over all finite

_{places p}

of K

_by

where

_Frp

is the

_geometric

Frobenius at _p.

Definition 1.2.2. We say that M has an

L-function, if

.for

all _p and

_{L(M, s)}

_converges

_for

Rs » 0.

Note that

_{L(M, s)}

is an E

_0Q

C-valued function. The order of

vanishing

of

_{L(M, s)}

at s = 0 is

conjecturally given

_as follows:

Conjecture

1.2.3. The

_{L-function of}

M exists and it has a

meromorphic

Let us define the

leading

coefficient of the

Taylor

series

_expansion

of

L(M, s)

at 0:

Note that

_{L(M(n), 0)*}

=

L(M, n)*,

if

M (n)

is the Tate twist of M. The

Bloch-Kato

conjecture

determines this value as follows: Assume that

con-jecture

1.1.2

_holds,

then one can define an

_isomorphism

recalled below

If

_conjecture

I.I.I

_b)

and

_c)

_holds,

then one can define an

_isomorphism

recalled below

Conjecture

1.2.4

_{(Bloch-Kato).}

The

_assumption

are as above then:

a)

There is an element

bp(M)

E

_{0 f}

such that

(Beilinson conjecture).

b)

For all

_prime

numbers p, the element

is a unit in

OE.

Remark. There is also an

_equivariant

version of the

_conjecture,

which

is _very

_important

if one wants to understand the relation to the Iwasawa

main

_conjecture.

This is due to Kato

_[23]

and

_[24].

A

_{generalization}

to non

abelian coefficients can be found in

[7].

Let us recall _very

briefly

how one defines and 6p: The

isomorphism

₆₀₀

is defined

_{by taking}

the determinants in the exact sequence in

_conjecture

1.1.2 and the determinants in

The definition of _Lp is

_slightly

more

complicated.

Fix a finite set of

_places

of K

_containing

the infinite

_places

and the ones over _p, such that

_M~,

is an

étale sheaf over

C7s

:=

OK[11S].

As in

[13]

4.5. one defines a

distinguished

triangle

Here one uses

fl9pjcxJ

HO(Kp,

Mp)

for

_{primes p}

_dividing

oo and an identification of

det Rrf (Kp,

Mp)

with

det-1 (tanm

as follows:

four E S’fin and p

p

I one has p

Mp)

p

MM

1

’

11,

where

₁

is p the inertia group at _{p and the}

_complex

is in

_degree

0 and 1.

_Similarly

for

pip

one has

Rr(Kp, Mp)

I"J =

[Dcris (Mp)

Dcris(Mp)

EÐ

_tanM]

where

Dcris(Mp)

:=

(Bcris,p

0 is the Frobenius on

_Dcris(~Mp)

(normalized

as in

_[13])

and ir is the canonical

_projection

of onto

tanm .

Note that the identification

_{of detEp}

_{Rr c(Os, Mp)}

with

_{det-1 (tanm}

P

introduces Euler factors at the

_primes

in

S’fin

(see

[13]

4.4. and

_[19]

the dis-cussion before definition

_1.2.3).

A

_calculation,

which can be found in

[13]

4.5.,

shows that this

_gives

an

_isomorphism

Now take _any stable

_OE

OZ

Zp

lattice

_Tp

in

_Mp.

Then the

lattice in is

_independent

of

_Tp.

Take _any

_generator

of this rank 1 lattice and use this to define

i.e.

_mapping

this

_generator

to 101.

1.3. The case of _pure motives. Let us discuss several

simplifications

which occur if we restrict to certain

_types

of

_motives,

in

_particular

if we

restrict the

weight w

of the motive M.

Definition 1.3.1. Let M =

(X, q, r)

be a motive. The

integers w

such that

0 are called the

weights

of M.

The motive M is _pure

of

_{weight w if w is}

the

_integer

such that 0.

Remark. It is of course

_equivalent

to define the

_weights

with the de Rham

or the étale realization.

Lemma 1.3.2.

a)

Assume that

_conjecture

_{c) is}

_true,

then

_{= 0 if 0 is}

not

a

weight of

M.

b)

Assume that

_conjecture

true,

then = 0

if all

weights

w

_of

M

satisfy

w &#x3E; -1.

Proof.

Clear from

_conjecture

I.I.I and the direct sum

_{decomposition}

of the

Soul[

_regulator

_rp. 0

Consider a motive M =

(X,

q,

r)

pure of

weight

w. We will discuss now

several

_special

cases of the Bloch-Kato

_conjecture

(cf.

[13]

_§

7).

For this

(1) (w

-3):

then has

_weight

-w - 2 &#x3E; 1 and the lemma

_implies

that = 0 = and the order of

vanishing

of

_{L(M, s)}

at s = 0 is

dM

= 0. Also

by assumption

and the lemma = 0. In

particular

the exact _sequence of

conjecture

1.1.2

reduces to

where roo is the Beilinson

regulator.

Note that

(A f)R

reduces to

(2) (w &#x3E; 1):

here =

dim ker aM

and the exact _sequence of

conjecture

1.1.2 reduces to

(This

is the Beilinson

conjecture

after

_{identifying Coker aM}

with

Deligne

cohomology.)

(3)

(w

=

-2):

The dual motive

_M’(1)

has

_weight

0 and

_consequently

the lemma

_implies

that

_M)

= 0 and

by

definition of

K-theory

Hf(K,M~(1))

= 0. In

particular

the order of

vanishing

of

L(M, s)

at s = 0 One

gets

and the fundamental line is

_accordingly:

(4)

(w = 0):

One

_gets

and the fundamental line is

(5) (w

=

-1 ) :

Here one

_conjectures

that

ker aM =

0 =

Cokeram,

i.e. that the _map

_{H f (H, ~VI ‘~ ( 1 ) )~}

- is an

_isomorphism.

This is

_equivalent

to the statement that it induces a

perfect height

pairing

The formula for the L-value is the

_{generalization}

of the

Birch-Swinnerton-Dyer conjecture.

Finally

we want to formulate a weak version of the Bloch-Kato

_conjecture,

which is

_useful,

if we can

only

construct a certain

_subspace

of the motivic

Conjecture

1.3.3

_(Weak

Bloch-Kato

_conjecture).

Let M be a _pure

motive

_of

_weight

w.

1) (w

-2):

Suppose

there is an E sub-vector _{space R(M)} C

_Hf(K,M)

such that the _sequence

is exact. Then the Bloch-Kato

_{conjecture 1.~.l~}

is true

_if

one

_replaces

ev-erymhere

M)

by

R(M).

2)

(w &#x3E;

0):

Suppose

there is an E sub-vector _space

of

such that the _sequence

is exact. Then the Bloch-Kato

_{conjecture 1.,.1.4 is}

true

_if

one

_replaces

eve-°

Let us

finally

make a remark about the

compatibility

with the

expected

functional

_equation

of the L-function. In fact one

_conjectures

that the

L-function

_{L(M, s)}

has a

_meromorphic

continuation to C and satisfies

for certain r factors and an -

factor e(M, s).

Conjecture

1.3.4. The Bloch-Kato

_{conjecture for}

M

_implies

the one

for

MV (1).

Remark. Perrin-Riou

_developed

a

_theory

of

_p-adic

L-functions of motives

starting

from the Bloch-Kato

_conjecture.

Decisive is her

_exponential

_map,

which

_interpolates

certain

_exponential

_maps of Bloch and Kato. For this

she

_conjectured

an

"explicit reciprocity

law" _proven

by Benois,

Colmez,

Kato-Kurihara-Tsuji

(alphabetical order).

This

_{"explicit reciprocity}

law" also allows to deduce Kato’s

_{explicit reciprocity law,}

which can be used to prove the above

conjecture

in certain cases

(see e.g.

[19]

_appendix

B).

2. A _survey of results

Here we collect the known results of the Bloch-Kato

_conjecture

to the best of the

_knowledge

of the author. As we do not

_attempt

an overview

of the Beilinson

_conjecture

_(i.e.

_part

_a)

of

_conjecture

_1.2.4)

we

concen-trate

_only

on the cases where

_part

b)

is known

(at

least for some

_primes

p) .

Moreover we do not _say

_anything

about the

_weight

_{-1 case,} which is

essentially

the

_{(generalized)}

_{Birch-Swinnerton-Dyer conjecture.}

The

_organization

of this section is as follows: we first treat Dirichlet characters and abelian number

_fields,

where the

_conjecture

is known in

only

very

special cases

are

_known,

which follow from the class number

for-mula. The second _group of

examples

concerns

_elliptic

curves with

_complex

multiplication

and Hecke characters of

_{imaginary quadratic}

fields.

2.1. Artin motives. Consider the

_category

_AQ(E)

of Chow motives over

Q

with coefficients in E. An

_object

M :=

(Spec

F,

_q,

0)

of where

F/Q

is a number field is called an Artin motives. We also consider Tate

twists

_M(T)

:=

(Spec F, q, r).

An Artin motive is called

_abelian,

if

_F/Q

is Galois with

_Gal(F/Q)

abelian.

Definition 2.1.1.

_{Let X :}

_{(ZINZ)X -7}

(C" be a Dirichlet character and

G :=

Suppose

that E contains all values

of

Dirichlet

char-acters

_of

_(71/NZ)x.

Let

be the

_projector

onto the

_{x-I-eigenspace}

in

_{End(ho(Q(p,N ))).}

We call

the associated Dirichlet motive. Note that the element a E

_Gal(Q/Q)

acts

via

_X(a)

on

h(x)et.

We norrnalize the

reciprocity

_map so that the

_geometric

Frobenius

_X(Frp)

=

x(p).

We also

_{denote by}

_h(X)(r)

the r-th Tate twist

_of

h(X).

Note that we

_get

a

_{decomposition}

(Dx

h(X),

where X

runs

through

all Dirichlet characters of In

_particular,

_every abelian Artin motive is a direct sum of Dirichlet motives. The E

0Q C-valued

L-function of

_h(X)

exists and is

_{given by}

for ~s &#x3E;

_1,

where

_f

is the conductor of _x. This has a

meromorphic

contin-uation to C. Note that

_h(X-1).

We collect what is known about the

_conjectures

_{in §}

1,

necessary to formulate the Bloch-Kato

conjecture:

Theorem 2.1.2. Consider the motive

_h(X)(r).

1) (Borel):

Conjecture

1.1.1

_a)

and

_c)

is true.

2) (Soul6):

Conjecture

1.1.1

_{b) is}

true.

3)

Conjectures

1.1.2 and 1.2.3 are true.

Concerning

the Bloch-Kato

_conjecture

we have:

Theorem 2.1.3. Let

_h(X)

be a Dirichlet motive and r E Z. Then there is

an elements such that

and

b) (Burns-Greither,

[6],

Huber-Kings,

[19]):

for

all

_prime

numbers

_{p ~}

2

i. e. the Bloch-Kato

conjecture

is true up to powers

of 2.

Remark.

(1)

The case is due to Bloch and Kato

_[5]

Theorem 6.1 _{up to} a

conjecture

(6.2

loc.

_cit.),

which has been

_proved

in

_[20]

and later

with a different method in

[18]

(2)

A

_proof

of the

_{equivariant conjecture}

for Tate motives

_Q(r)

and r G 0

over abelian number fields and

p #

2 is

_{given by}

Burns and Greither

in

_[6].

(3)

In

_[23]

_§

6 Kato _proves the

_p-part

of the

_{equivariant conjecture}

for

))+

for r &#x3E; 2 even.

(4)

The

_proof

of Beilinson of

_part

_a)

of

_{conjecture 1.2.4,}

had some _gaps,

which were filled

_by

[35]

and

[12].

(5)

Fontaine

_[13]

_§

10 mentions the case of Dirichlet motives

h(X)

but

he assumes that

p2

_{resp. p} does not divide N or

cp(N) (where

N is

the conductor of

_X)

_depending

on

_parity

conditions and no

proof

is

given.

(6)

The Bloch-Kato

_conjecture

for with coefficients

_{in Q}

and

r 0 is

_easily

seen to be

_equivalent

to the

_{cohomological}

Lichtenbaum

conjecture

(see

[18]

corollary

1.4.2.).

The case where F is

totally

real and r 0 is odd is is a direct _consequence of the main

conjecture

due to Wiles

_(see

2.1.5

_below).

In

_[30]

this

_conjecture

is

proven for abelian number fields for all

primes

p, which do not divide

the

_degree

of the number field over

Q. However,

the formula

_given

there has erroneous Euler factors.

(The

Euler factors are fixed in

[3]

and

_[29]).

(7)

In recent work

_[4],

_Nguyen

and Benois show how to reduce the Bloch-Kato

_conjecture

for

_{h° (F)}

and r &#x3E; 1 to the case of r 0

_{by showing}

compatibility

under the functional

equation.

(8)

Burns and Greither

_[6]

proved

even an

_equivariant

version of the

above theorem.

For

_general

Artin motives we have the

_following

results: The first is a

reformulation of the class number formula.

Proposition

2.1.4. The Bloch-Kato

_conjecture

is true

_for

ho(F)

and

ho (F) (1)

-This is mentioned in

_[13]

with some indications of the

proof.

A full

proof

can be found in

_[19]

2.3.

_(at

least for

_{p #}

_2).

Remark. The

_{equivariant conjecture}

for

_hO(F),

where

_F/Q

is a _very

_special

non abelian extension and r = 0 is considered in

[7]

§

7. We refer to this for the exact

_description

of their result.

The next result is a _consequence of the main

conjecture

for

totally

real

fields

_by

Wiles:

Theorem 2.1.5

_(Wiles,

_[38]

_1.6.).

Let F be a

totally

real

field

and r &#x3E; 1 odd. Then the Bloch-Kato

_conjecture

is true

_for

_h°(F)(-r)

and all

_prime

numbers 2.

2.2.

_Elliptic

curves with

_{complex multiplication}

and Hecke

char-acters of

_imaginary

_quadratic

fields. There are two

_types

of results

in the case of Hecke characters of

_{imaginary quadratic}

fields: there is the

critical case for

_{weights w}

-2 and the case of all L-values at

_negative

integers

(recall

that we do not treat the

_{Birch-Swinnerton-Dyer}

_case).

Let us start with the motives in

_question.

Let K be an

_imaginary

qua-dratic field and

_X/K

be an

_elliptic

curve with

_{complex multiplication by}

OK,

the

_ring

of

_integers

in K. Note that this restricts the class number of

K to 1. The motive

_h(X)

:=

(X, id, 0)

over K has a

_{decomposition}

with

respect

to the zero section

where

_{(Spec K, id, 0)}

and

_{h,2 (X ) "-_r}

Consider

as

object

in the

_category

of Chow motives over K with

coefficients in I~. This _{is pure} of

_weight

1. We fix an

embedding

A :

K C C such For _{any w} &#x3E;

_1,

the motive

has

_{multiplication by}

_K(1)w,

where the tensor

_product

is taken over

Q.

The

_idempotents

in

K&#x26;Jw

are

_{parameterized by}

the

Aut(C)

orbits of the

embeddings

(C).

These in turn are

_given

_by

for all

a, b &#x3E;

0 with a + b = w. Denote the motive associated to this

_projector

by

This is a motive in which is _pure of

_weight

w. Fix an

isomorphism 8 :

EndK (X)

such that the

composition 0 :

I~ is the natural inclusion.

_{Let cp :}

Kx

be the Serre-Tate character associated to X and

_f

be its conductor. We will write

and for its Tate twists.

_According

to

_Deninger

_[10]

1.3.2. we

have

where the

_product

runs

_through

all

_primes

_{p of K} and Rs

_{&#x3E; ~}

+ 1. This function has a

_meromorphic

continuation to C. Note that

has a first order zero at s = -l and 1 _E Z for those 1 such that

(5)

-1

_{min(a, b)}

if

_{a ~}

b and

(6)

-l a

= b = 2

otherwise.

The L-function

_s)

is critical at s = --L and 1 _E Z if

Remark. The above construction can be carried out mutatis mutandis if

the curve X is

_already

defined over

Q.

The

_resulting

motives are

then in

_(see

_{[10] §}

_4).

To

_get

the

_right

L-function one has also to

descend the coefficients from K to

_Q.

This is carried out in loc. cit. The result in the critical case is:

Theorem 2.2.1. Let

_X/Q

be an

_elliptic

curve with

complexe

multiplication

by

OK

and consider E

_MQ(Q),

with k

_{&#x3E; j}

&#x3E; 0. Assume that

L(~(~+~)(&#x26;),0) ~0.

Then:

a)

(Goldstein-Schappacher, [15]

thm.

_1.1):

There is an element

such that

b)

(Guo, [16]

thm.

_1,

_Harrison,

_[17]):

For all

_prime

numbers p &#x3E; k + 1 where X has

_{good ordinary reduction,}

Remark.

(1)

Note that

(2)

Harrison

_[17]

considered the

_{case j}

= 0 and k _&#x3E; 1 over K =

Q(i).

(3)

The result of

_{Goldstein-Schappacher}

concerns all critical values and

treats

_{imaginary quadratic}

fields.

We turn to the non

critical, K-theory

case: Let for a

subspace

Theorem 2.2.2. Let

_{X/ K}

be an

_elliptic

curve with

_{complex multiplication}

by

OK.

Consider the motive with

r &#x3E;

0. Therc

_{L(h(ëp)(-r),s)}

has a zero

_of

order 1 at s = 0 and:

a) (Bloch

r =

_0,

Deninger,

[9]

11.3.2):

There is a

_subspace

R(h(p)~(r))

of

i. e. the weak Beilinson

_conjecture

is true.

b) (Bloch-Kato

r =

0,

Kings,

[27]

theorem

1.1.5.):

Assume that

for

_p

~’

6ff

the _group

_h(cp)(2+r))

is

_{finite. Then,}

_for

all

_prime

numbers

p not

dividing

6f f,

-i. e. the weak Bloch-Kato

_conjecture

is true.

Remark.

(1)

In the case where X is

_already

defined

_{over Q}

and r = 0

_part

b)

is

due to Bloch and Kato

_[5]

at least for all

_primes

_p, which are

_regular.

(2)

The case X

_already

defined

_{over Q}

and r &#x3E; 0 was deduced from

_part

b)

in

_~1~.

(3)

Deninger’s

result is much more

_general.

The methods for

_proving

b)

generalize

to this case.

(4)

If we fix _p, the finiteness of

h(cp)(2

+

_r))

is known for

almost all r

(see

the discussion in

[27]

1.1.4).

2.3.

_Adjoint

motives of modular forms. In this section we describe

briefly

the results

_{by Diamond,}

Flach and Guo

_[11]

about the Bloch-Kato

conjecture

for the

_(critical)

values

_{L(A, 0)}

and

_L(A,1),

where A is the

adjoint

motive of a newform

f

of

weight

k &#x3E; 2. We will omit a lot of

details because

_they

are _{very technical and}

give only

a

rough

sketch.

Let

_f

be a newform of

weight

k &#x3E; 2 and level N with coefficients in the

number field K. Scholl

_[37]

has constructed a motive

_Mf

with coefficients in

K,

such that for all

_{places A}

of

_K,

the Galois

_{representation}

_Mj,À

coincides with the one associated to

_f.

The

_adjoint

motive

_Af

is then

_essentially

defined to be the kernel of the canonical _map

_{Mf 0}

_{M’ -+}

K. Define a

set of

_places

_Tf

of K to consist of the and the ones where the

Ga-lois

_{representation}

_{(fl4 j, x a}

lattice in

_{Mj, x)}

is not

_absolutely

irreducible when restricted to

_GF,

where F =

Q( vi (-I)(l-1)/2l)

and

All.

Theorem 2.3.1

_(Diamond,

_Flach,

_{Guo). a)}

For i =

0, l,

there is _an

ele-ment

such that

(This

is

_essentially

due to Rankin and

_Shimura)

b)

For all

_places

_{A 0}

_Tf

The

_proof

is

_mainly

a

_computation

of a Selmer _{group, which} can be

computed

because it arises as the

_tangent

_space of a deformation

problem

as in the

_theory

of

_{Wiles, Taylor-Wiles.}

3.

_Ingredients

in the

_proofs

Here we want to

_explain

the most

_important

_ingredients

in the

_proofs

of

the above results. We concentrate on the more difficult non critical case. 0

Then not

_only

Iwasawa

_theory

enters but also the motivic

_{polylogarithm}

classes and their

_twisting

_property.

The idea of

_proof

is the same in all cases and follows the line of ideas

_already

used in

[5]:

One describes the

Galois

_cohomology

_groups

_H’(K,Mp)

with the

_help

of certain Iwasawa

modules. These modules are

independent

of the Tate twist. The

_image

of the in

_{H) (K, Mp)}

extends to an Euler

_system

and the

conjecture

can be reduced to the main

_conjecture

in Iwasawa

_theory.

To make this reduction

_possible

one needs a twist

compatibility

of

6f (M).

This is the most difficult

part

of the

_proof

_(besides

the use of the main

conjecture).

The twist

_{compatibility}

can be shown in the known cases

because the

_{elements 6f (M)}

are

_{given by}

motivic

polylogarithm

classes.

We divide this section into two

_parts,

the first treats Dirichlet motives and the second CM

_elliptic

curves.

3.1. Dirichlet motives. In the case of Artin motives one can not

_only

reduce the statement of the Bloch-Kato

_conjecture

to the Iwasawa main

conjecture

but also use the Bloch-Kato

_conjecture

_{to prove} the main

con-jecture.

This uses ideas of Kato

[23]

and follows Rubin’s

_arguments

(see

[36])

to _prove the main

_conjecture.

Let us cite from

[19]

how the

_argument

works:

(1)

One _proves the Bloch-Kato

_{conjecture directly}

in the case of the full

motive

_h°(F)

where F is a number field and r = 0. This is the class

number formula.

_By

the

_{compatibility}

with the functional

_equation

(which

can be checked

directly

in this

case)

the

_conjecture

also holds

for and r = 1.

(2)

Then one uses the Euler

_system

methods to establish a

_divisibility

statement for Iwasawa modules in the case

_{X(-l) =}

(-1)’’-1.

_By

the class number trick - with the class number formula

replaced by

the Bloch-Kato

_conjecture

for F =

Q(J.LN)

and r = 1 -

one _proves the

main

_conjecture

of Iwasawa

_theory

from it.

(3)

Using

Kato’s

_{explicit reciprocity}

_law,

one deduces the Bloch-Kato

conjecture

for r &#x3E; 1 and

_{X(-1) _ (-I)’}

from the main

_conjecture.

The necessary

computation

was

_already

used in the "class number trick" in the

_previous

_step.

’

(4)

Using

the

_precise

_{understanding}

of the

_regulators

of

_cyclotomic

from the main

_{conjecture. However,}

this

_argument

does not

work for r = 0 and

X(p)

=

_1,

the _case of "trivial zeroes".

(5)

Using

the

_{compatibility}

of the Bloch-Kato

_conjecture

under the

func-tional

_equation,

one deduces the

_remaining

cases for r &#x3E; 1 and r 0.

(6)

From the Bloch-Kato

_conjecture

at

_positive

and

_{negative integers,}

we can deduce different versions of the main

_conjecture.

Conversely,

these new versions allow to _prove the Bloch-Kato

_conjecture

for r = 0

and r = 1 unless there _are trivial _zeroes.

(7)

The last

_exceptional

case r = 0 and

X(p)

= 1 follows

again by

the

functional

_equation.

We

_explain

the most

_important

_steps

in the next section in more detail.

There we

_only

consider the case of

h(x)(r)

with

X(-1) _ (-1)’’

and r E Z. The functional

_equation

1.3.4

_implies

then

_(except

for r = 0 and _r =

1)

that the result is also true for

_{h(X-1)(1 -}

_r)

for all r E Z. This takes care

of the characters of the other

_parity.

3.2. The

_{cyclotomic polylogarithm.}

Let us consider the motive

h,(x) (r)

with r 0 and

X(-I) = (-1)T.

The fundamental line is

Choose a lattice of

h(x)s

and let

Tp(X)

:= _0z

_Zp

C

h(X)p.

We let

_tB(X-1)

the dual basis element in

_h(x-1)B.

Recall the

following

theorem of Beilinson

_(see

theorem 5.2.3. of

_[19]):

Theorem 3.2.1

_(Beilinson

_[2],

Neukirch

_[35],

Esnault

_[12]).

There is an

el,ement

such that the element

maps to

_{(L(x, r)*)-1}

under the

_isomorphisme

too.

Define an element in

_by

One can now reformulate

_part

b)

of

_conjecture

1.2.4 as follows:

Proposition

3.2.2. The element

_{cp(b f(h(X)(r)))}

is a unit in

_OEp

i.e.

_part

_b)

_of

the Bloch-Kato

_{conjecture 1.~.l~}

is

_true,

_if

and

_{only if}

Cl-r(X-1)

is a

_generator

ofdetoEpRr¡(Q,Tp(X-1)(I-r))

via the

To

_study

we use Iwasawa

_theory.

Let

be the maximal

_Zp-extension

inside

_Q(Cp.)

and

_Q

C with

= We let A :=

p

be the Iwasawa

algebra.

Let G :=

_Gal(Q((N)/Q)

and X :

G -+ ex be a Dirichlet character

of conductor N and

_Tp(X)

be a Galois stable lattice in

h(X)p.

Let be

the

_ring

of

_integers

in the

_{Z/p’Z-extension}

_{Qk of Q}

and define

where the limit is taken with

_respect

to the corestriction _maps. Let

be the

_cyclotomic

character and write x _6’oo

_according

to the

decomposition

_{~~ ^’}

(Z /pZ) x Zp.

We write for the

_Op

module

of rank 1 with Galois action

_{given by}

Then

The elements extend to Euler

systems

and in

_particular

we

_get

elements

_(also

denoted

_by

_cl-,(X-1)

_by

abuse of

_notation)

The miracle is:

Theorem 3.2.3

_{(Huber-Wildeshaus,}

_[20]).

Under the above

_isomorphism

the element _maps to

_{c2_r(X 1~-1).}

Remark. The theorem consists in

_showing

that the

_C1-r(X-1)

are

special-izations of the motivic

_cyclotomic

_{polylogarithm.}

A different

_approach

via the

_{elliptic polylogarithm}

is

_developed

in

_[18].

3.2.1. Reduction to the Iwasawa main

_conjecture.

Write

then we

_get

_by

taking

coinvariants under

Gal(Qoo/Q)

and the

_image

of

_cl-,(X-1)

is the old

_cl-,(X-1).

Now one can use the

property

of the elements

_cl-,(X-’)

E

_r))

that

_they

are

compatible

under

_twisting

to define them for all r E Z

_by

_taking

_simply

the twist for

_positive

r. It turns out:

Proposition

3.2.4

_(theorem

3.3.2. in

_[18]).

The Bloch-Kato

_conjecture

for

for

r &#x3E; 1 and

_{X(-l) =}

_{(-1)’ is}

_true,

_if

and

_{only if}

_C1-r(X-l)

Thus the

_propositions

3.2.2 and 3.2.4

_give:

Proposition

3.2.5. The Bloch-Kato

_conjecture

_for

_h(X)(r)

for

all r E Z and

_{X( -1) = (-I)’}

is

_equivalent

to

Remark.

(1)

Note that the statement is

_independent

of r. For r = 0 it is _a

restate-ment of the main

_conjecture

of Iwasawa

_theory

in the form

at least if the conductor

_{of X}

is not divisible

_by

_p and

_{x(P) ~}

1. Here

2°§

is

_{the X}

_component

of the inverse limit of the

_global

units and

is the inverse limit of the ideal class _groups

_(see

_[36]

_3.2.8).

(2)

The above

_proposition

turns out to be the

_right

formulation of the

Iwasawa main

_conjecture

_{ifp )}

_I

conductor _X. The main

_point

of the

proof

of this more

general

main

conjecture

is that the formulation

is

_independent

of the choice of lattices

_TB(X).

In

_particular

the role

of the

_cyclotomic

elements is

_{explained by}

the fact that

_they

_span

8f(h(X)(r)),

i.e. _map to the

_complex

L-value under

3.3.

_Elliptic

curves with

_{complex multiplication.}

The

_proof

follows

the same lines as in the case of Artin motives.

We consider the motive

_h(~p)(-r).

Note that if we knew that the motivic

cohomology

+

₂₎₎

had the

_right

_dimension,

we would have

The cokernel can be identified with

HI

_(X(C),

Fix as in

[27]

theorem 1.2.2 a

_generator

of this group.

Theorem 3.3.1

_(Beilinson,

_Deninger

_[9]).

There is arc element

such that

maps under

Remark. Note that

_{the ~}

here is not the one from

_[27]

1.2.2. because there

some Euler factors are excluded from

L(ë¡5, -T).

Let

_-r)

be the Euler factor

_(or

the

_product

of Euler

_factors)

at the

_primes

_dividing

_{p in} K. Define

Let

_{X ~p’~~}

be the n torsion

_points

of X and let

_Tp(X)

:=

_Urn

S :_ ~

primes

in

_Kdividing

_{pf f }.}

the

_cyclotomic

case we can define

and

The element e-r extends to an Euler

_system

and

gives

in

particular

elements

Assume that is finite. One has:

Proposition

3.3.2. The Bloch-Kato

_{conjecture for}

_{h(~p) (--r)}

is true

_for

_p,

if

Note that the

_right

hand side is

_independent

of r.

_Again

a miracle

happens:

Theorem 3.3.3

_(Kings

_[27]

_5.2.1).

The

_isomorphisms

_{of A}

modules

maps e-r to ei-r.

This theorem allows to reduce to the case r =

0,

which is

just

_a

refor-mulation of Rubin’s main

_conjecture

for

_{imaginary quadratic}

fields.

Remark.

1. This last theorem is the most difficult

_step

in the

_proof

of the Bloch-Kato

_conjecture

for

_{h(§5) (-r)}

and the main contents of

_[27].

2. The idea of

_proof

is

_roughly

as follows: first one uses the result from

[18]

theorem 2.2.4 that the element e-r is

essentially

the

specialization

of the

_{elliptic polylogarithm.}

Then one has to

_compute

the

_p-adic

real-ization of the

_{elliptic polylogarithm.}

For this a

_{"geometric" approach}

to the

_polylog

sheaf is

_{developed in §}

3 of

_[27].

The

_polylog

extension

can then be written as a

_projective

limit of one-motives

_(section

4.1. in

loc.

_cit.).

0 The

specialization

can then be

expressed

in terms of

elliptic

units. This

_gives

the twist

_{compatibility}

and the relation to Rubin’s

main

_conjecture.

References

[1] F. _BARS, On the _Tamagawa number conjecture for CM elliptic curves _defined over Q. J. Number _Theory 95 _(2002), 190-208.

[2] A. BEILINSON, Higher regulators and values of L-functions. Jour. Soviet. Math. 30 _(1985), 2036-2070.

[3] J.-R. BELLIARD, T. NGUYEN _{QUANG Do,} Formules de classes pour les corps abéliens réels. Ann. Inst. Fourier _(Grenoble) 51 _(2001), 907-937.

[4] D. BENOIS, T. NGUYEN QUANG DO, La _conjecture de Bloch et Kato pour les motifs Q(m)

sur un _{corps abélien. Ann. Sci.} Ecole Norm. _Sup. 35 _(2002), 641-672.

[5] S. BLOCH, K. _{KATO, L-functions} and _Tamagawa numbers of motives. The Grothendieck

Festschrift, Vol. I, 333-400, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990.

[6] D. BURNS, C. GREITHER, On the equivariant Tamagawa number conjecture for Tate motives, preprint, 2000.

[7] D. BURNS, M. FLACH, Tamagawa numbers for motives with non-commutative coefficients.

Doc. Math. 6 _(2001), 501-570.

[8] P. DELIGNE, Valeurs de _fonctions L et _périodes d’intégrales. Proc. _Sympos. Pure _Math.,

XXXIII, Automorphic forms, representations and L-functions _{(Proc. Sympos.} Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part _{2, pp. 313-346,} Amer. Math. Soc.,

Provi-dence, R.I., 1979.

[9] C. DENINGER, Higher regulators and Hecke L-series of imaginary quadratic fields I. Invent. Math. 96 _(1989), 1-69.

[10] C. DENINGER, Higher regulators and Hecke L-series _{of imaginary quadratic} fields II. Ann. Math. 132 _(1990), 131-158.

[11] F. _DIAMOND, M. FLACH, L. _{Guo, Adjoint} motives _of modular _forms and the _Tamagawa

number _{conjecture, preprint,} 2001.

[12] H. ESNAULT, On the _{Loday symbol} in the _{Deligne-Beilinson cohomology. K-theory} 3 _(1989), 1-28.

[13] J.-M. FONTAINE, Valeurs spéciales des _fonctions L des motifs. Séminaire Bourbaki, Vol.

1991/92. Astérisque No. 206, (1992), Exp. No. 751, 4, 205-249.

[14] J.-M. _FONTAINE, B. PERRIN-RIOU, Autour des _conjectures de Bloch et _{Kato, cohomologie}

galoisienne et valeurs de fonctions L. Motives _(Seattle, WA, 1991), 599-706, Proc. Sympos.

Pure Math., 55, Part 1, Amer. Math. _{Soc., Providence,} RI, 1994.

[15] C. GOLDSTEIN, N. _{SCHAPPACHER, Conjecture} de _Deligne et 0393-hypothese de Lichtenbaum sur

les corps quadratiques imaginaires. C. R. Acad. Sci. Paris Sér. I Math. 296 _(1983), 615-618.

[16] L. Guo, On the Bloch-Kato _{conjecture for} Hecke L-functions. J. Number Theory 57 _(1996), 340-365.

[17] M. HARRISON, On the conjecture of Bloch-Kato for Grössencharacters over _Q(i). Ph.D.

Thesis, Cambridge University, 1993.

[18] A. _HUBER, G. KINGS, Degeneration of l-adic Eisenstein classes and _of the _elliptic poylog.

Invent. Math. 135 _(1999), 545-594.

[19] A. HUBER, G. _KINGS, Bloch-Kato _conjecture and main _conjecture of Iwasawa _{theory for}

Dirichlet characters, to appear in Duke Math. J.

[20] A. HUBER, J. _WILDESHAUS, Classical motivic _{polylogarithm according} to Beilinson and

Deligne. Doc. Math. J. DMV 3 _(1998), 27-133.

[21] U. _{JANNSEN, Deligne homology, Hodge-D-conjecture,} and motives. In: _Rapoport et al _(eds.): Beilinson’s conjectures on _special values of L-functions. Academic Press, 1988.

[22] U. _JANNSEN, On the l-adic _{cohomology of} varieties over number fields and its Galois

coho-mology. In: Ihara et al. _(eds.): Galois groups over Q, MSRI Publication, 1989.

[23] K. _KATO, Iwasawa _theory and _{p-adic Hodge theory.} Kodai Math. J. 16 _(1993), no. _1, 1-31.

[24] K. KATO, Lectures on the _approach to Iwasawa _{theory for} Hasse- Weil _L-functions via _BdR I. Arithmetic _{algebraic geometry (Trento, 1991), 50-163,} Lecture Notes in Math. 1553,

Springer, Berlin, 1993.

[25] K. KATO, Lectures on the _approach to Iwasawa _{theory of} Hasse-Weil L-functions vis _BdR, Part II, unpublished preprint, 1993.

[26] K. KATO, Euler _systems, Iwasawa theory, and Selmer groups. Kodai Math. J. 22 (1999),

313-372.

[27] G. KINGS, The Tamagawa number _{conjecture for} CM _elliptic curves. Invent. math. 143

(2001), 571-627.

[28] G. _{KINGS, Higher regulators,} Hilbert modular _surfaces and _special values _{of L-functions.} Duke Math. J. 92 _(1998), 61-127.

[29] M. KOLSTER, TH. NGUYEN QUANG DO, Universal distribution lattices for abelian number

fields, preprint, 2000.

[30] M. KOLSTER, TH. NGUYEN QUANG DO, V. FLECKINGER, Twisted _{S-units, p-adic} class

num-ber _formulas, and the Lichtenbaum _conjectures. Duke Math. J. 84 _(1996), no. 3, 679-717.

Correction: Duke Math. J. 90 _(1997), no. 3, 641-643.

[31] K. _KUNNEMANN, On the Chow motive _of an Abelian Scheme. Motives _(Seattle, WA, 1991),

189-205, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

[32] Y. I. MANIN, Correspondences, motives and monoidal _{transformations.} Mat. Sbor. 77, AMS

Transl. _(1970), 475-507.

[33] B. MAZUR, A. WILES, Class fields of abelian extensions of Q. Invent. Math. 76 _(1984),

179-330.

[34] J. NEKOVAR, Beilinson’s _conjectures. Motives _{(Seattle, WA, 1991),} 537-570, Proc. _Sympos.

Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.

[35] J. NEUKIRCH, The Beilinson conjecture for algebraic number fields, in: M. Rapoport et al. (eds.): Beilinson’s _conjectures on _Special Values _{of L-functions,} Academic Press, 1988.

[36] K. RUBIN, Euler systems. Annals of Mathematics Studies, 147, Princeton University Press, Princeton, NJ, 2000.

[37] A.J. SCHOLL, Motives _for modular forms. Invent. math. 100 _(1990), 419-430.

[38] A. WILES, The Iwasawa main _{conjecture for totally} real _fields. Ann. Math. 131 (1990),

493-540. Guido KINGS NWF-I Mathematik Universitat Regensburg 93040 _Regensburg Germany E-rra,ail : _{guido.kings0mathematik.uni-regensburg.de}