An intro duction to Katos Euler systems

A J Scholl

to Bryan Birch

Contents

KatoSiegel functions and mo dular units

Review of mo dular forms and elliptic curves

KatoSiegel functions

Units and Eisenstein series

Norm relations

Some elements of K theory

Level structures

Norm relations for structure

n

Norm relations for structure

Norm relations for pro ducts of Eisenstein series

A App endix Higher K theory of mo dular varieties

A Eisenstein symb ols

A Norm relations in higher K groups

The dual exp onential map

Notations

The dual exp onential map for H and an explicit recipro city law

Fontaines theory

Big lo cal elds

Pro of of Theorem

The RankinSelb erg metho d

Notations

Adelic mo dular forms

Eisenstein series

The RankinSelb erg integral

Lo cal integrals

Putting it all together

Reprinted from Galois Representations in Arithmetic Arithmetic Geometry

c

edited by A J Scholl R L Taylor Cambridge University Press

A J Scholl

The Euler systems

Mo dular curves

Elliptic curves

References

Intro duction

In the conference there was a series of talks devoted to Katos work on the

Iwasawa theory of Galois representations attached to mo dular forms The

present notes are mainly devoted to explaining the key ingredient which is

the Euler system constructed by Kato rst in the K groups of mo dular

curves and then using the Chern class map in Galois cohomology This

material is based mainly on the talks given by Kato and the author at the

symp osium as well as a series of lectures by Kato in Cambridge in In

a companion pap er Rubin explains how given enough information ab out

an Euler system one can prove very general niteness theorems for Selmer

groups whenever the appropriate Lfunction is nonzero see x of his pap er

for precise results for elliptic curves

Partly b ecause of space and partly b ecause of the authors lack of un

derstanding the scop e of these notes is limited There are two particular

restrictions First we only prove the key recipro city law Theorem b e

low which allows one to compute the image of the Euler system under the

dual exp onential map in the case of a prime p of go o d reduction actually for

stupid reasons explained at the end of x we also must assume p is o dd

Secondly we say nothing ab out the case of Galois representations attached

to forms of weight greater than For the most general results the reader

will need to consult the preprint and Katos future pap ers

Katos K Euler system has its origins in the work of Beilinson see

also for a b eginners treatment Beilinson used cuppro ducts of mo dular

units to construct elements of K of mo dular curves He was able to compute

the regulators of these elements by the RankinSelb erg metho d and relate

them to the Lfunction of the mo dular curve at s in partial conrmation

of his general conjectures relating regulators and values of Lfunctions

Kato discovered that by using explicit mo dular units one obtained norm

compatible families of elements of K These mo dular units are the values

at torsion p oints of what are called here KatoSiegel functions These are

canonical no indeterminate constant functions on an elliptic curve over

any base scheme with prescrib ed divisors which are normcompatible with

resp ect to isogenies Such functions were over C rst discovered by Siegel

the asso ciated mo dular units were studied in depth by Kub ert and Lang

Over C generalisations of these functions were found by Rob ert It

An intro duction to Katos Euler systems

was Kato who rst found their elegant algebraic characterisation In x I

have given an arithmetic mo dular construction of these functions which is

more complicated than Katos but at least reveals the key fact b ehind their

existence namely the triviality of the th p ower of the sheaf ! on the

mo dular stack The Picard group of the mo dular stack was computed by

Mumford many years ago

In x we turn to K theory and give a fairly general construction of the

Euler system in K of mo dular curves and the norm relations It is relatively

formal to pass from this to an Euler system in Galois cohomology of say

a mo dular elliptic curve The hard part is to show that the cohomology

classes one gets are nontrivial if the appropriate Lvalue is nonzero In

his Cambridge lectures Kato explained how this can b e regarded as a

consequence of a huge generalisation of the explicit recipro city laws Artin

Hasse Iwasawa Wiles to lo cal elds with imp erfect residue eld This

is the sub ject of the preprint At the Durham conference he sketched a

slightly dierent pro of using the FontaineHyo doFaltings approach to padic

Ho dge theory In x we give a stripp eddown pro of of a weak version of one

of the recipro city laws in in the case of go o d reduction using a minimal

amount of padic Ho dge theory

In x we explain how Kato uses the RankinSelb erg integral very much

as Beilinson did to compute the pro jection of the the image of the dual

exp onential into a Hecke eigenspace Finally in x we tie everything together

for a mo dular elliptic curve

The app endix to x which is the authors only original contribution to

this work is an attempt to extend Katos metho ds to other situations We

construct an Euler system in the higher K groups of a suitable op en part

of KugaSato varieties This is a precise version of the construction used

in see also x of for a summary to relate archimedean regulators

of mo dular form motives and Lfunctions The padic applications of these

elements remain to b e found

I have many p eople to thank for their help in the preparation of this

pap er Particular mention is due to Jan Nekovar He encouraged me to think

ab out norm relations in although in the end that work was overtaken

by events and all that remains of it is the app endix to x It is only b ecause

of his insistence that xx exist at all and his careful reading of much of

the pap er eliminated many errors although he is not to b e held resp onsible

for those that remain I am also grateful to Amnon Besser Sp encer Blo ch

Kevin Buzzard John Coates Ofer Gabb er Henri Gillet Erasmus Landvogt

and Christophe Soule for useful discussions Karl Rubin read the original

draft of the manuscript and made invaluable suggestions Ab ove all it is

a great pleasure to thank Kato the creator of this b eautiful and p owerful

mathematics for encouraging me to publish this account of his work and for

A J Scholl

p ointing out some blunders in an earlier draft

This pap er was b egun while the author was visiting the University of

Munster in winter as a guest of Christopher Deninger and completed

during a stay at the Isaac Newton Institute in It is a pleasure to thank

them for their hospitality

Notation

If G is a commutative group or group scheme and n Z then n G G

G

is the endomorphism multiplication by n written simply n if no confu

sion can o ccur We also write G and Gn for the kernel and cokernel of n

n

resp ectively

Throughout this pap er we use the geometric Frob enius and normalise the

recipro city laws of class eld theory accordingly see x b elow for precise

conventions as well as the remarks following Theorem

The symb ol is used to denote equality or canonical isomorphism We

use the usual notation to indicate that the righthand expression is the

denition of that on the left and for the reected relation

KatoSiegel functions and mo dular units

Review of mo dular forms and elliptic curves

We review some wellknown facts ab out the mo duli of elliptic curves See

for example Chapter For any elliptic curve f E S with

zerosection e we have the standard invertible sheaf

! f e

E S E S E S

From the second description as the conormal bundle of the zerosection of

E S we have the isomorphism ! e O e Because is free

E S E

E S

for any section x E S along the bres of f in fact ! x

E S

E S

The formation of ! is compatible with basechange in fancy language

E S

! is a sheaf on the modular stack M of elliptic curves

A meromorphic mo dular form of weight k is a rule which assigns to

k

each E S a section of ! compatible with basechange By denition this

E S

k

is the same as an element of M ! The discriminant E S is a

nowherevanishing section of ! compatible with basechange and it denes

E S

an invertible mo dular form of weight From this it follows in particular

that

d d

The set of nowherevanishing sections of ! is f g for any integer

d

An intro duction to Katos Euler systems

Let N b e an integer The mo dular stack M classies pairs

N

E S where E E is a cyclic isogeny of degree N of elliptic curves

over S When N is not invertible on S the denition of cyclic can b e found

in x The functor E E denes a morphism c M M

N

k

A meromorphic mo dular form on N of weight k is a section of c !

over M Equivalently it is a rule which asso ciates to each cyclic N

N

k

isogeny E E of elliptic curves over S a section of ! compatible

E S

with arbitrary basechange S S As well as one has the mo dular form

N

of weight dened by

N

E E E

It is invertible exactly where is etale In particular it is invertible on

S ZN

Supp ose N p is prime The reduction of M mo d p has two irre

p

ducible comp onents one of which parameterises pairs E S where is

Frob enius and the other those pairs where is Verschiebung On the rst

p

comp onent vanishes and on the second it do es not

p th

Let m b e the denominator of p Then is the m

p ower of a mo dular function u M O which is invertible away

p p

from characteristic p by the previous remarks It is a classical fact that

M Q O hQ u i

p p

and therefore

M O fg

p

Recall the Ko dairaSp encer map see eg if E S is an elliptic

curve and S is smo oth over T one has an O linear map

S

KS KS !

E S ST

E S

If T Sp ec Q and E S is the universal elliptic curve over the mo dular

curve Y N N the denition is recalled in x b elow then KS is an

isomorphism

If S S E E is an extension of E to a curve E S of genus not

necessarily smo oth and the identity section e E S extends to a a section

S E whose image is contained in the smo oth part then ! e e

E S E S

S extending ! If S is smo oth over the base is an invertible sheaf on

E S

S of a divisor S with relative normal scheme T and S is the complement in

crossings the Ko dairaSp encer map extends to a homomorphism

KS log S !

E S

S T

E S

If S X N for N and E is the regular minimal mo del of the universal

Q

elliptic curve then is an isomorphism

A J Scholl

KatoSiegel functions

If D is a principal divisor on an elliptic curve over say a eld there is in

general no canonical function with divisor D For certain sp ecial divisors

such canonical functions do exist In their analytic construction they have

b een used extensively in the theory of elliptic units Kato observed that they

have a completely algebraic characterisation Here we give a slightly more

general mo dular description of such a class of functions

Theorem Let D be an integer with D There is one and only

one rule which associates to each el liptic curve E S over an arbitrary

D

E S

base a section O E ker D such that

D

E S

i as a rational function on E has divisor D e ker D

D

ii if S S is any morphism and g E E S E is the basechange

S

E S E S

then g

D D

iii if E E is an isogeny of el liptic curves over a connected base S

whose degree is prime to D then

E S E S

D D

iv and If D MC with M C then

D D

C M

and M M

M C C D D

In particular D

D

v if C with Im and E C is the el liptic curve whose points are

E C

C Z Z then is the function

D

D

D

u D u

where

Y

n n

u q q t q t t t

n

i iu

and q e t e

Remarks i We do not require that D b e invertible on S

ii Lo cally for the Zariski top ology any elliptic curve may b e obtained by

basechange from an elliptic curve over a reduced base It is therefore enough

to restrict to reduced base schemes S

An intro duction to Katos Euler systems

E S

iii Prop erties i and iii alone already determine uniquely any

D

other function with the same divisor is of the form u for some u O S

and applying ii for the isogenies would give u u u whence

u

iv In downtoearth terms if S Sp ec k for an algebraically closed eld

k then for a separable isogeny E E the prop erty iii is just the

distribution relation

Y

E k E k

x y for any y E k

D D

xE k

xy

v Over C this theorem was obtained by Rob ert who proves rather

more he shows that for any elliptic curve E C and any nite subgroup

P E of order prime to there is a certain canonical function with divisor

P e P and prop erties generalising those of One can prove his more

D

general result in a manner similar to the pro of of in place of the mo dular

D P

form one should use E E P where E E P is the

quotient map

Proof We b egin with the rst two conditions First observe that if S is a

sp ectrum of a eld then the divisor ker D D e is principal b ecause D

is o dd the sum of ker D D e in the Jacobian is zero To give a rule

satisfying i and ii is equivalent to giving for any elliptic curve E S

D

an isomorphism of line bundles on E

O ker D O D e

E E

compatible with basechange We have just observed that the line bundles

are isomorphic when restricted to any bre of E S Since we can assume

by remark ii ab ove that S is reduced the seesaw theorem tells us that

to give an isomorphism is equivalent to giving an isomorphism of the

restriction of the bundles to the zerosection In other words the existence

of is equivalent to nding for each E S a trivialisation of the bundle

D

e O ker D e O D e e D O e e O D e

E E E E

D

e O e

E

D

!

E S

compatible with basechange Note that D implies D mo d

D

There are then exactly two nonvanishing sections of ! compatible

E S

D

with arbitrary basechange namely E S Cho ose one of them

E S

and let b e the corresp onding function on E ker D So the rule

A J Scholl

E S

E S satises prop erties i and ii In a moment we will see

that exactly one of satises iii See also remark b elow

By the basechange compatibility ii we are free to make any faithfully

at basechange in order to check iii There exists such a basechange over

which factors as a pro duct of isogenies of prime degree It is therefore

enough to verify iii when deg p is prime The quotient

E S E S

g E S O S

p

is compatible with basechange It therefore denes a mo dular unit g

p

M O and so g E S fg for every E S Moreover the

p p

sign dep ends only on p

To determine the sign evaluate g E F F for an elliptic curve over F

p p E p

and its Frob enius endomorphism The norm map F E E is

E

then the identity map so g E F and therefore if p is o dd we have

p E

E S E S

g Notice that replacing by do es not change g for p

p p

o dd but replaces g by g Therefore for exactly one choice it will

D

b e the case that g so exactly one of these choices satises iii

Now for prop erty iv Evidently also satises the characteristic prop

D

erties i and iii hence Also for the same reason The

D D

function M has divisor M C e ker C and is compatible with

D

M

for some Now prop base change so we can write M

C D

erty iii gives

M

M M

C D D

M M M

M

C C C D

and so The same calculation works for M D by writing

C

b oth have divisor If D MC then the functions M and

M C D

C ker M ker D

hence their ratio is a unit compatible with basechange The norm compati

bility iii then shows that this unit equals as in Remark iii ab ove This

proves prop erty iv

Finally we check v Classical formulae as can for example b e found in

the function is essentially the same as the Jacobi theta function

show that

D

F u u D u

is a function on E with divisor D e ker D and is SL Zinvariant

E C

Hence F u is a constant multiple indep endent of of As a

D

   

a b u

a b

1

The SL Z action is u 7!

2

c d

c d c d

An intro duction to Katos Euler systems

formal p ower series

n D n D

Y Y

q t q t

D D D

F q t

n D n D

q t q t

n

n

D

is a unit in the ring of Laurent q series with co ecients in Zt t t

So by the q expansion principle the constant has to b e To determine

the sign consider any elliptic curve E dened over R with real connected

comp onents For such a curve one can assume that Re To b e

denite take E to b e the curve

Y X X

for which i The real comp onents of E are the images of the line seg

ments and in the complex plane We compute F u

for such a curve The explicit formula for u shows that the rst non

vanishing uderivative of F u at the origin is real and p ositive On the

interval F u has simple p oles at u k D k D and so by

D

calculus F On the segment F is real

nite and nonzero hence the pro duct F F is p ositive

Therefore at the origin

u u u u

F u F F F F

D D

p ositive real u

D

and so F u F u

Remark As we saw in the pro of corresp onds to one of the two

D

nowherevanishing mo dular forms of weight D Using v it is easy

to determine which The form arises by restriction to the zerosection of the

comp osite isomorphism

O D e D O e O ker D

E E E

E S

D

since e D e Therefore the q expansion is D times the leading co e

E C

cient in the expansion of in p owers of t which from v is easily seen

D

to b e

Y

D D D D

n D

q q

n

Remark Supp ose that E S is an elliptic curve over an integral base S and

that P E S is a nite group of sections Let

X

D m x ZP

x

xP

A J Scholl

P P

b e a divisor with m and m x e In the case when S is the

x x

sp ectrum of a eld D is principal but in general this will not b e the case

For example supp ose that P fe xg for a section x of order disjoint from

e Then D x e is principal if and only if ! e O D is trivial

E S

Consider a Dedekind domain R containing and an ideal A R which

has order in Pic R Let A a and let E R b e the elliptic curve given by

the ane equation

y xx a

over the eld of fractions of R Take an op en U Sp ec R over which A

b ecomes principal lo cally generated by say Then a for some unit

O U and an equation for E over U is

y x x

Therefore ! is lo cally generated over U by

E R

dx dx

y y

ie ! A So the divisor e is not principal on E R

E R

Units and Eisenstein series

Let E b e an elliptic curve over an integral base S let D b e an integer

prime to and x E S a section If x is disjoint from ker D then one

obtains a unit x x O S on the base In particular supp ose that

D D

x is a torsion section of order N with N D Since S is integral x

has order N at the generic p oint Under either of the following conditions it

is automatic that x ker D

N is invertible on S then x has order N in every bre or

N is divisible by at least two primes

In the classical setting one takes S to b e a mo dular curve over C and the

functions x are the Siegel units studied extensively see for example

D

There are at least two ways to form a logarithmic derivative from the pair

x The simplest is to form

D

dlog x S

D S

which in the classical setting gives weight Eisenstein series The other

way which leads to weight Eisenstein series is to rst form the vertical

logarithmic derivative

dlog E ker D

v D E S

An intro duction to Katos Euler systems

see x we obtain Since ! x

E S

E S

Eisx EisE S x x dlog S x S !

D D v D E S E S

a mo dular form of weight one Notice that in this construction one can start

with any function whose divisor is D e ker D since it will b e of the

form g for some g O S and dlog g

D v

From prop erty iv we have

D D

D D

D D D D

and therefore

D dlog D dlog D dlog D dlog

v D v D v D v D

Hence Now D is multiplication by D on global sections of

E S

gives

D EisE S x D EisE S D x

D D

D EisE S x D EisE S D x

D D

It follows that for any D mo d N the section

EisE S x S Z ! EisE S x

D

D D D D

is indep endent of D Now if p j N there exists D D mo d N with

D and p j D D so one can glue the various EisE S x for the

dierent D to get a section EisE S x S ZN ! For any D one

then has

EisE S x D EisE S x D EisE S D x

D

Supp ose E C is an elliptic curve over C with Z Z Let u

D

b e the variable in the complex plane Using the function z D z

Weierstrass function in place of gives

D

dlog D u D D u dz

v D

and if x E C feg is the torsion p oint a a N N with

N D then

X

EisE C x du

s

m m jm m j

s

a

i

Z m

i N

A J Scholl

N

On the Tate curve Tate q over Z q there is the canonical

N N

dierential dtt and the level N structure

a

a N

ZN Z ker N a a mo d N q

N

a

a N

If x is the p oint q then by explicit dierentiation of the innite pro duct

N

in Theorem v

X X

a

a d

nN

sgnd dtt q EisTate q x B

N

N

N

n

dZ djn

n

a mo d N

d

if a n a n Here B X X is the Bernoulli p olynomial

In the case a a the constant term is somewhat dierent In

particular EisTate q x is holomorphic at innity

N

One can also compute the logarithmic derivative of the unit x

N D

The result is most interesting if one works with absolute dierentials that is

in the mo dule of q adically separated dierentials

N

dq I d

Z N N N N

N

where I is the annihilator of d and equals the ideal generated by

N N N

the dierent of Q  The key p oint is that the logarithmic derivative of

N

a typical term in the innite pro duct for x is

D

a

ma N

dlog q

N

a

ma N

q

N

N

a dlog a mN dlog q

N

a

ma N

q

N

whereas the corresp onding term for the vertical logarithmic derivative is

a

ma N

q

N m

dlog t q dlog t

a

a

a N

ma N

q t

q

N

N

Comparing gives the following striking congruence

a

a N

Prop osition If x q Tateq then

N N

EisTate q x

N

D N

a dlog a dlog q mo d N dlog x

N D

dlog t

An intro duction to Katos Euler systems

Norm relations

Some elements of K theory

For a regular separated and no etherian scheme X the Quillen K groups

K X i together with the cuppro duct

i

K X K X K X

i j ij

dene a graded ring K X which is a contravariant functor in X for any

morphism f X X of regular schemes there is a graded ring homomor

phism f K X K X If f is proper then there are also pushforward

maps f K X K X group homomorphisms which satisfy the pro jection

i i

formula

f f a b a f b

For i there is a canonical monomorphism

O X K X

For arbitrary f the restriction of the pullback map f to the image of

is pullback on functions if f is nite and at then the pushforward map f

restricts to the norm map on functions

In this section we are concerned with K The cuppro duct in this case is

the universal symbol map

O X O X K X

u v fu v g

which is alternating and satises the Steinb erg relation fu ug if u

u O X If X Sp ec F for a eld F then the symb ol map induces

an isomorphism

K X K F F Steinb erg relation

by Matsumotos theorem

Returning for a moment to the general situation let Y b e a smo oth not

Y Y Q necessarily prop er variety over a numb er eld F Write

F

j

G GalQ F and let p b e prime Then if H Y Q n has no G

F p F

invariants there is an AbelJacobi homomorphism

j

Y Q n K Y H G H

p nj F

j

The condition that H G H Y Q n can often b e checked just by

F p

considering weights if for example Y is prop er then by considering the action

A J Scholl

of an unramied Frob enius and using Delignes theorem Weil conjectures

one sees that it holds if j n

In the case of interest here Y is a curve and j and n Then the

Ab elJacobi map is even dened integrally

AJ K Y H G H Y Z

F p

It is constructed as follows There is a theory of Chern classes from higher

K theory to etale cohomology these are functorial homomorphisms for each

q and n Z

nq

c K Y H Y Z n

q n q p

Here the cohomology on the righthand side is continuous etale cohomology

These maps are not multiplicative but can b e made into a multiplicative map

by the Chern character construction All we need to know here is that if

K Y then

c c c

see for example p One writes ch c

The etale cohomology of Y is related to that of Y by the Ho chschildSerre

sp ectral sequence

ij

i j ij

E H G H Y Z n H Y Z n

F p p

Let Y X b e the smo oth compactication of Y so that Y X Z for a

Y Z H X Z is free of rank equal nite Z X The Z mo dule H

p p p

to the numb er of comp onents of X and H X Z H X Z The

p p

X Z is the Tate mo dule of the Jacobian of X hence is free mo dule H

p

There is an exact sequence

H X Z H Y Z H Z Z

p p p

X Z H Y Z H

p p

The map is the Gysin homomorphism mapping the class of a p oint z Z

to the class of the comp onent of X to which it b elongs Therefore all the

j

mo dules H Y Z are free Moreover if is an eigenvalue of a geometric

p

j

Y Q at a prime v j p of go o d reduction then is Frob enius acting on H

p

an algebraic integer satisfying

if j

j j

N v or N v if j and

N v if j

2

In the case to b e considered later Y is actually ane in which case one even has

2

Y Z H p

An intro duction to Katos Euler systems

j

g of the sp ectral sequence van Therefore when n the rst column fE

ishes The exact sequence of lowest degree terms then b ecomes

Y Z H Y Z H G H

p p F

e

H G H Y Z H G H Y Z

F p F p

Comp osing the edge homomorphism e with ch c denes the Ab el

Jacobi homomorphism the minus sign is chosen b ecause of

Notice also that the last group H G H Y Z is zero if p is o dd and

F p

killed by in general see for example

We also need the Chern character into de Rham cohomology For a No ethe

rian ane scheme X Sp ec R there are homomorphisms for each q

q

dlog dlog K R

R q

RZ

satisfying

i dloga b dlog a dlog b

ii If b R K R then dlog b b db

RZ

iii On K R dlog is the degree map

iv If R R is a nite at extension of regular rings then tr dlog

R

R R

K

dlog tr

R

R R

K

In iv tr K R K R is the prop er pushforward for Sp ec R Sp ec R

q q

R R

q q

also called the transfer and tr is the trace map for

R R

RZ

R Z

dierentials Since this compatibility do es not seem to b e do cumented in the

literature we make some remarks ab out it What follows was suggested in

conversation with Gillet and Soule

To check the compatibility we can work lo cally on Sp ec R and thus assume

that R is lo cal Therefore R is a free R mo dule of rank d say Cho osing a

M R We get for every n basis gives a matrix representation R

d

corresp onding inclusions GL R GL R which in the limit give an

n nd

inclusion GLR GLR This induces by functoriality the transfer on

K B GL

q q

One way to dene the map dlog is to use Ho chschild homology see

for example There is a simplicial R mo dule C R with

q

C R R tensor pro duct over Z whose homology is Ho chschild ho

q

q q

q

HH R mology HH R There is also a pair of R linear maps

q

RZ

q

whose comp osite is multiplication by q The map is given by

q

RZ

r r r r dr r

q q

A J Scholl

There is a map Dtr H GLR Z HH R the Dennis trace see

which maps r R H GLR Z to the homology class of

r r C R Assume that q is invertible in R Then comp osing on

one side with the Hurewicz map K R H GLR Z and on the other

q q

q

with q denes the map dlog K R for any q It is

q q

RZ

not to o hard to check directly an exercise from Ch that if a b R

then dlogfa bg ab da db which is the only part of i needed in what

follows

HH

There is a trace map tr on Ho chschild homology the representation

R M R induces by functoriality a map HH R HH M R and

d d

by Morita invariance we have an isomorphism HH M R

d

HH R

Still under the hyp othesis that q is invertible the maps and q

q q

q

make a direct factor of HH R One can then dene the trace map

q

RZ

HH

tr as the comp osite q tr This approach to trace maps is

q q

R R R R

due to Lipman see also Hubls thesis It now is a simple exercise

to check the compatibility iv the essential p oint b eing the transitivity

E of the generalised trace

We need all of this only for q This means that in the recipro city law

and all its consequences we need to assume that p is o dd

Level structures

Let E S b e an elliptic curve Then for every p ositive integer N which is

invertible on S there exists a mo duli scheme S N which is nite and etale

over S and which represents the functor on S schemes T

level N structures on E T

S

S N T

ker N ZN

T

T

More generally for pairs M N of p ositive integers invertible on S there is

a scheme S M N which represents the functor

monomorphisms of S group schemes

S M N T

ZM ZN E

T T

The group GL ZN acts freely on S N on the right with quotient S

One has S N N S N in general S M N is a quotient of S N where

N lcmM N One usually writes S N for S N and we will also

write S N for S N Of course S N and S N are isomorphic but

they are dierent as quotients of S N We have a lattice of subgroups of

GL ZN Z and a corresp onding diagram of quotients of S N

3

To avoid overloading the notation we do not include the dep endence on E in the notation

An intro duction to Katos Euler systems

S N

S N S N

A

A

A

A

N S N S

A

A S N

A

A

AU

S N S N

S N

S N

S N S

S N S

R

S

Over S N there is a canonical level N structure ZN ker N

N

E and we let y y E S N b e the images of the generators

N N

Then y already b elongs to E S N and y to E S N

N N

S N is canonically isomorphic to the op en subscheme of ker N con

sisting of p oints in the kernel whose order is exactly N and

a

ker N S M

M jN

The scheme S N parameterises N structures on E S in other words

S N T is functorially the set of cyclic subgroup schemes of rank N of

E T In the case T S N the morphism S N S N

S S N S N

classies the cyclic subgroup generated by y

N

If M jM and N jN then there is a canonical levelchanging map

S M N S M N

S M N S M N

induced by the inclusion ZM ZN ZM ZN One has y

M

and likewise for y M M y

M

We also recall that all the ab ove mo duli schemes can b e dened for integers

M N which are not invertible on S using Drinfeld level structures see

passim They are nite and at over S

Recall nally that for a p ositive integer N which is the pro duct of two

coprime integers there is a universal elliptic curve with level N structure

over the mo dular curve Y N Z We shall use the standard notations Y N

Y M N without comment

Norm relations for ()structure

Now x an integer D which is prime to On each basechange E T

S

where T is one of the ab ove mo duli schemes there is the canonical function

A J Scholl

E T E S

T

which by Theorem ii is simply the pullback of Since

D D

E and D will b e xed in the discussion that follows we shall write all these

functions simply as

Consider the case of prime level Write y y y y and abbreviate

S S or Fix x E S such that D x do es not meet the zero

e

section of E Let E S E b e the quotient by the canonical subgroup

S

e

scheme of rank generated by y Let x E S b e the comp osite

x

e

E E S S

S

e

E S

e

Write E ker D O

D

Lemma

N x y xx N

S S

Y

N x y x x ay N

S S

aZ

N x y x x N

S S

N x y xx N

S S

N x x x N

S S

Proof N By there is a Cartesian square

xy x

S q S E

y y

x

S E

Hence

N x y x N x y x

S S S qSS

x since the square is Cartesian

x by iii

x

N The same argument applied to the Cartesian square

a

xy xay

S S q E S

S

aZ

y y

x

E S S

S

An intro duction to Katos Euler systems

N This comes from the Cartesian square

xy x

S q S E S

S

y y

x

e

E S

The remaining relations N and N are obtained by combining NN

and using

Y

x ay x y N

S S

aZ

Lemma The norm relations NN hold without the hypothesis

that is invertible on S

Proof Cho ose an auxiliary integer r prime to Then after replacing S

by an etale basechange there exists a level r structure Zr ker r

r

univ

on E with r invertible on S Let E Y r b e the universal elliptic

curve with level r structure over Zr Then there is a unique morphism

univ

S E ker D which classies the triple E S x there is a

r

Cartesian square

univ univ

E E E ker D

Y r

pr

y y

univ

S E ker D

univ

such that is the inverse image of the canonical level r structure on E

r

univ univ univ

and x is the pullback of the diagonal section E E E

By the basechange compatibility of it is enough to verify the norm

D

relations in this universal setting but the inclusion Y r Z Y r

induces an injection on O Thus we reduce to the case in which is invertible

on S

Now consider two auxiliary integers D D with DD and write

E S E S

and Following tradition we write N for the push

D

D

forward maps on K but the group op eration in K will b e written

additively for consistency with the higher K theory case to b e considered

b elow

Prop osition In K S the fol lowing identity holds

N fx y x y g

S S

fx x g fx xg N fx x g

S S

A J Scholl

Proof Compute using the pro jection formula and the norm relations

N fx y x y g

S S

x y g N fx y N

S S S S

N fx y x x g by N

S S

N fx x x x g by N

S S

N fx x g fx x g

S S

x x g x g fN fx N

S S S S

N fx x g fx x g

S S

fx x x g fx x x g by N

N fx x g fx x g fx x g

S S

Now supp ose that S is a mo dular curve of level prime to and E is the

universal elliptic curve Therefore S Y Y N H for some subgroup

H

f

univ

H GL ZN E E Y and S Y Y N H It

H H

is then p ossible to rewrite the ab ove norm relation using the Hecke and the

diamond op erators whose denitions we briey recall

univ

The centre ZN GL Zn acts on Y and E on the right dening

H

the diamond op erators hai Aut Y hai Aut E for a ZN In mo d

H E

ular language the B valued p oints of Y are pairs XB where XB

H n H

is an elliptic curve and is an H equivalence class of level N structures

N H

ZN X Then hai XB XB a is an automor

N H N H

univ

phism of Y The B valued p oints of E are triples XB z with

H N H

univ

z X B and the automorphism hai of E is given by XB z

E N H

XB a z

N H

Recall also that the e pairing denes a morphism

N

det H

e Y Sp ec Z

N H N

N

E S e

N N N N

N

det H a

to Z is then the map since and the restriction of hai

N

det hai a

The is a commutative diagram

hi

E

univ univ univ

E E E

f f f

y y

R

Y Y

H H

hi

An intro duction to Katos Euler systems

univ

and if x Y E is any N torsion section hi x x hi Therefore

H E

by the basechange prop erty ii

x x x hi hi x h i x

D D E D D D

The scheme S is the quotient Y Y N N H and Y N N classies

H

triples X C with C X a subgroup of rank One then has the

N

standard diagram

w

E



c c pr

v

E E

univ univ univ univ

univ

Y E E E E Y E

H Y H Y

H H

y y y y y

Y Y Y Y Y

H H H H H

c w c

y y

H H

Sp ec Z Sp ec Z

N N

in which the rst third and fourth squares in the top row are Cartesian The

Hecke op erator T is by denition the corresp ondence c cw on Y and the

H

univ

corresp ondence c c w on E All the horizontal arrows are compatible

E E E

univ

with the level N structure on E and the quotient level N structure on

univ

E hence c w x x c w and therefore

E E

x x x c v c w x

E

using as always the basechange prop erty ii Therefore

T fx x g c c w fx x g N fx x g

S S

Finally write z x y z x y so that x z and x z h i

Observe that the norm relation is invariant under the action of GL Z so

that y y can b e replaced by any basis for the torsion of E This yields

the following reformulation of

Prop osition If S Y is a modular curve of level prime to and z

H

z are torsion sections of E S whose projections onto ker are linearly

independent then

N fz z g T hi hi fz z g

S S

A J Scholl

n

Norm relations for ( )structure

m n

We now consider norm relations in the tower fS g

m n

Lemma If m n and x E S is any section then

N x y N x y

m m n m n m

S S

Proof If is invertible on S this follows from the Cartesian square

xy

m

m n

E S

y y

xy

m

m n

E S

In the general case one reduces to the universal situation exactly as in

Prop osition If DD and m n then for al l x x

m n

E S

x y g N fx y

m n m n m n

S S

g fx y x y

m

n

Proof This follows from the lemma since

N fx y x y g

m n m n m n

S S

N x y g N fx y

m n m n m n m n

S S S S

g x y N fx y

m n m

n

S S

g x y fx y

m

n

If E S is a mo dular family over S Y of level prime to then E Y

H H

is nite of order prime to Therefore

m n m n

E Y Z Z prime to

H torsion

so there is a welldened pro jection onto Z ker Computing as in

we get

Prop osition Suppose that S Y is a modular curve of level prime

H

m n

to and that z z are torsion sections of E over Y with m n

H

If the projections of fz z g into ker are linearly independent then

N fz z g fz z g

m n m n

Y Y

H H

An intro duction to Katos Euler systems

Norm relations for pro ducts of Eisenstein series

We shall rep eat the construction of the last paragraph for pro ducts of the

form

EisE S x S ! EisE S x

D D E S

If g S S is a nite and at morphism of smo oth T schemes and E

E S then there are trace maps

S

O g tr tr g O

S g S S S S T ST

as well as a trace map on mo dular forms dened to b e the comp osite

k k

tr tr S S g

g S S

E S

E S

tr

g

k k

S g O S

S O

E S E S

S

The Ko dairaSp encer map x and the trace are not compatible

Prop osition The diagrams below commute

KS

KS

E S

E S

!

!

S T

E S

S T

E S

x x

x

o g

tr

o

g

y

g KS

E S

deg g g KS

E S

g ! g

g ! g

ST

E S

ST

E S

Proof The rst commutes b ecause of the functoriality of the Ko dairaSp encer

map Then applying tr g deg g gives the second

g

Lemma The notation as in Lemma

tr Eisx y Eisx y

m n m n m m

S S D D

Proof Since we have tr dlog dlog But on global

D D D D

tr is multiplication by Therefore the diagram sections of

E S

xy

n

n n

S ! E ker D

tr

tr

n n n

S S

y y

xy

n

n

E ker D S !

commutes which gives the result

Corollary The notations being as in let g be the projection

m n m n

g Y Y Then

H H

Eisz Eisz tr Eisz Eisz

g D D D D

tr KS Eisz KS Eisz Eisz Eisz

g D D D D

Proof Follows from the preceding two lemmas since deg g

A J Scholl

A App endix Higher K theory of mo dular

varieties

A Eisenstein symb ols

Let f E S b e an elliptic curve and assume from now on that S is a regular

k

scheme For any integer k write E for the bre pro duct E E

S S

it is an ab elian scheme of dimension k over S

k

In Beilinson discovered a family of canonical elements of K E

k

More precisely he dened a canonical map

degree k

Q E K E Q

tors k

which he called the Eisenstein symb ol Here we make a mo died construction

which gives a normcompatible system

k

Let b e the semidirect pro duct of the symmetric group S and  which

k k

k

acts on E as follows

S acts by p ermuting the copies of E

k

th th

the i copy of  acts as multiplication by in the i factor of the

pro duct

There is a character  which is the identity on each factor  and

k k

the sign character on the symmetric group

has a natural realisation in GL Z as the set of all p ermutation matri

k k

ces with entries Geometrically it is the group of orthogonal symmetries

of a cub e in nspace In terms of this representation is just determinant

k

For any Z mo dule M write M for the isotypical comp onent of

k k k

M Z k

For x E S we shall consider the inclusion

k k

i E E

x

u u x u u u u u u

k k k k

whose image is the subscheme

k

X

k

E v x v v

i k

For any integer D such that x is disjoint from ker D we dene the

k

following op en subschemes of E

k k

U i E ker D

D x x

k k

U U

D x D x

k

An intro duction to Katos Euler systems

k k k

Observe that U and U are stable under translation by ker D and

D x D x

there is an etale covering

k k

D U U

D x D x

We prove b elow the following lemma

Lemma A If z E S is any section disjoint from e the inclusion

k k

U E fz g induces an isomorphism

z

k k

K E fz g K U

k z k

Using this lemma we dene K theory elements whenever D

k

k

K E ker D pr pr

D

k D k D

k k

k

K U x i

D x D k x D

X

k k

k

x K U x

k D k D x k D

k

k

k k

k

x D x K E fD xg

D D

k k

i k

We call x Eisenstein symbols For k we simply dene i to b e the

D

x

section x E S and the Eisenstein symb ol then b ecomes a Siegel unit

i

x x O S

D D

e

If E E are isogenies of degree prime to D then by rep eated

k

application of iii one get the normcompatibility

k k

A

D D

k

Actually this is only of interest when all the are equal

i

Proof of Lemma A For any T S

for all i u e z

i

k k

U T u u E T

z k

for all i j u u

i j

k k

The complementary divisor E U is the union of the two divisors

z

k

V fu j for some i u eg u j for some i j u u e

i i i i j

and

k

W fu j for some i u z g

z i i

theory As S is regular the K groups in the lemma can b e computed in K

k

From the lo calisation sequence it is then enough to show that K V

k

W vanishes This is a sp ecial case of the following

z k

A J Scholl

k

Lemma A Let V V be any invariant open subscheme Then

k

K V is trivial

k

Proof Dene a sequence of reduced closed subschemes

k k k k

V V V V

k

k k

inductively by writing V for the smallest closed subset of V such that

r r

k k k

V V is smo oth over S Write V V V Then from the denition

r r r r

k

of V it is easy to see that

k

is a union of closed subsets each given by the vanishing of a certain i V

r

collection of expressions u u u which are p ermuted by

i i j k

ii This gives a decomp osition of V V as a disjoint union V

r r r

of op en and closed pieces p ermuted by in such a way that for each

k

there is some which acts trivially on V and for which

k

r

k

for each r In fact if V V This forces K

k

r r

X

c c K V V Z k

r r

M

K V Z k

r

c is evidently c c c whereas the comp onent of then

k

c by ii Now using the long exact sequences

K V K V K V V

r r r r

inductively b eginning with r k we deduce that K V

k

A Norm relations in higher K groups

Here we nd norm relations for the Eisenstein symb ols and for cuppro ducts

analogous to those in sections and

For the structure norm relations we use the same notation as in

k k

e

In addition write E E S for the isogeny dual to and for

S

k k

e

the isogenies on E E Consider the pushforward for the morphisms

k k

E S E

S S S

k k k

e

E S E

S S S

k k k

e

E E

S S

k i k k

Fix i f g and abbreviate x x The symb ol will

D D D

k

k

e

denote the analogue on E of D

An intro duction to Katos Euler systems

Lemma A The fol lowing relations hold in K

k

k k k

x y x x EN

S S D D D

k k k

k k

x y x x EN

D D D

S S

k k k

k

x x x EN

D D D

S S

k

Proof Here EN and EN are to b e understo o d on U and EN on

D x

k k

U The relations EN and EN are proved just as N and

D x

N by considering the Cartesian diagrams

i i

xy x

k k k

E S q E E

S

S S

y y

i

x

k k

E E

and

i i

xy x

k k k

E S E S q E S

S S S

k k

k

S S

y y

i

x

k k

e e

E E

k

and using the normcompatibility Applying to EN gives

S S

EN

k

We now consider cuppro ducts of the form in K Consider

k D D

the factorisation of multiplication by

k k

id

S S S S S S

k k k k

e

E S E E E S

S S

 

S S

We compute

k

k

x y x y

S S D D

k

k

x by N x x y

S S D D D

k k

k

x x x x by EN

D D

D D

k

We need to compute the image of this cuppro duct under Taking

S S

A J Scholl

the terms in turn

k

S S

k k k

x x x x x by EN

D D D

D D

k k

k

x x x x x by N

D D

D D D

k k

k

x x as deg x x

S S D D D D

Combining these gives the required generalisation of

Prop osition A

k k

x y x y x x

D D

S S D D

k k

k

x x x x

S S D D D D

Having got this far the analogue of presents no further diculty

n

Prop osition A If n and x x E S then

k

x y x y

n n n n

D

S S D

k

x y x y

n

n

D

D

The dual exp onential map

Notations

In this section K will denote a nite extension of Q with ring of integers

p

of K Write o for the integral closure of o We x an algebraic closure K

o in K and G for the Galois group of K over K We normalise all padic

K

b

b

valuations such that v p Let K b e the completion of K and write o

for its valuation ring Fix a uniformiser of o

K

n

We x for each n a primitive p th ro ot of unity in K such that

n

p

p

Write K K and denote by o the valuation ring of K

n n

n

p n p n n

p

Put d the relative dierent of K K

n n

i

For a top ological G mo dule M write H K M for the continuous Galois

K

cohomology groups

g

cycl

The cyclotomic character G Z is dened by g

n

n

p

cycl K p p

for every g G and n Its logarithm is a homomorphism from G to

K K

Z often viewed as an element of H K Z

p p

We normalise the recipro city law of lo cal class eld theory in such a way

that if LK is unramied then the norm residue symb ol LK equals

K

q

the geometric Frob enius inverse of the Frob enius substitution x x This

ab

implies that for any u o we have u K K N u

cycl K Q p

An intro duction to Katos Euler systems

The dual exp onential map for H and an explicit

recipro city law

Let V b e a continuous nitedimensional representation of G over Q Sup

K p

p ose that V is de Rham for generalities ab out padic representations see

G

K

b e the asso ciated ltered V for example Let DR V B

dR Q

p

i

K vector space with the decreasing ltration DR V induced from the l

tration on B Then Kato has dened a dual exponential map xI I

dR

exp H K V DR V

which is the comp osite

V H K Fil B DR V DR V H K V H K B

dR Q dR K

p

The last isomorphism comes from Tates computation of the groups

b

i

j H K K

b

i

H K K j unless j and i or and

log

cycl

b b

K H K K H K K

j j

b

j together with the isomorphisms B B K

dR dR

The group H K V classies extensions V V Q of

p

padic Galois representations and the extension V is de Rham if and only

if its class lies in ker exp This follows from remark b efore and

Lemma In particular the kernel of exp is the Blo chKato subgroup

H K V H K V

g

In some cases one can dene and study the dual exp onential map without

reference to B For example if V H A Q for an ab elian variety

dR p

AK it can b e dened just using the exp onential map for the analytic group

AK More generally if the ltration on DR V satises DR V then

one only needs to use the Ho dgeTate decomp osition

M

b b

i

K V i gr DR V K

Q K

p

iZ

since then by exp is the natural map from H K V to

M

b b

i

H K K H K K V i gr DR V

Q K

p

i

b

H K K DR V DR V

K

A J Scholl

Q for a In what follows we shall b e concerned with the case V H Y

p K

smo oth oscheme Y which is the complement in a smo oth prop er oscheme

X of a divisor Z with relatively normal crossings Write

i i

H Y o H X Z

dR Xo

the hyp ercohomology of the de Rham complex of dierentials with logarith

mic singularities along Z Then DR V is just de Rham cohomology with a

shift of ltration

DR V DR V H Y o K

dR o

DR V H X log Z K Fil H Y o K

Xo o dR o

DR V

Moreover the Ho dgeTate decomp osition has an explicit description essen

tially thanks to the work of Fontaine and Coleman To compute exp

one just needs to know the pro jection

b b b

K V K H X H X O log Z K

Q o o X Xo

p

b

K H X log Z

o Xo

It is the limit of the maps given by the diagram

1 2

n n

O p H Y H Y  O p H Y

n

o K Zar K p Zar

X

X

X

X

X

X

X

dlog

X

n

y

X

mo d p

X

X

Xz

n

log Z H X op

Xo

Remarks i The isomorphism lab elled comes ab out as follows Gener

ally let S b e a scheme on which m is invertible with  O An element

m S

of H S  is an isomorphism class of nite etale coverings S S Galois

m

with group  Given such an S S there is an op en Zariski covering fU g

m i

p

m

f It is easy to see that of S and units f O U such that S U U

i i i i i

ff g is a welldened element of H Y O m and moreover that the map

i K S

thus obtained ts into an exact sequence

1

H S  H S  H S O m H S 

Zar m et m Zar S Zar m

If S is irreducible as is the case here then every nonempty Zariski op en

subset is connected so  is asque for the Zariski top ology and the map

m

is an isomorphism

An intro duction to Katos Euler systems

n

Y induces an isomorphism H Y O p ii The inclusion j Y

o o K

n

O p denoted in the diagram To see this consider the eect H Y

K

n

of multiplication by p on the exact sequence

v

O Q j O

Y Y Y

o

K

k

the last map is the padic valuation along the sp ecial bre taking values

n n

p j O p It is in the constant sheaf Q This shows that O

Y Y

o

K

n n

p p b ecause then j O therefore enough to show that j O

Y Y

K K

n n n

p H Y O p H Y O p H Y j O

K o o Y

K

By passing to the direct limit we can replace K by a nite extension of

K Now consider more generally an op en immersion U S where S is a

separated no etherian scheme which is integral and regular in co dimension

i

Supp ose that m is invertible on U and that  O Then R j  for

m U m

i as  is Zariski asque The exact sequences

m

m

 O O

m

m

O O O m

give a short exact sequence

j O m j O m R j O

U U m U

But b ecause S is regular in co dimension one the divisor sequence

a

div

i Z O K

x S S

co dimx

is exact and therefore R j O

U

iii The simplest case which is however not enough for our purp oses is

when Y X is prop er when this recip e reduces to that given by Coleman

n

an element of H X  p is the class D of a divisor D on Pic X

n

K p K

n

X such that p D div g is principal One can assume that the divisor of

K

c

is precisely the closure D of D Put g on X

o

c

dlog g H X supp D

o X o

o

n

Then b ecause the residues of at supp D are mo d p one has

n n

mo d p H X op

Xo

n n

mo d p Coleman even denes such and this denes D mo d p

a map in the case of bad reduction Unfortunately we know of no reference

for this description of the Ho dgeTate decomp osition in the nonprop er case

A J Scholl

Now assume that X is a smo oth and prop er curve over o and that Y

is ane Then Z is a nite etale oscheme Recall see also the following

section that the dierent d d is the annihilator in o of If

n K K n o o

n n

and therefore is generated by K Q is unramied then o o

n

o o p n p

n

n

dlog and moreover d p o

n

p n p n

Theorem Suppose that K Q is unramied and that p There

p

exists an integer c such that for every n the fol lowing diagram commutes

c

up to p torsion

ch

H Y K  K Y o 

n

n

p

n p n

dlog

Ho chschildSerre

y y

 log Z H K H Y K H X o

n

p n n

X o o

n

n

mo d p

y

n

op Fil H Y o Fil H Y o H K

o dR dR n

o o

n

x

n

p log

dlog

n n o

cycl

y

p p

n

o d Fil H Y o o p Fil H Y o

n n o dR n o dR

Corollary The explicit reciprocity law The fol lowing diagram com

mutes

HS ch

H K H Y K K Y o  lim lim 

n

n

p

n n p

n n

dlog

y y

H K H Y K H X o log Z lim  lim

n

m n p

X o o

n

n

nm

o

y

Z Fil H Y o H K H Y K lim

p dR m

o o

n

n

exp

y y

tr

n

nm K K

n m

p

o d Fil H Y o lim K Fil H Y o

n n o dR m o dR

n

Remarks i The assumption that K Q is unramied is not essential for

p

the pro of and is only included to simplify the statement In general the

situation is completely analogous to b elow The case p is excluded

only b ecause we do not know a reference for the compatibility of the trace

maps in this case cf x

An intro duction to Katos Euler systems

ii The maps Ho chschildSerre comes from the Ho chschildSerre sp ec

tral sequence with nite co ecients cf x since Y is ane H Y

 K

n

p

iv For a discussion of the map dlog see x A priori its target is the

group H Y o We just explain why its image is contained

n

X o o

n

in the submo dule of dierentials with logarithmic singularities along Z By

making an unramied basechange one is reduced to the case when Z is a

union of sections Let A b e the lo cal ring of X o at a closed p oint of Z

n

and t a lo cal equation for Z Then by the lo calisation sequence one sees

that K At is generated by K A and symb ols fu tg with u o and

n

dlog fu tg u du dlog t

Proof First we explain precisely what are the transition maps in the various

inverse systems in the diagram In the Galois cohomology groups they are

n

given by corestriction and reduction mo d p The nite at morphisms Y

o Y o induce compatible trace maps cf x

n n

K Y o K Y o and

n n Y o o Y o o

n n

which are the maps in the rst and second inverse systems in the lefthand

the transition maps are trace side of the diagram In the system

n

o o

n

and in the remaining system o d the maps are tr For the

n n n K K

p

n n

compatibility of these various maps see b elow

From the discussion ab ove the diagram b elow commutes

b

H K H Y K Y o Q H K K Fil H

m dR p m o

exp

k

y

b

Y o Fil H Y o K Fil H H K K

dR o dR m o m

log

cycl

To deduce the corollary from the theorem it is thus only necessary to take

inverse limits and use the commutativity cf Prop osition b elow of the

following diagram

log

n

cycl

p

b

H K K K

n n

tr

cor

K K

nm

y y n m

p

log

m

cycl

p

b

H K K K

m m

Remarks i Consider the sp ecial case Y A fg Sp ec ot t Let

u lim o b e a universal norm By applying the corollary to the norm

n n

compatible symb ols fu tg K Y o one recovers a form of Iwasawas

n n

A J Scholl

cyclotomic explicit recipro city law which will b e proved more directly in

b elow

ii Theorem is proved in section b elow It is much easier than the

general cases considered by Kato in rst b ecause one is not working with

co ecients in a general formal group and secondly b ecause the assumption

that Xo is smo oth makes for considerable simplications In the nonsmo oth

case there is an analogous statement which is needed to compute the image

of Katos Euler system when p divides the conductor

Fontaines theory

We shall review here some of the theory of dierentials for lo cal elds devel

op ed by Fontaine and as a warmup for the next section show how it

gives a version of Iwasawas explicit recipro city law

Recall see for example xI I I that if K K is a nite extension

then its valuations ring o equals ox for some x o This implies that the

mo dule of Kahler dierentials is a cyclic o mo dule generated by dx

o o

and that its annihilator is the relative dierent d

K K

The mo dule equals the direct limit of taken over all nite ex

oo o o

In particular it is torsion tensions of K in K

Theorem There is a short exact sequence of omodules

a K

oo

where a a is the fractional ideal

oo

d o K a

oo p

K Q

p

and where K Z K olinear map satisfying is the unique

p oo

d

n

p

n

p dlog

m n

p m p

n

p

for any n

Remark In particular for any n the annihilator of dlog

n

p oo

n

o is p a

From we get the fundamental canonical isomorphism

b

T a

oo oo p

b

Z a to dlog olinear and maps which is

n n

n n p p p

An intro duction to Katos Euler systems

Supp ose that K K K are nite extensions Then there is an exact

sequence of dierentials

o

o o o o o o o

the rst exact sequence H which is exact on the left as well by

the multiplicativity of the dierent or alternatively by the argument in the

fo otnote on page Passing to the direct limit over K gives a short exact

sequence

o

oo oo o o o

At this p oint recall that for any short exact sequence X Y Z

of ab elian groups there is an inverse system of long exact sequences

n n n

X Y Z Xp Y p Z p

n n n

p p p

M for M X Y Z satisfy the MittagLeer If the inverse systems

n

p

condition ML then the inverse limit sequence

n n n

T X T Y T Z lim Xp lim Y p lim Z p

p p p

M satises ML is also exact a sp ecial case of EGA Note that

n

p

in two particular cases

M is surjec M the torsion subgroup of M is pdivisible then

n n

p p

tive

the pprimary torsion subgroup of M has nite exp onent then M

n

p

is MLzero

are divisible and and Applying these considerations to since

o o oo

is killed by a p ower of p we get an exact sequence

o o

T o T

p oo oo o o o p

Now pass to continuous Galois cohomology This gives a long exact sequence

since the surjection in has a continuous settheoretic section this is

o is discrete We are only interested in the connecting obvious here as

o o o

map and dene to b e the comp osite homomorphism

connecting

H K o H K T

o o o o o p K oo

n

The map reduction mo d p T induces a map on cohomol

n

oo oo p p

ogy which when comp osed with gives

K

n

mo d p H K

n

oo o o p K

A J Scholl

Lemma i The fol lowing diagram commutes

Kummer

o H K 

n

p

dlog dlog

y y

n

mo d p

K

H K

n

oo p o o

ii For any nonzero x o

n

dx mo d p x dlogKummerx

K

o with Proof i Simply compute if u o then x a sequence u in

m

p

u The comp osite dlog Kummer maps u to the class of u u u

m

m

the co cycle

g

g dlogu

n

n p oo

on dlog u rst lift dlog u in the exact sequence Now compute the eect of

K

then act by g to get the to the element dlog u T

oo m m p

desired co cycle So the commutativity is trivial

ii If x is a unit this is equivalent to i For the general case one simply

calculates as in i

Lemma Let n and assume that  K If p then the

n

p

diagram

log

n

cycl

p

n n

p H K a o p

oo

dlog

n n

o

p p

y y

n

mo d p

K

H K

n

p

oo

o o

n

commutes For p it commutes mod

Proof All the maps are o linear so it is enough to compute the image of

n

We have g mo d p for all g G hence log g

n

cycl K cycl p

n

mo d p and so if p then

n

log g g mo d p

cycl cycl

n n

p p

for all m and then In the pro of of one can take u

mn

m p

is represented by the co cycle dlog H K T

n

o o K p p

g

g dlog T

mn

m p oo p

An intro duction to Katos Euler systems

n

g p g

g

cycl cycl

and Applying the inverse of maps

m

mn mn

p

p p

this to the class of the co cycle

g a g

m

cycl p m oo

n

p

n

log g mo d p

cycl

n

p

The reader will make the necessary mo dications when p

We now need some elementary facts ab out cyclotomic extensions of lo cal

elds Our chosen normalisation of the recipro city law of lo cal class eld

theory identies the homomorphisms

log H K Z Hom GalK K Z

cycl p cts p

and

log N K Z

K Q p

p

As observed in the pro of of the previous lemma if  K then log

m

p cycl

m

mo d p

K Then for any nite extension K K Lemma Suppose that 

m

p

the diagram

log

m

cycl

p

b

o o H K

tr

cor

K K

y y

b

o H K o

log

m

cycl

p

commutes

Proof The statement follows from the pro jection formula for cuppro duct in

group cohomology since on H the corestriction

b b

o o H K o o cor H K

equals tr

K K

K Let b e the largest integer such that  Recall that K K

n

p n p

Then if n m direct calculation gives

nm

if m p o

m

p

o tr

n

p K K

n

n m

p o if m

Dene for any n m

K K tr t

n m K K nm

nm

n m p

A J Scholl

Prop osition If n m max the diagram

log

cycl

n

p

b

o o H K

n

p n

t

cor

nm

y y

log

cycl

m

p

b

o H K o

m

p m

is commutative If n the diagram

log

cycl

n

p

b

H K o o

n

n p

pt

cor

n

y y

log

cycl

b

o H K o

is commutative

Proof For n the second diagram commutes by with K K By

transitivity of trace and corestriction the lemma will b e proved if we verify

the commutativity of the rst diagram for n m Take the diagram

of for K K and factorise

m m

m

p log

cycl

b

'

o H K o

m

m p

tr

cor

K K

m

y y

m

m

p log

cycl

b

o po H K o H K tr

m

K K

m

p

p m m m

m

x

m

p

o

p log

cycl

&

o

m

p

K o Hom GalK The b ottom triangle commutes since H K o

m m cts m m

is torsionfree Hence the entire diagram is commutative and going round

the outside gives what we need

Now consider d d From the denition of d as the largest fractional

n K K n

n

ideal of K whose trace is contained in o it is an easy exercise to check

n

n

d p o

n

n p

By Propn the dierence v d n is b ounded so for some c indep en

p n

dent of n

c

o p o o

n

n n p

An intro duction to Katos Euler systems

Since is cyclic with annihilator d the homomorphism

o o n

n

xx dlog

n

p

n

p o

n

o o p

n

is welldened and its kernel and cokernel are killed by a b ounded p ower of

p by remark

Prop osition Let n m max Then the diagram

dlog

n

p

o

n

o o p

n

t

tr

nm

y y

dlog

m

p

o

m

o o p

m

commutes

Proof It is enough to compute what happ ens when m n Taking

p

for j p over o as basis for o

n

n n n

p

p p p

j j j

tr dlog trj d j dtr

n n n

n

p p p

p

and for j

dlog dlogN trdlog

n n n

p K K p p

n n

Therefore passing to the inverse limit gives a homomorphism

n

p lim o lim o lim

n n

o o p p

n

tr t t

which b ecomes an isomorphism when tensored with Q If K Q is unram

p

ied then is itself an isomorphism By Prop osition the

n

H K Zp is an isomorphism Invert canonical map H K Z lim

m m p

n

ing b oth of these arrows yields a diagram

Kummer

H K  Q lim o Q lim

n n n

n norm

n

dlog

p

y y

n

H K Zp Q lim Q lim

n o o

n

n trace

dlog

n

cor

o

p

y y

n n

H K Zp o p Q lim Q lim

n

m p

n t

t

nm n

y y

b

K H K K

m m

log

m

cycl p

A J Scholl

where down the righthand side all the inverse limits are with resp ect to the

n

corestriction maps and reduction mo d p We then have the following version

of the classical explicit recipro city law of ArtinHasse and Iwasawa Without

K loss of generality we can assume m chosen so that 

m

p

Theorem The diagram is commutative

n n

op one has the diagram Proof At nite level replacing Zp with

Kummer



H K  o

n

n p n

dlog dlog

y y

n

mo d p

K

n

H K

n

oo n p

n

o o

p

n

x x

dlog dlog

n n

p p

log

n

cycl

p

n n

p o op H K

n

p n

t

cor

nm

y y

log

m

cycl

p

n n

o p op H K

m

p m

n

This is for m for m the b ottom arrow should read p op

n

H K p op The top two squares commute by resp ectively

The b ottom square commutes up to ptorsion by All maps are com

patible with passing to the inverse limit As remarked after equation

has cokernel and kernel killed by the lefthand map lab elled dlog

n

p

a b ounded p ower of p and by the same is true for the one on the

right Therefore passing to the limit and tensoring with Q one obtains the

theorem

Remark One can use ii to describ e the image of an arbitrary element

of K under the Kummer map in a similar way

n

Here is the relation with the usual form of the explicit recipro city law Let

u u lim o b e a universal norm Its image down the left hand side

n n n

of the diagram equals with an obvious abuse of notation

dlog u

n

K tr u lim

m K K

nm

n m

n

p dlog

n

p

Going round the other way use the expression of the Kummer map in terms

of the Hilb ert symb ol which we write as a bilinear map K K

n n n

n

Zp given by

p

ab

n

xa

aK K

p

n

n n

x

n p

An intro duction to Katos Euler systems

n

Thus u is mapp ed to the co cycle in H K Zp which takes the norm

n n

ab

residue symb ol a K K to u a By the compatibility of the norm

n n n n

residue symb ol with norm and corestriction one gets that the image of the

family u in H K Z is represented by the co cycle ie homomorphism

m p

ab

a K K lim u a Z

m m n n p

n

Therefore the recipro city law says that this homomorphism and the homo

morphism

m

g p u log g

cycl

b

represent the same cohomology class in H K K

m

Prop osition I I IA ex Let c Hom K K K be the

K Q

p

unique map such that for al l T Hom K K and al l x K

Q

p

trx T tr x c T

K Q K

p

Then the diagram

c

K

Hom K K K

Q

p

log

y

Hom o K o log

cycl

cts

x

o

lo cal CFT

y

b

ab

K H G K Hom G

K K cts

is commutative

Remark Because of the normalisation of the recipro city law of lo cal class

eld theory used here see x this diers from the statement in by a

sign

Now the comp osite

c

K

Hom K Q Hom K K K

Q p Q

p p

is the inverse of the isomorphism K Hom K Q given by the trace

Q p

p

form Therefore for every a o

m

m

lim u a p tr u log a

n n K Q

m p

n

which is the limit form of the classical explicit recipro city law Ch

Thm

A J Scholl

Big lo cal elds

This section reviews the generalisation by Hyo do esp x and Faltings

x of Fontaines theory to lo cal elds with imp erfect residue eld We

consider elds L Q such that

p

L is complete with resp ect to a discrete valuation and

p r

its residue eld satises p

Fix such a eld L and write A for its ring of integers If R A is any

subring dene

n

b

lim p

AR AR AR

Fix also an algebraic closure L of L and let A b e the integral closure of A in

For any B with A B A and any subring R B set L

b b

lim

A A R B R

A

the limit running over all nite extensions A A contained in B

Let K L b e a nite extension of Q with ring of integers o and uni

p

formiser Then is prime in A if and only if Ao is formally smo oth

K K

by G and Theorems

Let L L b e a nite extension with valuation ring A Then A is nite

over A b eing the normalisation of a complete DVR in a nite extension

and is a relative complete intersection by EGA IV Therefore the

rst exact sequence of dierentials is exact on the left as well

A

A Ao A o A A

4 0

More generally if A A is a relative complete intersection of integral domains which is

generically smo oth then for any R  A the rst exact sequence is exact on the left For

0

an elementary pro of write A as the quotient B I of a p olynomial algebra B over A by an

ideal I generated by a regular sequence Then one has a split exact sequence

! B ! ! !

AR B R B A

as well as exact sequences for A or R

2 0

0

I I ! A ! !

B

B ? A ?

0

Applying the tensor pro duct A to and using and the snake lemma gives the

B

exact sequence

0

0 0 0

N ! A ! ! !

A

A A AR A R A A

2 0 0

0

where N kerI I ! A Since A A is generically smo oth the map

B

A A B A

2 2

0

I I ! is generically an injection hence N is torsion Now I I is pro jective

B A A A

0

since I is a regular ideal therefore N

A A

An intro duction to Katos Euler systems

As in we get an exact sequence of inverse systems

n n n

A p p p

n

p A A A Ao A o A A

is MLzero Since is a nite A mo dule the inverse system

n

A A A A p

and so passing to the inverse limit gives an exact sequence

b b

A

Ao A o A A A

b

Prop osition i is a nite Amodule generated by elements of

Ao

the form dy y A

ii If T T A are elements whose whose images in form a pbasis

r

b

then fdlog T g is a basis for the vector space L

i Ao A

b

iii If is prime in A then is free over A

K Ao

Proof By pp A is a nite extension of a complete DVR B

in which p is prime Then A B o is a complete DVR with uniformiser

and AA is nite and totally ramied Let k o o One knows

K K

loc cit Thm that the image of fdT g is an basis for and therefore

i k

L L

b

by Nakayamas lemma A dT A dlog T proving iii

A o i i

To deduce i and ii it is enough to apply the exact sequence to

AA o

Taking the direct limit of over all nite extensions L L one gets

an exact sequence

b b

A

Ao A A o AA

p p

of A mo dules Now apply again Since x dy pz x dz if y z

b

and are divisible Therefore since one sees using i that

o AA A

b

is nitely generated one can pass to the limit to get an exact sequence

Ao

b

of A mo dules

b

b b

T T A

A Ao p A o p AA

b

Because is a nite Amo dule the map has a continuous settheoretic

Ao

b

b

section write P N with P free and N torsion over A P one has

Ao

b

N is discrete so over it a continuous linear section of by freeness and A

one can take any section

One then has Hyo dos generalisation of see also

xb

A J Scholl

L be as in above and put a a A Prop osition Let a

oo Ao oo

Then there is an exact sequence of Amodules and Galoisequivariant maps

r

b

a L L

A A o o

where is given by the same formula as in and where the map is a

split surjection with right inverse

r

b

L

A o

X

n

p

n n

a A a p a p a dlogT

i

i r i

Remark Hyo do states this only in the case K Q but his pro of works in

p

general The key p oint which underlies Faltings approach to padic Ho dge

p

theory is that the extension L is almost unramied cf the T L

i

p

b

pro of of Prop osition b elow which shows that is generated as an

A o

n

p

b

dlog T mo dule by the forms dlog A

n

i

p

Corollary There is a unique isomorphism

b

b

T a

o p Ao A

Z to dlog which maps

n n

n n p p p

Remark Comparing and we have in particular

b

A

n n

oo o o p A p

Now consider as b efore the connecting homomorphism attached to the

Galois cohomology of for a not necessarily nite extension L L

contained in L

b b b

A H L T

L L Ao A p A o

For L L we write for If L L is nite the maps are

L L L L L L

related by a commutative diagram

b

A

Ao A

H

H

L L

H

canonical

y

H

Hj

L

b b

H L T

p A A o o

b ecause the exact sequence is functorial in A If L L is innite

b b

for as the direct limit of the maps H L T we dene

o L A o p A L

nite sub extensions L L L the analogue of still holds

The following lemma is proved just the same way as

An intro duction to Katos Euler systems

Lemma For any algebraic extension L L the fol lowing diagram

commutes

Kummer

A H L 

n

p

dlog dlog

y y

n

mo d p

L

b b

H L

n

o A p A o

Prop osition Let L L be an algebraic extension which contains al l

pth power roots of unity with valuation ring A and whose residue eld

p

extension is separable Suppose that r so that l l p Then for

j

b

is kil led by the maximal ideal m A and the j H L T

o p A

kernel and cokernel of

b d b

A H L T

o L L Ao A p A

are kil led by a power of p

Proof Initially there is no need to make any assumption on r Cho ose units

T T A whose images in l form a pbasis Consider the extensions

r

p p

and M ML Let B B b e the valuation T M LT

i i

rings of M M Then the residue eld of M is p erfect so Tates theory

b

i

applies in particular the groups H M A are m torsion for i

Therefore using the Ho chschildSerre sp ectral sequence and the fact that

m m the ination map

b b

j j j

d

H L A H M L B H M L H M A

is an isomorphism up to m torsion Now by Kummer theory and the hy

r

p othesis on the residue elds GalM L Z the isomorphism b eing

p

determined by the choice of fT g Therefore if r

i

j

d

H M L B for all j and

d d

H M L B B

Z

p

Now by there exists a noncanonical isomorphism of GalL L

b

b

A mo dules T Combining this and equations and

o p A

j

b

is killed by m for all j one sees that that H L T

o A p

For the second part we compute the coinvariants in First observe

n

p

that the ring A AT is nite over A and that A is a maximal ideal A

A J Scholl

in it Therefore A is a discrete valuation ring hence is the valuation ring of

n

p

d

It follows that any element of B has the form LT

X

a

b b T

a

aQ Z

p p

d

where T T and b A with b as jaj Let GalM L

a a

p

r r

p p

T for each r If b e the top ological generator for which T

r

p

b is divisible by then b b b where

p

X

r

xp x

d

B T b b

r r

xp p

r

xp Q Z

p p

d d

From this one sees that the inclusion A B induces an injection

d d

A H M L B

whose cokernel is killed by Now there is a diagram

p

d d

A H M L B

in

y

b

H L A dlog T

y y

L L

b b d

H L T A

p A Ao o

in which the vertical arrows have kernel and cokernel killed by a p ower of

p by and It remains to check that it is commutative

which having got this far is an easy exercise

A similar computation can b e carried out for all r using the isomor

r

phism GalM L Z and the Koszul complex In this way Hyo do

p

b

computed the cohomology of L j over L generalising Tates result His nal

result not needed here is

Theorem Theorem There are canonical isomorphisms

q

b

Q if j q

Ao

b

q

q

b

j H L L

Q if j q

Ao

otherwise

compatible with cupproduct For j q it is given by cupproduct

with log and for q j by and cycl

An intro duction to Katos Euler systems

Pro of of Theorem

Theorem is proved by reducing to the setting of the previous section

Recall that X is a smo oth and prop er curve over the ring of integers o of

a nite unramied extension K Q Assume that X is connected and that

p

X O o otherwise rst replace K by an unramied extension Let

X

X b e the generic p oint of the sp ecial bre Write also

A O L eld of fractions of A

X

A integral closure of A in L L L

n

n n n p

The elds L L satisfy the hyp othesis with r There is an obvious

n

lo calisation map Sp ec A Y Note that since Ao is formally smo oth we

b

actually have A A o and by is a free Amo dule of rank

n n Ao

Now use the fact that the map

b

Fil H Y o H X log Z

dR Xo Ao

is injective and its cokernel is torsionfree this holds b ecause the bres of

Xo are connected This means that the diagram in Theorem can

b e lo calised to Sp ec A without losing information We shall write down the

lo calised diagram and then explain why it implies

Prop osition There exists an integer c such that for every n the

c

fol lowing diagram commutes up to p torsion

ch

K A  H L 

n

n

p

n n p

dlog

Ho chschildSerre

y y

 H K H LK

n

p n

A o

n

n

b

H K A o p

o Ao n

o o

n

dlog

o n n dlog

y y

p p

n

p log

cycl

n

b b

H K o d op

n Ao n n o Ao

is simply Remarks i Since Ao is formally smo oth the valuation ring of LK

Ao

ii We have written

H LK ker res H L  H L  

n n n

n p n p p

 K A  ker ch K A  H LK

n n n

p n n p p

A J Scholl

The map marked Ho chschildSerre is then the rst edgehomomorphism

from the Ho chschildSerre sp ectral sequence

iii Concerning the b ottom righthand corner the natural map is

n n

b

o p dlog A p

A oo

but as Ao is formally smo oth

b b

o A

oo Ao oo A o

and the second summand is divisible

iv To deduce Theorem from the prop osition it is enough by what

has already b een said to show that there is a map from the diagram in

to the diagram ab ove Since the comp osite K Y o K A

n n

factors through H Y K one obtains the map H LK

n n

p p

K Y o K A The only remaining thing to check is that the

n n

diagram

  H K H LK H K H Y K

n n

p p n n

n

n

mo d p H K A o p

n

dlog

y y

n n

b

Y o H K op Fil H op H K

dR n Ao o n

commutes but this follows from the description of given in x

Proof of We reduce the diagram to the smaller diagrams in the fol

n

b

lowing three lemmas By p of rank one and is free over A

n

A p o

n

b

by we mean its tensor square as A p mo dule

n

p

A o

Lemma For any m n the diagram below commutes

ch

K A H L 

n

p

m m

dlog

y y

L

m

b b

H L

n

m p

A o

o A

m

b

in which the unlabel led arrow is induced by dlog 

n n

o A p p

Proof Since A is lo cal the symb ol A A K A is surjective Since

m m m m

the Chern character is compatible with cuppro duct the compatibility follows

by Lemma

An intro duction to Katos Euler systems

This reduces the computation to Galois cohomology Write

h i

b b b

ker H L H LK H L

n n n

p n p n p

A A o o o A

Lemma i The composite map

L

n

b b b

H LK H L

n n

p n p A o

o o A A n

equals zero

ii The fol lowing diagram is commutative

L

n

b b b

H L

n

n p A o o o Ao

A o

n n

id

K K

y

n

b b

H K Ho chschildSerre

n

n p oo o Ao

H id

y y

LK L

b b b

H K H LK H LK H K

n n n

A p n p oo o o n p

A o

has kernel and cokernel kil led by a bounded iii The map H id

L LK

power of p

Proof i The cuppro duct factorises as

L

n

L L K

n n

b b b

H L H K H L

n n n

oo o n p n p o o Ao n p A

A o

n

is zero it factors through and so its comp osition with the restriction to LK

b

H K H LK

n n

A p p oo o

ii The b ottom equality comes from The commutativity is a

general fact We have groups

GalL L GalLLK GalLK L GalK K

n n n

and two exact sequences of mo dules

A B C

A B C

given by and resp ectively On the rst acts trivially So we

A J Scholl

have the following diagram

H C H C H A H A

y

id ker res H A A H A A

HS

y y

H A H C H H A A

x

y

H id

H A H C H A H A

and it is a simple if tedious exercise to check this commutes

iii Follows from Prop osition applied to L LK

Lemma The fol lowing diagram commutes

dlog dlog

b

H LK  H LK

n

n

p p

o A

x

o

Kummer

n

b b

A o p H LK

n n

oo o o A p p

x

id

dlog

LK L

y

x dlog x

n n

p p

n

b b

op

n

oo o Ao p Ao

Proof This follows from Lemma

n

As K Q is unramied we have a a o by so d o p a

oo p n p

n

and od o ap

n

We now can make a big diagram

n



 H K H LK K A Zp H L 

n n

p p

n n n

y y y

b b b

H K H LK H L

n n

p n n p

A o

A o o A

n

o

b b b

H K H K H LK

n n n

oo oo o o o o Ao n p p A o Ao n p

n

x

o

dlog o

n n

y

p p

y

n n

b b



o d H K ap H K A o p

n n Ao n o Ao n

An intro duction to Katos Euler systems

To save space we have not lab elled most of the arrows they can b e found

in the corresp onding places in the sub diagrams apart from the

n

arrow lab elled y which is p log The top left square commutes

cycl

by and the top right square by functoriality of the Ho chschildSerre

sp ectral sequence The rectangle in the middle commutes by and the

b ottom left square by The remaining part of the diagram the right

hand hexagon commutes by

Going round the outside of the diagram in b oth directions gives two maps

n n

b

K A Zp H K ap

n n o Ao

and it is enough to show that their dierence is killed by a b ounded p ower

of p This follows from the commutativity of the diagram since the kernel of

the arrow marked is killed by a b ounded p ower of p by iii

The RankinSelb erg metho d

In this section we calculate the pro jection of the pro duct of two weight one

Eisenstein series onto a cuspidal Hecke eigenspace using the RankinSelb erg

integral In order to separate the Euler factors more easily we work semi

adelically regarding mo dular forms as functions on C R GL A The

f

passage from classical to adelic mo dular forms is wellknown but we review

the corresp ondence briey in x since there is more than one p ossible nor

malisation The same applies to the discussion of Eisenstein series in section

x

Notations

G denotes the algebraic group GL with the standard subgroups

a

P U Z

a

If R is a ring and H is G or any of the ab ove subgroups write H for the group

R

of R valued p oints of H If R R then H denotes fh H j deth g

R R

Q

The ring of nite adeles of Q is A Z Q If A Q C is

f Z p f

a character continuous homomorphism and M is a multiple of the conductor

of then ZM Z C denotes the asso ciated not necessarily

mo d M

primitive Dirichlet character for a Z a a mo d M Of

mo d M

course this means that if p M then p mo d M p

mo d M p

Write nite idelic and padic mo dulus as jj jj and archimedean

f p

absolute value as jj If there can b e no confusion we drop the subscripts

A J Scholl

and dene the standard congruence H Write also H H in place of H

Q f p A

p f

subgroups

o n

mo d p K p h K j h G K G

p p p Z

p

o n

K p h K j h mo d p

p

Haar measure on all the groups encountered is to b e normalised in the usual

way on Q the additive measure dx gives Z measure and on Q the

p p p

multiplicative measure d x gives Z measure On G G the subgroups

p f p

K have measure G

p

Z

Q

Fix additive characters Q C A C by requiring

p p f p f

n

n ixp

xp e for every x Z

p

H denotes the upp erhalf plane and H C R The group G acts on

R

H by linear fractional transformations Put

a b

G j det c d if H

R

c d

so that j

C of compact for the space of lo cally constant functions A Write S A

f f

supp ort The group G acts on S A by the rule

f f

g x S A x A g x

f f

If A and S A write for the function

f f

x x

So in particular if is the characteristic function of an op en compact subset

X A then is the characteristic function of X

f

Adelic mo dular forms

In the adelic setting a holomorphic mo dular form of weight k is a function

F H G C

f

which is holomorphic in the rst variable and satises

k

i For every G F g j F g

Q

ii There exists an op en compact subgroup K G such that F g h

f

F g for all h K

An intro duction to Katos Euler systems

iii F is holomorphic at the cusps

Any mo dular form F has a Fourier expansion

X

im

F g a g e H g G

m f

mQ

where a g when m this is the meaning of condition iii Put

m

Ag a g the Whittaker function attached to F Then A is a lo cally

constant function on G which satises

f

b

A g bAg for all b A

f f

One can recover the remaining Fourier co ecients apart from the constant

term from Ag by

m

k

g if m Q a g m A

m

It is convenient to intro duce the normalised Whittaker function

k

A g Ag jdet g j

f

The group G acts on adelic mo dular forms by right translation and the

f

translates of F by G generate an admissible representation call it From

f

the denition is isomorphic to the representation generated by Ag the

k

representation generated by A g is isomorphic to the twist jdet j If

f

is irreducible it has a factorisation and the centre of G acts on

p f

the space of via a character With the normalisation used here there is a

k

C with such that Q nite order character A

f

a

k

ajaj for all a A

f f

a

This means that

a

g aA g for all a A and g G A

f f

a

If F comes from a newform on N then Ag is factorisable there are

Q

functions A G C satisfying A A g Ag for all g

p p p p p

g G and such that

p f

a b

k

g ajaj bA g for all a Q and b Q A

p p p p p p p

a

A J Scholl

Supp ose A is K invariant Then is unramied and its lo cal Lfunction

p p p

is given by

r r

X X

p p

r k s r k s

p p A L s A

p p p

r r

s s

p p

p p

where

p

k k

and pp p A

p p p p p p

this is the normalisation of Lfunctions which gives the functional equation

for s k s it diers from that of JacquetLanglands by a shift

Complex conjugation of Fourier co ecients denes an involution of the

space of mo dular forms In representation theoretic terms this b ecomes the

isomorphism

If A Q C is any character of nite order and F is an adelic mo dular

f

form of weight k so is

F g det g F g

To go from adelic to classical mo dular forms let K n b e the standard level

Then n subgroup of G

Z

K n Y nC G nH G K n G nH G

Q f Z

Z

where the last isomorphism is normalised in such a way that the p oint h

H GL ZnZ corresp onds to the elliptic curve E C Z Z with

level structure

n n h v v

h

Z ZZ Z ker n ZnZ

E

n n

Write z for the co ordinate on E and let F b e an adelic mo dular form which

is invariant under K n It corresp onds to the classical mo dular form over C

k k

E F hdu H E !

h

is any lifting of h where h G and h G

ZnZ

Z

The map e Y N Sp ec Q  is then given on complex p oints

N N

by

e g det g N if g G

N f

Z

the sign dep ends on the normalisation of the e pairing N

An intro duction to Katos Euler systems

Eisenstein series

Here we establish notations for Eisenstein series in the framework of the

previous section The results quoted can b e obtained easily from those found

in classical references probably Chapter VI I is closest to what is found

here

Let S A with the action of G The series

f f

X

k s

m m jm m j g m E g

k s

Q m

is absolutely convergent for k Res with a meromorphic continuation

and satises

k s

jj j E g for all G E g j

k s Q k s

The functions E E are holomorphic and therefore mo dular of

k k

weight k if k or k if k they are holomorphic provided that

R

A

f

The map E is G equivariant In particular if d Q then

k s f

in the notation of

k s

E d d jdj E

k s k s

One can rewrite the Eisenstein series as a sum over the group If f G C

f

is a lo cally constant function satisfying

a b

k s

f g a jaj f g for all a d Q b A

f

d

then dene

X

k s

f g j jj j E g

k sf

nG P

Q Q

The relation b etween the two denitions is that E E with

k s k sf

X

x

k s

f g g x jxj

xQ

Moreover every E is an E for some

k sf k s

In the normalisation used here the Whittaker function of E is

k

Z

k

X

i

x

k s

xy g dx lim y jy j B g

f

y

s

k

y Q

A f

A J Scholl

which can b e obtained without to o much diculty from the classical formulae

see eg pp

One can decomp ose the Eisenstein series under the action of the centre of

G It is more convenient to replace f and B by the normalised functions

f

cf ab ove

k s k

f g f g jdet g j B g B g jdet g j

f f

and then to write

X X

f g f g B g B g

k

where the functions f g B g are zero unless in which

case

Z

a

g d a af f g

a

Q A

f

Z

x

k s

k s

jdet g j xjxj g d x

f f

A

f

and

Z

a

g d a B g aB

a

Q A

f

Z

k

i

x

k

k s

jdet g j y jy j xy g dx d y

f

f f

y

k

s

A A

f

f

Q

is factorisable with equal to the characteristic function of Z If

p p p

for almost all p then the expressions ab ove factorise and one has

k

Y Y

i

f g f B g B g for g g G g

p p p f

p p

k

p p

where the functions f are given by lo cal integrals B

p

p

Z

x

k s k s

d x xjxj g f g jdet g j

p p p p p p p

p

Q p

An intro duction to Katos Euler systems

Z

x

k s k

dx d y xy g y jy j g jdet g j B

p p p p p p p p

p

y

s

Q Q

p p

Z

xy

k s k

dx d y y jy j xg jdet g j

p p p p p p p

y

s

Q Q

p p

In fact the integral in is a nite sum b ecause the xintegral is a nite

linear combination of integrals of the form

Z

xy dx

p

tp Z

p

which vanish if y is suciently close to So one can omit s from the formula

Because of and the functions f and B are determined

p p

by their restrictions to the subgroup

Q

p

K G

p p

and these are given by

m

k s

h mjmj f h f

p

p p

Z

x m

k k

dx d y y jy j mxy h h jmj B

p p p p p

p

y

Q Q

p p

Z

xy

k k

dx d y y jy j mxh jmj

p p p p p

y

Q Q

p p

The RankinSelb erg integral

Let F G b e adelic mo dular forms of weights k l k resp ectively at least one

of which is a cusp form and let E b e the Eisenstein series The

l sf

pro duct

k l s k l s

E G F y jdet g j d d

l sf

is a left G invariant form on H G and the aim of this and the following

Q f

sections is to compute the inner pro duct

Z

hE G F i dg

l sf

G nHG

f Q

A J Scholl

which is a RankinSelb erg integral

Prop osition Let Ag B g be the Whittaker functions of F and G

Then

ik l s

hE G F i

l sf

k l s

Z

m m m

k l s

A h jmj d m dh f h B h

G A

f

Z

Proof A very similar calculation is done in x here we simply write the

equations with little comment

Z

X

l s

hE G F i f g j jj j

l sf

nG P

Q Q

nHG G

f

Q

k l s

k l s

G g F g y jdet g j d d dg

f

Z

k l s k l s

i f g G g F g y jdet g j dx dy dg

f

P nHG

f

Q

Z

X

m

A g i f g B

mQ

P nHG

f

Q

my k l s k l s

e y jdet g j dx dy dg

f

Z

k l s

y k l s

i Ag e y jdet g j dx dy dg f g B g

f

Z U nHG

Q Q f

Z

ik l s

k l s

Ag jdet g j dx dg f g B g

f

k l s

Z N nRG

Q Q f

To get the nal result use the parameterisation

A A fg G Z G

f f Q f

Z

m b

h b m h

in terms of which integration is given by

Z Z

g dg b m hjy j db d m dh

G Z G fg A A

f Q f

f Z

An intro duction to Katos Euler systems

b

g the integral in the last line Ag is invariant by g Since f g B g

ab ove splits as a pro duct

Z Z

m

h d m dh A f B dx db

G fg A QnRA

f

f

Z

and the rst factor equals by choice of Haar measure

Now supp ose

F is a cusp form b elonging to an irreducible with central

p

Q

character whose Whittaker function Ag A g is factorisable

p p

G E is an Eisenstein series and f f for factorisable functions

k

Q Q

S A

f p p

Then the integral in the previous prop osition can b e decomp osed under the

action of the centre and then factorised giving

Prop osition Under the above hypotheses

Y X

I hE E F i C

p p p l s k

p

k

i k l s

where C and

k l s

k

Z

m

I f B h jmj d m dh A

p p p p

p p

K Q

p p

and f are as in above The sum is over al l pairs of B

p

p

l

characters A Q C such that and

f

Remark It will b ecome clear in the computation that follows that the

sum over characters is actually a nite sum

Lo cal integrals

X

for the characteristic function of a subset X Y A Write char

f

Y

The next prop osition will compute the lo cal integral I for almost

p p p

all primes

A J Scholl

Prop osition Suppose that

Z

p

char

p p p

Z

p

and that A is K invariant with A Then

p p p

L k l s L l s if is unramied

p p p p

I

p p p

otherwise

Remark Since has a K invariant vector is unramied at p There

p p

fore since either b oth or neither of are unramied

p p

Proof See x If then by for h K m Q

p p p p p

Z

m

l s l s

xjxj d x h mjmj f

p p

p

Z fg

p

L l s if is unramied

p p

otherwise

Moreover by

Z

m

k k

h jmj B y jy j mxy dx dy

p p

p

Z Z fg

p p

where the xintegral equals is my Z and vanishes otherwise The y

p

integral then vanishes if giving

p

Z

p

P

r k j r j k

p p p

p

j r

m

h B

if r v m and is unramied

p

p p

otherwise

Thus I unless b oth are unramied at p which we now

p p p

are K invariant and assume Then f B

p

p

p

r

X

p

r

I p f A B

p p p p

p p

r Z

r

X

p

r r l s

A L l s p p B

p p

p

r

An intro duction to Katos Euler systems

Now from

r

X

p

r

T B

k k

p

pp T p T

p

r

and therefore by Lemma one gets

L k l s L l s

p p p p p

I L l s

p p p p

L l s

p p p

Since and the result follows

Corollary Under the hypotheses of Proposition let S be a nite

set of primes such that for every p S A is K invariant

p p p p p

and A Then

p

hE E F i

l s k

Y X

L l s I C L k l s

S p p p S

pS

where the sum is over characters unramied outside S with

l

and

Here L denotes the Lfunction with Euler factors at all p S removed

S

At other primes we use the following choice for

p

Prop osition Let t Q with v t Suppose that

p p

t Z

t p

char

p p

Z

p

a b

K Then Let m Q h

p p

c d

l s l s

p amtjmj

p

p

m

h f

p

if cond and h K p

p

otherwise

Proof Straightforward calculation from

At the bad primes for F we are going to cho ose in such a way as to

p

make the lo cal factor b e simply a constant

A J Scholl

The standard way to achieve this is to use a suitable AtkinLehner op erator

n

to replace the co ecient of q in the q expansion by zero whenever pjn In

representationtheoretic language this means to use the vector in the Kirillov

mo del which is the characteristic function of Z See Thm and

p

also compare For Eisenstein series it is easy to write down a

parameter which do es the trick although p ossibly this do es not give the

p

b est constant in b elow

Prop osition Suppose

t Z t p Z

p t p

char char

p p

p

Z Z

p p

a b

where t Q K p K p v t Then for al l h

p p

c d

Z

amt y y d y

p p p

m

p Z

h B

p

if cond and m Z

p

p p

otherwise

Remark In the sp ecial case this b ecomes

p Z Z

p p

char char

p p

p

Z Z

p p

and for all h K p and all m Q

p

if is unramied and jmj

m

p

h B

p

otherwise

Z t

p

Proof First consider what happ ens when char For every

p

Z

p

h K p one has

at Z

p

h char

p

Z

p

which gives

Z

m

k

y mxy dx d y h jmj B

p p

p

at Z Z

p p

An intro duction to Katos Euler systems

and the xintegral vanishes for m Z and equals amt y otherwise

p p

Therefore for m Z b ecomes

p

Z

k

y amt y d y jmj

p p

Z

Z

k p

jmj amt y y d y

p p p

mt Z

p

Now if B G C is any Whittaker function ie satises and is

p

lo cally constant which is invariant under K p some then the

function

X

p x

e

B B B

p

x mo d p

satises for every h K p K p

m m

e

h mB h char B

Z

p

Since

a

p x

t Z Z t p Z Z

p p p p

x mo d p

the result follows

From Prop ositions and one obtains

Prop osition Suppose that A is invariant under K p and that

p

A Let t t Q with v t v t and

p p p p

t t

Then if and

p p p p

Z

l s

p t t y y d y

p p p p

p

I

p Z

p p p

p

if cond

p

otherwise

Remark Since one has cond cond

p p

Proof If cond then I Otherwise if h K p and

p p p p

A is K p invariant one has A h aA aA so that

p p p p p p p

A J Scholl

I equals

p p p

Z Z

l s

p y y d y A h dh at at

p p p p p

p

K p

p Z

p

Z

l s

y y d y t t p vol K p A

p p p p p

p

p Z

p

and vol K p K K p p p

p

There is just one more case to consider in order to compute the image of

the Euler system in the cyclotomic tower

Prop osition Suppose that

Z

tr t

p

char v t

p p p

t Z

p

a b

K p and h Then for al l m Q

p

c d

k k

p jmj dt mbd

p p

p

m

h B

p

and v m if cond

p p

otherwise

bt Z

p

char Proof One has h and dt Z dt p Z therefore

p p p

dt Z

p

by

Z Z

k k

mxy dx d y B y jy j jmj

p p

p

bt Z dt p Z

p p

if v m

p

Z

k k

y mbdy d y if v m jmj dt p

p p p p

p Z

p

If v m and y p Z then mbdy mbd and the

p p p p

result follows

Corollary Suppose there exists a character Q C such that

p p

is unramied and that A is the twist of the spherical vector that

p p p

is A g det g A g is K invariant and A Put

p p p p p p

tr t t

with v t v t

p p p p p p

An intro duction to Katos Euler systems

Then if is unramied and cond

p p p

k l s

I p t t L l s

p p p p p p p

p p

and otherwise I

p p p

Proof The central character of is times an unramied char

p p p p p

acter so one of is unramied if and only if b oth are By

p p p p

Prop ositions and I whenever cond or cond

p p p p p

Otherwise

k l s

p t t I

p p p p p

p

Z

m

sl k

h d m dh A am djmj

p p p

Q K p

p

k l s

p t t

p p

p

r

X

p

r r l k s r

A p p p

p p p p

r

Z

a ddet h dh

p p

K p

The integral in the last expression vanishes unless and are un

p p p p

b ecomes ramied in which case I

p p p

k l s

p t t

p p

p p

r

X

p

r l k s r

p A p

p p p p

r

k l s

p t t L l s

p p p p

p p

giving the desired expression from

Remark One can do exactly the same computation merely assuming

that A is K p invariant I vanishes unless is unram

p p p p p p p

ied in which case one gets the formula

A J Scholl

Putting it all together

We change notation slightly from the previous sections Begin with a cusp

form F of weight k l generating an irreducible and whose Whit

p

Q

taker function Ag A g factorises with A for all p Let b e

p p p

the character of the centre of G on A Assume that the following data is

f

given

i Disjoint nite sets of primes S and T such that if p S then A is

p

K invariant In particular this means that is unramied outside S

p

ii A character A Q C unramied outside T

f

iii For each p S elements t t Q with v t v t

p p p p p p p p p

p

invariant such that and A is K p

p p p p

iv For each p T elements t t Q with v t v t where

p p p p p p p p

maxcond

p p

Put

Y Y Y

p p p

N p M p R p

pS pS pT

Denote by t t the nite ideles whose comp onents at primes p S T are t

p

t and which are elsewhere Pick y Z such that

p

p

p t

p

p

y mo d p for all p S

N t

p

thus y is welldened mo d M note that the righthand side b elongs to Z

p

The integral to compute is

hE E F i hE E F i

l s k l s k

where

F g det g F g

are given as follows is the twist of F by and

t t

p p

For p S and

p p

p p

t t

p p

tr

and For p T

p p

p p

For p S T

p p p

An intro duction to Katos Euler systems

We can then assemble the previous calculations Put and

for a variable character thus the central character of A

Then only those satisfying the following conditions contribute to the sum

k

If p S then is unramied

p

If p S then cond

p p

So cond jM cond jN and cond jR This gives

hE E F i C L l k s

l s k S T

X Y

L l s I

S T p p p

k

pS T

cond jM

by and where

Z

Y Y

l s

t t y y d y I N

p p p p p p p p p

p

pS pS

p

Z p

p

with

Y Y

p p

t t p t t t

p p p p p p p p p p

pS pS

Y

p

y t p MN

mo d M p p p p

pS

and

Y

I

p p p

pT

Y

k l s

R t t L l s

p p p p p p

p p

pT

and for each p T t t t t t This gives

p p p p p p p p p

hE E F i

l s k

Y Y

l s k l s

C N R

p p

pT pS T

Y

t M N t t L l k s

p p p S T

pT

Z

X Y

p

y y d y p

p p p

k

pjM

p

Z p

p

cond jM

y L l s

mo d M S

A J Scholl

In the third line of this expression the pro duct over pjM can b e rewritten in

terms of a classical Gauss sum as

X

ixM

xe M

mo d M

xZM Z

Here is Eulers totient function The sum over characters in then

b ecomes combining o dd and even characters

X X X

k

M

m

cond jM xZM Z

mN

ixM l s

xy m e a m

mo d M m

X

k l s

my M my M a m

f f m

m

mN

by the character orthogonality relations

For A write

f

X

s

L s m a m

S f m

m

mN

for the twisted Dirichlet series

Theorem Under the above hypotheses

l s k l s

hE E F i CN R GL ZR Z

l s k

Y Y

L l k s t M N t t

S T p p p

p

pT pS

k

L l s y M L l s y M

S S

with C as in Proposition

The Euler systems

Mo dular curves

We can at last give Katos construction of an Euler system in the Galois

cohomology of the mo dular curve Y N over a family of ab elian extensions

of Q We assume throughout that p is a prime not dividing N

An intro duction to Katos Euler systems

Pick auxiliary integers D D which are prime to N p and put

R fsquarefree p ositive integers prime to N pD D g

p

m

R fr r p j r R m g

p p

univ

We supp ose that for each r R we are given p oints z z E Y N r

p r r

ZN r such that

If r and r s R then sz z ie one has elements of the inverse

p r s r

limit

Y

Z z z lim ker N r Z

r r r r p

r R

j N pD D

p

of torsion p oints on the universal elliptic curve

For every r R the p oints N z and N z generate ker r in partic

p r r

ular the orders of z z are multiples of r

r r

m

If r p then the orders of z z are divisible by a prime other than

r r

p

such that Remarks i The rst condition implies that there exists e Z

p

m

for every r r p R the Weil pairing of z and z is

p r r

er

primetop ro ot of e z z

m

p

N r r r

ii The third condition is really only added for convenience It ensures

that for every r the p oints z are not of prime p ower order which means

r

univ

that they do not meet the zero section of E Y N r in any characteristic

z It follows from ii that the mo dular units z actually b elong

D r D r

to O Y N r for any r R Dene

Z p

Y N r z e f z g K

r D r D r

and also

Y N Q  N e K

r r Y N r Y N Q r

r

by what was just said these b elong to the images of K of the mo dels over

Sp ec Z

Let T T hai hai denote the Hecke corresp ondence and dia

Y N Y N

mond op erators as in x ab ove If r write Frob GalQ  Q for

r

the geometric Frob enius automorphism so that Frob where

r

is the arithmetic Frob enius substitution For every nite eld extension r

A J Scholl

L L write simply N for the norm map K Y N L K Y N L

L L

Notice that if j N r then

N T T N

Y N r Y N Q Y N r Y N Y N r Y N Q

r r

N hi hi N

Y N r Y N Q Y N r Y N Y N r Y N Q

r r

since T acts as on the constant eld and hi acts as by x Therefore

from and one obtains

Theorem Let r R Then

p

i N

Q Q r p r

r p r

ii If is prime and NDD r then

N T hi Frob hi Frob

r Q Q r

r r

Now write

Q Z T H Y N

p pN Q

m

and consider for r r p the homomorphisms

lim K Y N Q  

n

n

p

r p

nm

AJ

y

Q  H Y N H Q  lim

n n

p Q r p

nm

cor

y

H Y N H Q  lim Q 

m n

Q r p p

nm

H Q  T

r pN

n mg is an element of the rst By i the family f

n n

p r p

group This twisting of elements of K was used rst by Soule Let

Y N H Q  T

r r r pN

b e its image On the one hand GalQ  Q acts on H Q  T since

r r pN

T is a Gal Q Q mo dule on the other the level N Hecke op erators T

pN

hi act by functoriality Also Frob acts as on  By Theorem the

n

p

classes therefore satisfy Euler systemlike identities r

An intro duction to Katos Euler systems

Corollary i For al l r R cor

p Q Q r p r

r p r

ii If is prime and r r R then

p

cor T hi Frob hi Frob

r Q Q r

r r

In the next section we will pass to an elliptic curve and get an Euler system

in the sense of x of

Recall from x the denition of the weight Eisenstein series for any

N and any D which is prime to N

Eisz z dlog H X N !

D v D

univ

dened for any z E Y N ZN Z The form Eisz extends

D

to X N provided the order of z is divisible by at least primes

Z

Recall from x the Ko dairaSp encer isomorphism

KS KS H X N ! H X N log cusps

N Y N X N Q

identifying holomorphic mo dular forms of weight and dierentials with at

ord

worst simple p oles at cusps Let Y N b e the complement in Y N of the

Z

nite set of sup ersingular p oints in characteristic dividing N The scheme

ord

Y N is smo oth over Z by Cor

N

Prop osition The KodairaSpencer map divided by N extends to a

ord

homomorphism of sheaves on Y N

KS !

ord

univ ord

N

Y N Z

E Y N

N

N

with logarithmic singularities at the cusps

N

Proof The Ko dairaSp encer map takes the mo dular form f q dtt to

N n N

the dierential f q dq q N f q dlogq So on q expansions it is

divisible by N The result follows by the q expansion principle

Remark One knows that always assuming that N is the pro duct of two

coprime integers each the scheme X N is regular Therefore the mor

phism e X N Sp ec Z is a lo cal complete intersection b eing a at

N N

morphism of nite typ e b etween regular schemes EGA IV Therefore

the sheaf of relative dierentials extends to an invertible sheaf on X N

Z

namely the relative dualising sheaf sheaf of regular dierentials and one can

then show that N KS extends to an isomorphism of invertible sheaves

N

on all of X N

Z

reg

KS ! log cusps

N

X N Z

N

N

This is not needed in what follows

A J Scholl

ord

Because Y N is smo oth over Z one has

N

ord ord

Z Z

Y N Z Y N Z

N

N

and is killed by N and generated by dlog

N

Z Z

N

univ

Prop osition Let z z E Y N be disjoint from ker DD

Z

In the identity

ord

Y N Z

dlogf z z g KS Eisz Eisz dlog e z z

D D N D D N

N

holds

Proof This can b e checked on q expansions Supp ose that on the completion

univ

of Y N along a cusp we have xed an isomorphism of E with the Tate

a

N a N

curve Tateq over Z q and that z z are the p oints z q

N

N

b

b N

q Applying the congruence and the fact that e z z z

N N

a b a b

one get the desired result We have normalised the e pairing as

N N

in

We can now give Katos description of the image of the Euler system f g

r

under the dual exp onential map see x ab ove

exp H Q  T Q Q  Fil H Y N Q

p r pN p Q r Q dR

recalling that Fil H Y N Q H X N log cusps

dR

X N Q

Dene the following dierentials on the mo dular curve in terms of the

weight Eisenstein series

e Eisz KS Eisz H X N r log cusps

r N r D r D r

N r

tr e H X N Q  log cusps

r X N r X N Q  r r

r

Theorem For every r R

p

e

exp

r p r

r

where e Z is as in

p

ord

Proof By we have in H X N r log cusps the identity

X N r Z

dlog e e dlog e z z

r r N r r r

m

Now take r r p and tensor with Z Then by

p

ord

dlog e r H X N r Z log cusps e e dlog

m

r p r p

An intro duction to Katos Euler systems

Taking the trace to X N Q  gives using the compatibility x of

r

trace and transfer

dlog r e dlog

m

r r p

Let o b e the ring of integers of Q  By the explicit recipro city law

n

n p p

e r

tr lim exp

n

Y N o Y N o r p p r

n

n m

n

p

nm

But since tr e p e by this gives the desired

n n

Y N r p Y N r r p r

formula

Elliptic curves

Supp ose that E Q is a mo dular elliptic curve of conductor N with a Weil

E

parameterisation

X N E

E E

Cho ose a prime p not dividing N and write T E H E Z of

E p p

course this is the same as the Tate mo dule of E but it is b etter to think in

terms of cohomology esp ecially if we were to work more generally with any

weight eigenform with character Let the Lseries of E b e

X

s

LE s a n

n

n

again this is b est thought of here as the Lseries attached to the motive

h E Let N b e any p ositive multiple of N with N p The actual

E

choice of N is to b e made later Consider the comp osite morphism

E

X N X N E

E N E

There are Galoisequivariant maps of restriction and direct image

restriction

H X N Q Z H Y N Q Z T

Q p Q p pN

E N

y

Q Z T E H E

p p Q

Now the ManinDrinfeld theorem or rather its pro of implies that there is

cusp

an idemp otent in the Hecke algebra with rational co ecients which N

A J Scholl

induces for every p a left inverse to the map lab elled restriction So for

some p ositive integer h indep endent of p the comp osite map

E

cusp

h T T E

N

E E N pN p

is welldened Cho ose D D prime to N p and systems z z as in the

r r

previous section

Theorem Dene for r R

p

cusp

E h Y N H Q  T E

r E E N N r r p

Then the family f E g is an Euler system for T E that is

r p

For every r R cor E E

p Q Q r p r

r p r

If is prime and NDD r then

E cor E a Frob Frob

r Q Q r

r r

where Frob GalQ  Q is the geometric Frobenius

r

Remark Actually Rubin considers cohomology classes not over Q  but

r

m

rather over the subeld Q  where r r p and Q Q is the unique

m r m

m

extension of degree p contained in the cyclotomic Z extension of Q To

p

get an Euler system in the precise sense of x one should therefore

take the corestriction of E to Q  Note that his formula for the

r m r

norm relation diers from that here as we are using geometric Frob enius as

Nekovar has explained to us the relation ii can b e rewritten more concep

tually as cor Q Frob where Q x det Frob x j T E

r r p

Writing P x Q x one gets the same formula as in loc cit

Proof The rst statement follows directly from the corresp onding statement

i for Y N The second follows from ii together with the

r p

fact that hi and T a on T E

p

cusp

On dierentials the pro jector is the identity on cusp forms and

N

annihilates Eisenstein series Put

cusp

cusp

H X N Q 

r N r r

Then Theorem gives

eh

cusp

E

exp E

p r E N r

r

To compute this in terms of the Lfunction use the RankinSelb erg integral

from x Fix a dierential on E Q such that is a newform on

E E E

X N which we write as iF g d for a weight cusp form F whose

E

Whittaker function satises

An intro duction to Katos Euler systems

A is K invariant if q j N and is K q invariant if ord N

q q E q E

A for all q

q

This means that A q q a for every q j N and that LE s L s

q q E

where is the representation of G generated by F

f

At this p oint it would b e wise to recall that we have in x normalised

the recipro city law of lo cal class eld theory to take uniformisers to geometric

Frob enius This gives the classical isomorphism

global CFT

ab

GalQ Q Z A

Q

a

a

n n

If is any idele class character of conductor M with asso ciated Dirichlet

character ZM Z C we then have

mo d M

X

s

L s LE s a mm

mo d M m mo d M

mM

We also dene the incomplete and twisted Lseries

X

s

L E s a mm

N mo d M m mo d M

mM N

X

im s

L E s e a m

N m

mN

as in x ab ove Now put

numb er of connected comp onents of E R

E

fundamental real p erio d of

E

E

fundamental imaginary p erio d of

E

E E

so that

Z

iR

E E E E

E C

th

ro ots of The set of complex p oints Sp ec Q  C is the set of primitive r

r

ixr

unity fe g in C which we identify with Zr Z Write Q  C

x r

ixr

for the corresp onding emb edding e Supp ose A Q C is

r f

A J Scholl

a character of conductor dividing r Then we can compute

Z

X

cusp

x

mo d r x E N r E

xZr Z

E C

Z

cusp

x

mo d r x r E N E

Y N C Zr Z

Z

x

mo d r x r E N E

Y N C Zr Z

since cusp forms and Eisenstein series are orthogonal

Z

e e

mo d N r N r r E N r E

Y N r C

where the map e Y C Sp ec Q  C ZN r Z is that dened

N r N r N r

in

At this p oint we need to cho ose the parameters z z of the Euler system

r r

in such a way that the expression can b e computed using Theorem

r

In fact it will b e necessary to replace E by a certain linear combi

r

nation of Euler systems The choices to b e made are b est broken down into

a numb er of steps

Step Fix a prime p with p j N and fg We will restrict to

E

characters with

Step If y M Q the value of the twisted Dirichlet series at s

is a p erio d integral

i

Z

X

n

a q i d L E

n MN

E

nM N

E

and one knows that this is a rational multiple of a p erio d along a closed path

P

n

in X N C for suitable N Moreover the cusp form a q dq q is

n

nN

obtained from the eigenform by applying a suitable Hecke op erator It

E

follows that for any Q

L E L E

MN MN

E E

is a rational multiple of Moreover one can nd with denominator

E

prime to any chosen integer for which is nonzero by

We cho ose an y M with M and M y M p and for

which is nonzero By what has b een just said there will b e a nite

An intro duction to Katos Euler systems

collection of such y M which will cover all p ossible choices of p We then

take

Y

q

max ord N ord M N q

q q E q

q jMN

E

Step Fix auxiliary integers D D with DD pN and D

E

m

D mo d M Let r r p R thus m and r is squarefree

p

and coprime to pD D N In the notation of x we put R r T fq jr g

S fq jN g and cho ose the ideles t t A to have lo cal comp onents

f

if q j M r

if q j N r

t t

r y jM j if q jM

q q

q

N r if q jN r

M r if q jr

Then holds and t N r Z t M r Z In x this data then

Let Z b e the nite unit idele determines functions S A

f

D if q jN r

q

otherwise

and set by analogy with

D D

D

in the obvious way in the notation of Likewise dene and

D

Since N r D if cond jr we have

Y

D D

q mo d r

q jN r

Step We have chart Z Z charN r Z Z Cho ose

univ

z E Y N r to b e the p oint which in complex co ordinates is

r

Z ZZ Z ZN r Z

N r N r N r

For dierent r the p oints z are compatible z z We then can use

r r r

to write the Eisenstein series in terms of the complex parameterisation as

Eisz E du

D r D

A J Scholl

Step The function has lo cal comp onents

char Z Z if q j N r

q q

char Z M r Z if q jr

q q

q

t Z t q Z

q q q q

char q char if q jN

Z Z

q q

The last expression can b e rewritten as

t Z q t q Z

q q q q

char q char

Z Z

q q

t q Z q t q Z

q q q q

q char q q char

Z Z

q q

univ

Now by there exist a nite set of p oints z E Y N r and

rj

constants b N Z which are indep endent of r such that

j

X

b Eisz E du

j D rj D

j

will b e N torsion and in z Moreover the dierences z z and z

ri rj rj rj

complex co ordinates N z will b e the p oint

rj

Z ZZ Z Zr Z N t mo d Z

r r

It follows that

M r

primeto p ro ot of e z z

m

p

N r r rj

and thus that the constant e of equals M Z

p

z Step Put e f z g and let E b e the asso ciated

rj D r D rj rj

Euler system for T E The required Euler system is then

p

X

b E H Q  T E c

j rj r p r

j

We can now compute the dual exp onential of c Put e for the dier

r rj

ential on Y N r constructed from z z The Ko dairaSp encer map takes

r rj

dtt to dq q and therefore du to i d Therefore

X

i

b e E E d

j rj D D

N r j

An intro duction to Katos Euler systems

We then get

X

exp c

mo d r p r

Gal Q QZr Z

r

Z

X

M h

E

b e e

E j mo d N r N r rj E N r E

r

E E

j

Y N r C

h GL ZN r Z

E

F i hE E

E D D

M N r

E E

By Theorem taking k l and s

Y

hE E F i C r GL Zr Z M N t t

q q q

q jr

L E L E y M L E y M

N r mo d r N N

for some C Q dep ending only on E M and N Moreover using

and the hyp othesis that D D mo d M

F i C r GL Zr Z hE E

D D

Y

M N t t DD D D D D

q q q mo d r mo d r

q jr

L E L E y M L E y M

N r mo d r N N

Q

Now for q jr we have t t M N r so M N t t since cond jr

q q q q q

q jr

Combining everything one gets the nal result

Theorem Let E Q be a modular el liptic curve of conductor N Fix

E

a nonzero form E Q with real and imaginary periods

E E

E

Let p be a prime not dividing N Then there is an integer M prime to

E

p and for every pair of integers D D with DD pN and

E

D D mod M an Euler system

m

c c E p D D H Q  T E r r p r squarefree and

r r r p

prime to pM N m

E

such that for each r and each character GalQ  Q Zr Z C

r

with

X

exp c

p r

Gal Q Q

r

L E

r M N

E

C DD D D D D

E E

E

for some constant C depending only on E

E

m

In the sp ecial case r p this is with minor mo dications of notation

Theorem of

A J Scholl

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baki exp ose Lect notes in mathematics Springer

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M Rap op ort Les schemas de modules des courbes el liptiques Mo du

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C Deninger A J Scholl The Beilinson Conjectures In Lfunctions

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