Classical and overconvergent modular forms

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

R

OBERT

F. C

OLEMAN

Classical and overconvergent modular forms

Journal de Théorie des Nombres de Bordeaux, tome

7, n

o

_{1 (1995),}

p. 333-365

<http://www.numdam.org/item?id=JTNB_1995__7_1_333_0>

© Université Bordeaux 1, 1995, tous droits réservés.

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

Classical and

_{overconvergent}

modular forms

par ROBERT F. COLEMAN

The purpose of this article is to use

_rigid

_analysis

to

_clarify

the rela-tion between classical modular forms and Katz’s

_{overconvergent}

forms. In

particular,

we _prove a

_conjecture

of F. Couv6a

[16,

_Conj.

3]

which asserts

that every

overconvergent

p-adic

modular form of

_sufficiently

small

_slope

is classical. More

_precisely,

let _p &#x3E; 3 be a

_{prime, K}

a

complete

subfield of

Cp,

N be a

_positive

_integer

such that

(N, p)

= 1 and k an

_integer.

Katz

[21]

has defined the space

Mk(f1 (N»

of

overconvergent

p-adic

modular forms of level

_ri(N)

and

_{weight k}

over K

_{(see §2)}

and there is a natural map from

weight k

modular forms of level

r1(Np)

with trivial character

at p to

_Mk(rl(N».

We will call these modular forms classical modular

forms. In

_addition,

there is an

_operator

U on these forms

_{(see [15,}

_Chapt.

II

_§3])

such that if F is an

_{overconvergent}

modular form with

_q-expansion

F(q)

=

_anqn

then

.

(In

fact,

all this exists even when _p = 2 or 3

_{(see [21]}

or

[9])).

We _prove,

Theorem

_6.1,

that if F is a

_generalized

_eigenvector

for U with

_eigenvalue

A

_(i.e.,

in the kernel of

_{(U -}

_A)’

for some

_positive

_integer

rt~

of

_{weight k}

and a has

_p-adic

valuation

_strictly

less than k -

_1,

then F is a classical modular form. In this case the valuation of A is called the

_slope

of F. In the case when F has

slope

_0,

this is a theorem of Hida

[20]

_and,

more

generally,

it

_implies

Couvba’s

_conjecture

mentioned above

_(which

is the above conclusion under the additional

_hypothesis

that the

_slope

of F is at

most

_{(k -}

_2)/2).

This almost settles the

_question

of which

_{overconvergent}

eigenforms

are

_classical,

as the

slope

of _any classical modular form of

_weight

J~ is at most k - 1. In Section

_7,

we

_investigate

the

_boundary

case of

overconvergent

modular forms of

_slope

one less than the

_weight.

We show that non-classical forms with this

_property

exist but _{that any}

_eigenform

for the full Hecke

_algebra

of

_weight

k ~ 1 is classical if it does not

_equal

(see

below)

where G is an

_{overconvergent}

modular form of

_weight

2 -1~. In

Section

_8,

we _prove a

generalization

of Theorem

_6.1,

Theorem

8.1,

which relates forms of level

ri

_{.Np)

to what we call

_{overconvergent}

forms of level

r1(Np)

and in Section

_9,

we

_interpret

these latter as certain Serre

p-adic

modular forms with

_{non-integral weight}

_{[30] .}

The central idea in this paper is

expressed

in Theorem 5.4 which relates

overconvergent

modular forms to the de Rham

_cohomology

of a coherent

sheaf with connection on an

algebraic

curve. More

_precisely,

we show that there is a map for

non-negative

J~ from modular forms of

_weight

-k to modular forms of

_{weight k}

+ 2 which on

_q-expansions

is

When N &#x3E;

_4,

the k-th

_symmetric

power of the first relative de Rham

cohomology

of the universal

elliptic

curve with a

_point

of order N over the modular curve is

_naturally

a sheaf with connection. Theorem 5.4 is the assertion that the cokernel of

Ok+1

is the first de Rham

_cohomology

group of the restriction of this sheaf to the

complement

of the zeroes of the

modular form on

X,

(N)

-The above

_result,

Theorem

_6.1,

is

_intimately

connected with the

conjec-tures of Gouvea and Mazur on families of modular forms in

_[171

and

_[18].

Indeed,

in a future article

[9]

we will use it to deduce

_qualitative

versions of these

_conjectures.

_(We

will also

_explain

how to

_handle

_p = 2 or 3 in

_{[9 .)}

We thank Barry Mazur for _encouraging us to think about this _problem and Fernando Gouvea for _helpful conversations.

1. The

_rigid

_subspaces

associated to sections of invertible sheaves Let v denote the

complete

valuation on the

_p-adic

numbers

_Qp

such that

v(p) =

1 and let

_Cp

denote the

_completion

of an

_algebraic

closure of

_Qp

with

_respect

to the extended valuation

_(which

we still call

_{v~ .}

We _also fix a non-trivial absolute

value [ on Cp

compatible

with v.

Suppose

R is the

ring

of

_integers

in a

complete discretely

valued subfield K of

_Cp.

Suppose X

is a reduced _proper flat scheme of finite

_type

over R and ,C is an invertible sheaf on ~. Let s be a

_global

section of ~C.

_{Suppose x}

is a closed

point

of the subscheme X := Y (9 K of ~. Let

Kx

denote the residue field

of x,

which is a finite extension of

K,

so the absolute value on K extends

uniquely

to

_Kx,

and let

Rx

denote the

_ring

of

_integers

in

_{Kx .}

_Then,

since X is proper, the

_morphism

_{corresponding}

to x extends

to a

morphism

X . Since K is

discretely

_valued,

is

generated by

a section t. Let

¡;s

= at where a E

_Rx

we set =

lal.

rigid

space over K. We

claim,

for each r &#x3E; 0 such that r E there is a

_unique

_rigid

subspace

XT

which is a finite union of _open affinoids whose

closed

_points

are the closed

_{points x}

of X such that

_{Is(x)1 ?}

r.

_Indeed,

there exists a finite affine _open cover C of X such that the restriction of

to Z for each Z in C is trivial. For each Z E

_C,

let tz be a

_generator

of

G(Z)

and suppose

slz

=

fztz

where

fz

_E

Let 2

denote the fiber

product

of the formal

_completion

of Z

_along

its

_special

fiber and

_SPec(K).

Then

Z

is an affinoid over K and n

Z

=

{x

_E

Z : ~

I fz (X)

I

&#x3E;

r}.

Since this is known to be the set of

_points

of an affinoid

[2, §7.2],

we have

established our claim.

Now,

for r E

_R,

r &#x3E;

_0,

the set of closed

points

of x such that such that

Is(x)1

[

&#x3E; r is also the set of closed

_points

of a

_rigid

space

_X(,)

as E &#x3E;

_r}

is an admissible cover. ,

Alternatively,

if L is the line bundle whose sheaf of sections is G then there is a natural metric on L and if we

regard s

as a section of L 2013~

_X,

Xr

is

the

_pullback

of the

_{rigid subspace}

_consisting

of

_{points greater}

than or

_equal

to r.

_Or,

if a E R and

_Jo)

= _r, then

Xr

may be identified with the fiber

prod-uct over R of

Spec(K)

and

thep-adic completion

of

SPecX(Syrn(G)/(s-a))_

(One

can deal with sections of

locally

free sheaves

_just

as

well.)

Now suppose X is an irreducible curve and r e

_ICpl.

_Then,

either

Xr

is an affinoid or

_Xr

=

X,

because such is true for any finite union of affinoids

in an irreducible curve. If the reduction 9 of s is not zero and

r #

0,

then

XT

is an affinoid. In

fact,

if X is

_{smooth, 9}

has

_only

zeros of

multiplicity

one and 1 &#x3E; r &#x3E; 0 we claim

XT

is the

complement

of a finite union of wide _open disks.

_Indeed,

suppose all the zeroes of s are defined

over R

(this

is not

_really

_necessary, it

_just

makes

_things

easier to

_visualize).

Let C etcetera. be as above and let Z E C.

_{Then tz}

is a local

_parameter

at

_Q

for each zero

Q

of s in Z

and,

in

particular,

the restriction

_tQ

of

_tz

to the residue disk

_containing

_Q

_gives

an

_isomorphism

onto the unit disk

B(0,1).

Moreover,

Xr

n Z = Z -

U

r).

This establishes the above

claim. It follows that is the

complement

of a finite union of affinoid disks and so is a wide _open

by

definition.

(See

[6]

and also

[28].)

Such spaces are

_quasi-Stein

_spaces

[24].

2.

_Application

to

_{overconvergent}

modular forms

Let p

&#x3E; 3 be a

_prime

and let N &#x3E; 4 be an

_integer

such that

_{(p, N)}

= 1.

Let X denote the model with

_good

reduction of

_X1(N)

over R and C the

subscheme of _cusps.

_(We

will also use C to denote the

_degree

of the

di-visor C when no confusion will

arise).

Let E -~ X denote the universal

sub-scheme of E

_consisting

of

_points

smooth over X and

13 =

Let and W =

Now w is an invertible sheaf and we have a section of Since the reduction of

_{Ep- i}

vanishes

_simply

at each

_{supersingular}

_point,

the

_rigid

spaces

_{X(,) =}

&#x3E;

_r}

are each the

complement

of a union of affinoid

disks,

one in each

supersingular

residue disk

by

the discussion of the

_previous

section.

Let Z =

Xi,

Wl -

X(p-P/(P+l»

and

W2

= Then

_W2

_C

_Wi

and

_Z,

the

_{ordinary locus,}

is the

_unique

minimal

_underlying

affinoid

_(see

~6~ )

of either

_WI

or

_W2

_containing

the _cusps. Let

_Mk

:=

:=

w*’(Wi)

for k E Z. Then

Mk

may be described in terms of Katz’s

overconvergent

forms of

_weight

k.

_Indeed,

if r E

_R,

_may be identified with

_{S(R, r, N, k)}

_(see

_{[21, §2.9]])}

and

Mk

=

a

where s

_approaches

from above. We call the sections of

úJk

on Z

convergent

modular forms and those of

_Xs

_{for any s} 1

_{overconvergent}

modular forms.

Remark. We

point

out that Katz’s

_{overconvergent}

modular forms are

only

defined for

_integral

_weights

while Serre’s

_p-adic

_{forms -may have}

_weights

in

Zp

x

Z/ (p -

I)Z.

Katz’s discusses the

relationship

between the two

_types

of

_objects

in

_{[21, §4.5].}

In

_particular,

he shows that Serre’s forms of

_weight

(k, k)

for

_{integral k}

are his

_convergent

forms of

_weight

l~. In a future

_article,

we will introduce a notion of

"overconvergent"

p-adic

modular forms with

weight

in x

Z/(p -

I)Z

which

_incorporates

the forms of both Katz and Serre

_(see

also

_§9).

Let

Ei

denote the

_pullback

of Earn to

Wj .

Katz describes

_[21,

Sect.

_3.10~

a commutative

_diagram

where 4i

_{and §}

are finite

morphismes.

There is a canonical

_family

of

_subgroup

schemes IC of rank p over

_tVi

in the

family

of

elliptic

curves

_El

over

_WI

and

₀

is a

morphism

such that the

family

over

_W2

is

_canonically

isomorphic

to the

_pullback

.E~~ ~

of

.~1

to

T~f~2.

Then 16 is the

_composition

of the

_isogeny

7r:

_{EZ -~}

E2/JCE2

and the natural

_projection

from

E~~ * }

to

Using

this we will make

_{0* -linear}

transformations

Vk:Hk(W2)

-&#x3E;

llk(W1)

for k E

_Z,

k &#x3E; 0

_(note

that

1-lo

=

Ox).

We first observe that the

_complex

_C)

is

_{naturally isomorphic}

to the

_complex

_(logC).

What this means is that we

_get

a natural map of

complexes,

which takes

_g4!*(a)

to

_{where g}

is a section of and a is a section of for an open set V of

El.

Second,

as

We

_get

a _map

Similarly

we

_get

a _map

which we also call

Yk .

We have a natural _map of

_complexes

yields

maps and

which we denote

by

F~ .

Now we have the Gauss-Manin connection

Vk

with

log-poles

at C on

1tk.

I.e. and it is easy to see that

as _maps from

xk(W2)

to

_{(0B (log C)}

and from to

(Ok ~?L~~(W2)

respectively.

From this it is _easy to see that if h E

_{and g}

then

This is also true with H

_{replaced by}

_Sl’

01-£.

Remarks. The restriction of

Hk

to Z

_(or

better the

_dagger

_completion

of

Z)

is an

F-crystal

in the sense of Katz

[22]

if one takes the _map from

to

rlk

to be

_{(~* ~ ~ .}

We

_{will, henceforth,}

use ** to denote

_{(jt* )*’ .}

3.

_Hodge

and U

0 the

_Hodge

filtration on

1ik

is a

_descending

filtration

such that, for 0 ::; i ::; r and 0 ::; j $ s,

as coherent sheaves. It is clear that

Vk

and

_respect

these filtrations.

We observe that

1,

is self-dual with

_respect

to a natural inner

product ( , )k:

x

_1ik

---~ which when J~ =

1,

_away from the _cusps,

is

_just

the _cup

_product

and more

_generally

satisfies

for

_fi,

_gi local sections

_Of1il.

This

_leads,

in

_particular,

to the exact _sequence

which

_together

with

_(3.1)

_implies

Gri1-lk

= _is

We will

identify

_Mj

with

_Gro1tj(W1)

_{when j}

&#x3E; 0 and with

when j

0. We let be the

_operator

on

_Mj

We will

_frequently

drop

the

_subscript

_(j)

from

_U(j)

when the context makes

it clear on which space we are

_acting

and sometimes abuse notation and

allow to mean

when j

&#x3E; 0.

Remark. The

_operator

extends to _any of the spaces for any 1 &#x3E; s &#x3E; _{but any}

_{overconvergent}

_eigenvector

for

_U(k)

_analytically

continues to

_Wl.

_(See

the

_proof

of

_{[16, Cor.II.3.18~.)}

The value of

consid-ering

these

_larger

_spaces is that for s E

_{IKI they}

are Banach spaces and one can

_apply

the

_theory

of Serre

[29].

Suppose g

E

Mk.

We set

_equal

to

_Fk(g)/pk

= when

k &#x3E; 0 and

_Fk(g)

when k 0. So that E

Wk(W2).

It then follows from

(2.4)

that if h E

wi

_(W2)

and

_{kj &#x3E;_}

_0,

In

_particular,

if h E

Mk

and a is a

_rigid

function on

_W2

then

Equation

(3.3)

will

_follow,

in

_general,

from the

_following

_proposition.

Remark. _{The map} Q is what is called cp in

[21, §3]

(and

Frob in

[16,

Chap.

2

_§2]),

which is

_only

defined there when k &#x3E; 2 in

_general

and when k = _{1 in}

some cases.

_Indeed,

we _may

regard

an element h of

_Mk

as a function which

assigns

to

_pairs

_{(G/ B, w)}

where B is a K

algebra,

G is a fiber of

_E1/W1

over a B valued

_point

of

WI

and w is a differential on G which

_generates

the invariant differentials on G over

_B,

an element

H(GIB, w)

of B

_by

the rule

_hIG

=

h(G/B,

w)wk.

Then if

G/B

is the fiber _{over a} B valued

point

of

PROPOSITION 3.1.

_{Suppose k &#x3E;}

fl. The map induced

by

~~

o

Resy~’j’j,.2

on

Mk-2i is

for

0 ~ i k.

Proof.

We pass to the

_underlying

affinoid Z and

_repeat

all

previous

con-structions in this context. This

_gives

us the

_advantage

of not

_having

to worry about the fact that

W2.

We will let A denote the

pullback

of E to Z.

We will now follow

Appendix

2 of

[21] . (Note:

What is denoted

_by

the

symbol

p there is what is called

4&#x3E;*

here.)

Suppose v

is an invariant differential on ~4

_generating

w.

Then,

**v =

Av(4* )

and

_{~-* v-1 - ~ {v~ x } t~* ~}

for some invertible A.

_Suppose

_f

is a section of such that

_f

=

_au(h)(A,

_modulo

Filk+l-i11.k

where

h is a

weight k -

2i modular form and a is a

rigid

function on Z.

(Here

we

are

_using

the

_equation

_(3.1).)

_Then,

Now

_then,

using

(2.3),

while

Thus

and _so, on

_Grz,

_Vk

acts _as

_using

_(3.4).

4.

_{Kodaira-Spencer}

and the theta

_operator

We have a

Kodaira-Spencer

_map

K,: CJ.)2

2013~ of coherent sheaves

where on the

right

we

regard w

and v as sections of This _{map is} an

isomorphism.

More

_generally,

we have an

_injection

of sheaves

.. - .. . determined

_by

the

correspondence

where 77

is a local section of

c~~

and on the

right

we

regard

it as a local section of

_1-lk.

PROPOSITION 4.1.

Proof. Suppose

the one

hand,

Then on

On the other

_hand,

The

_proposition

follows from this and the

_{definitions. I}

LEMMA 4.2. The _map

is an

_{isomorphism of}

sheaves

of

vector _spaces.

Proof.

The map shifts the filtrations

by

1

(Grif-fiths

_{transversality)}

and induces

_isomorphisms

on the

_graded

_pieces

_by

what we know about the

Kodaira-Spencer

_{map. I}

This

implies

that we

_{get a}

natural _map

M_k -&#x3E;

_Mk+2.

_Indeed,

_let w E

M_kw-k(W1).

Lift it to a section w of

71,.

Let s be a section of

Fi117-lk

on

_Wl

such that Vail = _{Os modulo}

Then the _{map is} t~ 2013~

V(n) 2013

s)).

At the _{cusp oo,} one

_computes

(see [7, §91)

that this _{map is}

- . 7 I 1 -where 0 =

qd/dq.

In

particular,

PROPOSITION 4.3. There is a linear _map

_from

M-k

to

_Mk+2

which on

q-expansions

is

We

_{will, henceforth,}

denote this map

by

the

_expression

ok+l -

Katz tells us

_[21,

_Appendix

1]

that we can

"canonically,

but not

_{functorially," regard}

for l~ &#x3E;

_0,

as

." - .

Suppose

f

E

Mk.

Suppose

first k &#x3E; 0.

_By

the above we can consider it as a section

_{Then Vf}

is a section of

(log C).

_By

virtue of Katz’s

decomposition

(4.1),

we can

_project

onto which

_by

virtue of

_{Kodaira-Spencer isomorphism}

we can

_identify

with

_Mk+2-

Call

this element

_{6k f .}

Now

_{suppose k}

0.

_By

_(4.1),

we can

_{regard f}

as an

ele-ment of

_1-l-k(Wl).

Then the

_projection

of

_V-kf

onto

Mk

can be

_regarded

as an element of

_Mk+2

and we call this element

_8kf.

In either _case,

_calculating

at oo we find

We will use this to show in

_[11~

that

_0(f)

is not

_{overconvergent}

when neither

f

nor k

_equals

zero. 5.

_Cohomology

For a

_rigid

_analytic

_open

_subspace

W of

_Xl(N)

set

Let SS denote the set of

_{supersingular}

_points

on

Xl (N)

_{mod p}

_{and let} SS be a set of

_liftings.

_(We

will also denote

_by

SS the

_degree

of the divisor

SS when

_{appropriate.)}

_By

_[5,

Theorems 2.1 and

_2.4),

via the natural _maps

H(k)(Wi)

and

_{H(k) (W2)}

are both

isomorphic

to

where Q

_{(1ik) (log}

_SS)

is the

_complex

We therefore let

_H(k)

denote _{any one} of these

_cohomology

_groups. The above and

_(2.4)

_imply

THEOREM 5.1. The space

H(k)

is

finite

dimensional and F and V induce

endorriorphisras

Frob and Trer

_of

_H(k)

such that

Let

_]C[

denote the union of the

_cuspidal

residue classes and

_]88[=

_W1-Z.

Then

_JSS[

is the inverse

_image

in

_WI

under reduction of SS and af-ter étale base extension is a

disjoint

union of wide open

annuli,

one for

each element of SS. We call these the

supersingular

annuli. Now let be the

_subcomplex

of

_{rf (1ik), ’H.k}

0

_xk,

where

Ic

is the ideal sheaf of the cusps. We let denote the kernel of the _map

_{H(k)(]C(U~SS[).}

This is

_naturally

_isomorphic

to the classical

_{weight k parabolic cohomology}

on

Xl(N)

which is the

_image

of

Also,

stable under Frob and V er and

THEOREM 5.2. There is a natural

perfect pairing

such that

Proof.

It is classical that the

_self-duality

of

_1ik

leads to a

perfect

pair-ing

on

(essentially

Poincar6

Duality

between

_compactly

and

non-compactly

supported

cohomology).

By

standard

_arguments

_(e.g.

see

[6,

Thm.

_4.5])

we can

_compute

it as follows:

Let h and g

be elements of

(1-lk

0 with trivial residues on the

_{supersingular}

annuli and let

_[h]

and

_[g]

denote their

_respective

_cohomology

classes in

_{Hpar(k) .}

It

follows,

in

_particular,

that for each

_{supersingular}

annulus

_A,

there exists a

ÀA

E such that

_{V ÀA}

=

IIA-

For each

supersingular

_{point x}

of

Xi (N)

let

_Ax

denote the

_{supersingular}

annulus above x. Then

where

_ResAx

is the residue _map associated to the orientation on

Ax coming

from

_WI

_{(see [6, §3]).}

Now let

_Tx

be an orientation

_preserving

_uniformizing

where the Then

and

We deduce that

Now we can

identify

the the space of horizontal sections of B7 on

Ax

with

K)

where is the

_{supersingular elliptic}

curve

correspond-ing

to x, the restriction of

( , )

to this space with the natural

pairing

and the _map from to

~r* a~~~ ~ ,

with the Frobenius

morphism

from to It follows

that

The result follows from this and the fact that =

pReSA.9- I

It follows

_immediately

from Theorems 5.1 and 5.2 that COROLLARY 5.2.1. We have

By

a

generalized eigenvector

with

eigenvalue

a, E K for a linear

operator

L on a vector _{space W} over

_K,

we mean a vector in W which is in the kernel of

_{(L -}

_a)’

for some

_positive

_integer

n.

Theorems

_5.1,

5.2 and the

_previous

_corollary

_imply

COROLLARY 5.2.2. The _map Ver is an

_isomorphisme

and

_{if a}

E K* then

the dimension

_{of the}

_{generalized eigensubspace}

_of

_Hpar(k)

with

_eigenvalue

a

_for

Ver is

_equal

to that

_of

the

_{generalized eigensubspace}

with

_eigenvalue

NOTE. We have not

_yet

shown that the

_{eigenvalues of}

Ver

_acting

on

H(k)

are

_integers

in This will

_follow

_frorri

Theorem 6.1.

Proof.

We take

_cohomology

of the short exact _sequence,

where

_Sk

is the

complex

of

_skyscraper

sheaves which makes this sequence

exact. First

where is the ideal sheaf of SS

_(via

a residue

map).

_Second,

the

boundary

map of the

complex

S.

takes

Sk

into

Hk

0

_OX(N;p)/Io

with

respect

to this

_{decomposition}

with a one dimensional cokernel at each _cusp

(as

one can deduce from the results of

[21, AI]).

Thus as

Hk

is

locally

free of rank k + 1 we see that

_dimK

= C ₊

(k

₊

1)SS.

The lemma

follows from the facts that

dimK

H2 (n

= 0 for l~ &#x3E; 0 and = 0 if k _&#x3E; 0 and 1 if k = 0.

Now on

_q-expansions

it is evident that 0 o U =

pU

o 0.

Using

this,

Proposition

4.1 and Lemma 4.2 we deduce:

THEOREM 5.4. The

_quotient

_naturally

_isomorphic

to

H(k)

and the

_{following diagram}

commutes:

Let

_P(k,T)

denote the characteristic series of U restricted to

_wk(Y)

where Y _{is any}

_underlying

affinoid of

Wl

strictly

containing

Z. This series exists since U is

_completely

continuous on this space as

_explained

in

_[15].

Then,

We will

_interpret

the

_polynomial

_det(1-

₂₎₎

in terms of

Hecke

_operators

and classical modular forms on in

_§7.

This will

generalize

Theorem 2 of

_[25].

6. Small

slope

forms are classical

For a module M and a rational number a, we set

Ma

equal

to the

_slope

a

_part

of

M,

that

_is,

the submodule of F E M such that there exists a

_polynomial

f(T)

E

_K[T]

_]

whose roots in

_Cp

all have valuation a such that

_{f (U}F}

= 0. We

say that a non-zero element of

MQ

has

slope

a. Let

_{X((N; p)}

denote the fiber

_product

of and

_Xo(p}

over the

_j-line.

Let

_Sk,ci

:=

_Mk,cl

:=

Mk(N;p)

denote the _spaces of cusp forms

and modular forms of

_{weight k}

on

X(N; p}.

We _may use JC to

_identify

_Wl

with a

_subspace

of

X(N; p)

and

_get

a natural Hecke

compatible

_injection

from into

.~k

under which

_U~

corresponds

to U. We call elements in

the

_image

classical modular forms.

THEOREM 6.1.

_{Every p-adic}

overconvergent

form of weight k + 2

and

_slope

strictly

less than 1~ + 1 in

_Mk+2

is classical.

An immediate _consequence of this theorem and

the

main result of

_[18]

is the

_{following generalization}

of a result of Koike’s

[25]:

COROLLARY 6.1.1.

_If

= det

(1

:.-

Up

TIMk,cl(N;p»)

then

if k,

k’

and n are

_integers

such that k’ &#x3E; k &#x3E;

_{2, n}

&#x3E; k-1 and k’ = k

mod pn-l

_{(p- 1),}

then

We now

begin

the

proof

of Theorem 6.1. It is vacuous for k 0.

There-fore _{suppose 1~ &#x3E;} 0.

Also,

suppose N &#x3E; 4. We will

_explain

how to handle

small levels at the end of this section. Let

Sk

denote the

_subspace

of _cusp

forms in

_Mk,

and let

₈₂

denote the

subspace

of

_sk

of forms with triv-ial residues on the

_{supersingular}

annuli

_(see

_{[7] ) .}

Then

_{S2+2/0k+l M-k}

is

naturally

isomorphic

to the

_parabolic

_cohomology

_{Hpar(k) .}

Let

_{82 cl}

:=

S2,cl(N¡p)

denote the space of

p-old (or equivalently

(by [7,

Thm.

_{9.1] )}

with trivial residues on the

supersingular

annuli)

_cusp forms on of

weight

k.

_{Then S2,cl}

_maps into

₈₂

and

Proof.

The dimensions of both _spaces are twice the dimension of the space of _cusp forms on

_{Xl(N). (For Hpar(k),}

this follows from the classical

Shimura

_isomorphism

_{[31]-) 1}

The next observation is

LEMMA 6.3.

_{If F is a}

non-zero element

_of

_Mk+2

with

_slope

_strictly

less than k +

1, it is

not in

Proof.

We _{may suppose} that F is an

_eigenform

with

_{eigenvalue -y}

such that

_V(/)

k + 1.

Suppose

G E

M-k

such that

øk+1G

= F and let

. . Then

This

_implies

_G(q)

has unbounded coefBcients which contradicts the suppo-sition that G E

COROLLARY f .3.1. The natural map

from

(Mk+2,cl)o:

to is an

in-jection

if

a k + 1. ,

In

_particular,

if a k + 1. Now it follows from

_Corollary

5.2.2 that

Let

_{fl, f2}

denote the two

_degeneracy

maps from

X(N; p)

to

_Xi(N)

and let F be a form of

_{weight k}

+ 2 on

X,

(N)

so that if

_{(E, P, C)}

_represents

a

point

on

X (N; p),

where E is an

_elliptic

curve P is a

_point

of order N on E and C is a

_subgroup

of order _p of

_E,

and

_{pc : E -+ pcE}

is the

_isogeny

with kernel

C,

Suppose

for d E

(Z/NZ)*.

Let u be a root of

x2 -

_Apx

+ Then the above identities

_imply

that G =

fi F -

is _an

eigenvector

for

Up

with

_eigenvalue

u. This

_implies

that for any A E

K*,

_any character

E:

(Z/NZ)*

-

K*

and _{any root U} of

x2 -

Ax ₊ we have a

homo-morphism hu

from the

_subspace

of

_{Sk+2 (N)}

which is the

_eigenspace

for

_Tp

with

_eigenvalue

A and character E for the action

_(Z/NZ)*

to the

sub-space

W (u, E)

of with

_{eigenvalue u}

for

_Up

and character E.

LEMMA 6.4. The

_{homomorphisme}

_{hu is an}

_{isorrcorphism.}

_Moreover,

_W(u,

_t)

is the kernel

_of

_{(Up -}

_u)2

in the E

_{eigencomponent}

unless

V(A, E) ~

0 and u = _=:

u’.

In this _case, the

kernel of

(Up -

U)2

in the f

eigen-component

equals

the kernel

_of

_{(Up - 1.1,)3}

in this

_component

and is

_strictly

bigger

than

_{W (u,}

_{E) .1}

Proof.

We will use the fact that

fiSk+2(1V)

fl

_{f2,Sx+2(N) _ ~0}.}

This

already implies

that

hu

is an

_injection.

Suppose

L =

fl* H + f2* G

_E

W(u, 6).

Then we see that

This

_implies

that

_HITP

=

(u

+ = AH and

huH

= L. Thus

H E and

hu

is an

_isomorphism.

Now _suppose

U)2

= 0 and

L is in the E.

_{eigencomponent.}

Then

by

what we now know there exists an F E

_{V(A, E)}

such that

This

implies

Hence,

UI

But since

_Tp

is

_{diagonalizable,}

we must have either ~’ = 0 or

u’

= u.

Finally,

if u = u’ the above

implies

that if H _E

Ii

u) =

So if

_{H 0}

_0,

_{Ii H}

is in the kernel of

_{(Up -}

_{u) 2}

but is not in

W {u, 6) ..

1 Wath Edixhoven, we have shown in "Simplicity of Frobenius _eigenvalues in the Galois

representations associated to modular forms" that this latter case does not occur when

COROLLARY 6.4.1.

_Suppose

0: :$

(k

+

_1)/2.

Then

and

where

_{Sk+2(r1(N»a is}

the

_slope

a

subspace

of

Sk+2(r1 (N»

when a

_(k

₊

_1)/2

and is the sum

of

the

subspaces of slope

at least

_(k

+

_1)/2

when a =

(k

₊

1)/2.

Proof. Using

the fact that

_Tp

is

_diagonizable

on

Sk+2(1V),

the lemma

_implies

that if a

_(k

+

_1)/2,

both the dimensions in

_(6.5)

_equal

The second statement is an immediate _consequence of the lemma if a

(k

+

_1)/2

and is a _consequence of the

_proof

if a =

(k

₊

1)/2. 1

Putting

(6.1~, (fi.2~, (6.3)

and

_(6.5)

_together

we deduce all the

inequali-ties in

_(6.2)

are

equalities

and so

LEMMA 6.5. The map

from

(S2+2,cl)a to

an

isomorphism if

a~+1.

’

This is

_enough

to prove Theorem 6.1 for elements of we shall see. To deal with

_arbitrary

elements of

_Mk+2

we will need

PROPOSITION 6.6. The map

from

(Mk+2,cZ)a:

to is ac?2

isomorplt2sm

Proof.

First we have

LEMMA 6.7.

_Suppose

a k + 1. Then

denote the _{space cusp} forms new at _p on

X(N; p)

identified with its

_image

in

_Mk(r1(N».

First the _map from

_+Sk+2,cl

to

is an

isomorphism.

Now

_Sk+2,cl

C and

by

[7,

Lemma

5.1]

has dimension

_SS(k

+

₁₎

unless k = 0 in which _case it has dimension S~’ -1.

Also

_{Ek (N; p)}

C

_(Mk+2,a)0

+

_(Mk+2,cl)k+l

and

_Sk+2,cl

n

_{Ek+2 (N; p)}

= o.

From this we deduce the lemma when

_Ek+2(N)

is an

eigenform

for

_Tp

with

_{eigencharacter}

_{E, its}

_eigenvalue

is 61 + where ,61 and E2 are roots of

_unity

of

_relatively

_prime

order such that eiE2 =

E(p)

(see

[14, §3]).

It follows that

are

_eigenvectors

for

_Up

in

_{Ek+2 (N; p)}

with

_eigenvalues

_{E I} and

_f2Pk+l.

. Since

Tp

is

_diagonizable

on

Ek+2 (N)

if k &#x3E;

_{0, dimK}

Ek+2 (N)

= C if 1~ _&#x3E; t~ and

dimK

Ek+2 (N; p)

= 2C if 1~ &#x3E; 0 the lemma follows as

long

as k &#x3E; o. Now

_{suppose k}

= 0. Then

_by

the above

_arguments

since

_dimK

=

C - 1

diMK

E~ (N; p) a &#x3E;

C - 1 if a = 0 _or 1. However the

pullback

W

of the one dimensional _space of Eisenstein series on

Xo(p)

to

_X(N;p)

lies in because

_Up

on

Xo(p)

acts on

_weight

2 forms as minus the

Atkin-Lehner involution. On the other

hand,

this _{space is} not contained in

¡; E2(N)

+

_{/2 E2(N).}

This can be seen

_by

_{considering weight}

2 forms as

differentials. Then if

_Coo

is the set of _cusps on

X(N; p}

_lying

over 00 on

Xo (p),

Respw

equals

zero if w E +

_{f2* E2 (N)}

but not if w E W.

_{Since dimK}

_{E2 (N; p)}

= 2C - 1 this

implies

the

remaining

cases of

the

_{lemma. I}

Now consider the

_diagram

with exact rows:

i )

Since,

for a 1~ +

_1,

the first vertical arrow is an

isomorphism

and the second is an

_injection

the third is an

_injection

as well. Thus the _map

is an

_injection.

_By

Lemmas 5.3 and 6.7 both these vector spaces have the same dimension so this _map is an

isomorphism.

It follows that the last

vertical arrow in

(6.7)

is an

isomorphism

and

_Proposition

6.6 follows from this and Lemma

6.5. ~

Since the

_eigenvaules

of

_Up

_acting

on classical modular forms are

alge-braic

_integers

we deduce

COROLLARY fi.7.1. The

_{eigenvalues of}

Frob and Ver

_acting

on

H(k)

are

atgebraic

integers.

Finally

suppose G is a

generalized

eigenform

for U in

_Mk+2

of

slope

a k + 1.

_Then,

_by

_Proposition

_6.6,

its class in

_H(k)

is

_equal

to the class of a classical

_generalized

_eigenform

F for

_Up

of

_slope

a. Hence G - F E

_(Mk+2)a:

and its class in is 0.

_By

Lemma 6.3 we see that G -- F = 0 which

proves Theorem

6.1. g

Remark. We can make Theorem 6.1 work for small levels.

Suppose

first N &#x3E;

_4,

_{{R, p)}

= 1 and The _map from

to

_Mk(rl(R»

is

_clearly

a U

_equivariant

_injection

_(look

at

_{q-expansions).}

We will now choose R

_large

_enough

so that all the levels mentioned in this

paragraph

divide R and

_identify

with its

_image

in

_Mk(r1(R».

One can show

(again

_using

q-expansions)

that if

(,M,

N)

&#x3E;

_4,

Now suppose

4,

A, B

&#x3E;

_4,

_{(AB, p)}

= 1 and

(A,

B)

= N.

Then,

it

follows from the above that the intersection of

_Mk(r1(A))

and

_Mk(r1(B))

is

_independent

of A and B and is stable under U. All this is

_compatible

with

_changing

R. We set

_(Mk(f1(N»,

_U)

_equal

_{to any} element in this

isomorphism

class. One can show this is

compatible

with Katz’s definition of

overconvergent

forms in small level. Since classical forms which are both of level and of level

_ri(B)

are of level

_{ri ((A, B))}

it follows from Theorem 6.1 that its statement is now also true for all ~V &#x3E; 1.

NOTES. 1. Since

_Frobenius,

it

_follows

that

2. The

_image

of

the

_form

F2

which has

_slope

k + 1 in

_(6.6)

in

_H(k)

is 0

_using

the

_fact

that

_{f2 /p}

is Robenius and thus there must be an

In fact, if ~F

=

Gk+2, k

_&#x3E;

0,

the classical Eisenstein series

of

level one and

weight

k +

2,

H is the

_p-adic

Eisenstein series

_{G* k}

_{(see [80,}

_{§ ~. ~J).}

7. The

_Boundary

Case

We now know that no

_{overconvergent}

forms of

weight k

of

slope

strictly

larger

than k - 1 are classical and all of

strictly

smaller

slope

are. In this

section,

we will

_investigate

the

_boundary

case of forms of

_{weight k}

and

Suppose

2. It follows from the results of

_§6

that

for 1~ &#x3E; 2

_(we

will _prove

_subsequently

that these two spaces are

isomorphic

Hecke

_modules).

_Hence,

it will follow from the

_following

_proposition

and Note 2 of

_§6

that there exist non-classical

_{overconvergent}

modular forms of

weight

k and

_slope

k -1.

Suppose

L is an

_{imaginary quadratic}

field of discriminant

D,

(1: is an

_embedding

_{and 1b}

is a Gr6ssencharacter of L with

_infinity

_type

0’ k-I

and conductor .II~.

_{Let 0}

be the Dirichlet character associated to

_L,

M the norm of ~I and E the Dirichlet character modulo

_{given by}

the formula

," . " ’B. L --- 1

Then there exists a

weight

k

cuspidal

newform on

X1(IDJM), G~,

_with

character E and

_q-expansion

where the sum is over

integral

ideals .A

_prime

to Jvl

_{(see (2?], [32~).}

_(This

is true even if k = 1 _as

long

as 1/J

is not the

composition

of a Dirichlet

character with the norm

(see

[26;

Thm.

4.8.2]).)

Now,

identify

C with

_Cp.

_{Suppose p splits}

in

_L,

and let _{p be the}

prime

of L above p such that = k - 1. Then the coefficient of

qP

in

G1jJ(q)

is + It follows that = _{is an}

eigenform

for

_UP

on

X (N; p)

with

eigenvalue

and so lies in PROPOSITION 7.1. The

_image

_{of F1/J}

in

_{H(k - 2) is}

zero.

Now choose a _sequence of

positive

integers I r,, I

such that _rn =

-1 where the limit is taken in

Z.

Then the _sequence

_{G 1jJrn}

_converges in the

sense of

[30]

to the

_weight

2 - k modular form H of

_slope

0 with

_q-expansion

Since H has

_slope

zero it lies in

M2-k

[15,

Prop.

11.3.22].

As

it follows from

_(7.1)

that = F. This

com-pletes

the

_proof.

Remarks. 1.

_Suppose

_{G(q) _}

is the normalized

_weight

2 cusp form on

Xo (49)

and p is a non-zero square modulo 7. Then

Ap

is a unit modulo p.

Using

the notation of

(6.4)

and

subsequent

lines,

if u is the non-unit root of

x2 -

_Apx

+ p and F =

fi G -

then since G

has CM

_by

the

_previous

_proposition

_implies

that the

_image

of F in

_H(0)

is zero.

2. It would be

_interesting

to know whether there are _any non-CM classical

weight k

cusp forms whose

image

in

H(k - 2)

is zero.

3. Another way to obtain non-classical forms of

slope k -

1 and

weight k

is to use the _map

ok-1.

Indeed,

So for

_{example, by}

standard

_procedure

we _{may deal with the A-Iiiie}

(which

sits between

_Xo(4)

and

_Xi(4)).

Then

_(S2,cl)1

= 0 and

dimK(So)o

=

(P-1)/2

by

[12].

Hence,

in this case,

(S2),

is a

(P-3)/2

dimensional _space of non-classical

_{overconvergent}

modular forms of

_weight

2 and

_slope

1.

_(See

also

_[1]

where the the dimension of the space of

overconvergent

forms of

weight

0 and

_slope

0 of level 3 is

computed

when p - 1 mod

_3.)

4. In

_{particular, if p = 13}

and A is Atkin’s

_{overconvergent}

_weight

0 form of

level 1

_{(see [21, §3.12]),}

8A is a non-classical

_{overconvergent}

form of level

1,

weight

2 and

_slope

1.

Let T be the free

_{K-algebra generated by}

the

_symbols

_p and

(d)N,

d E

_(Z/NZ)*

modulo the relations

_(djN(d’)N

=

(dd’)rr.

Then both

Hpa,(k -

2)

are

naturally

T[U]

modules via the

_homomorphism

which sends

_Ai

to

_T,

if I

_{Á Np}

and Ui

if

_IIN,

U to

_UP

and

_(d}N

to

_(d)N.

is

_compatible

with this

_action,

but as we have seen it is not

_generally

an

isomorphism

even

though

the two spaces have the same dimension.

How-ever,

THEOREM 7.2. The two

_T[U]-modules

and

_{Hpar(k -}

₂₎

are

isomor-phic.

’

Proof.

This will follow from Lemma 6.5 and

LEMMA 7.3. There exists a

_{non-degenerate}

_{(and non-canonical)}

variant

pairing

Proof.

We

_regard

_points

on

X (N, p)

as

_triples

(E, P, C)

where E is an

elliptic

curve P is a

_point

of order N on E and C is a

subgroup

scheme of rank _p of E. Fix a

_primitive

N-th root of

_unity

_~N

in _{K. Let w~ and} WN

be the

_{automorphisms}

of

_{X (N;}

_p)

where

_(P)

is the

_subgroup

scheme

_{generated by}

_P,

_pAE

is the

isogeny

with kernel A and

Q

=

p~ p~ ~~’~

where

_Q’

is a

_point

of order ~V on E which satisfies

_(P,

_Q’)Weil

=

(N. Then,

if ~’ is a modular form of

_weight

k on

and

First,

the

_operator

U is invertible on both and

_{82 cl}

and the

pairing ( , ) k

on

2)

satisfies

’

for h E T and

This map satisfies

ms i .. s i , , i

for h E T. In

_fact,

r o

_hu

=

(pu -

_on

E)

where u

and u’

_are

the two roots of

X2 -

Ax + In

_particular,

the restriction of r to an

isomorphism

onto We may now set

for m E

2)_1

and rt E That this

_pairing

is

T(U)-equivariant

follows from

_(7.2)

and

_(7.3).

That it is

_{non-degenerate}

fol-lows from the fact that the restriction of t to is an

_injection

onto

’

We do

_get

the

_following

_positive

result in the

_boundary

_case,

COROLLARY 7.2.1. 2.

_{If F is an}

_{overconvergent}

_{weight k}

Hecke

_{eigenform of}

_slope

J~ -1 such that

_{F ft}

_M2-k,

then F is classical.

Proof.

Since F is assumed to have

_positive

slope

it must be

_cuspidal,

the

_image

of F in

_{H(k - 2)}

is non-trivial and lies in

_2).

Let

F(q) =

The theorem

implies

that there

exists

a classical form

G such that

al G

for 1 a

_prime

not

_equal

_{to p} and

_apG.

Hence,

G has the same

_q-expansion

as F and so the two forms are

_{equal. I}

Remark. Richard

_Taylor

has asked whether an

_{overconvergent}

_eigenform

of

_weight

1 and

_slope

zero is classical if the

_image

of inertia at _p under the

representation

associated to F is finite.

Let

_{h(i) : T ~U~}

-

T [U~

denote the

homomorphism

such that

For a

T(U)-module

M and an

_{integer i,}

we define the "twisted"

_T(U)

module

_M(i)

= M

Q9h(i)

T[U].

COROLLARY 7.2.2. Tlae

_T[UJ

modude

_(Mk)k-i

sits in an exact sequence

where A = K

if k

= 2 and A = 0 otherwise.

for g E T(U).

It follows from

_[20]

and the results of

§6

that

where 6 is SS

_if

k = 2 and 0 otherwise.

Remark. Let F and notation be as in Remark 1 above. Let _mF be the maximal ideal of

_{T(U) corresponding}

to F. It follows from Theorem 7.2 that U does not act

_semi-simply

on the

_mF-isotypic

_component

of

~2(ro(49)).

Hence there exists an

_{overconvergent}

form H E

_s2(ro(49)),

which is not an

eigenform

and

p-adic

numbers ml for each

prime

number I such that

for

_{prime 1 :A}

7 and

At

_present

we have no further information about H. We do not even know

whether H is an element of

_{8Mo(ro(49»).}

More

_precisely,

Theorem 7.2 tells

us that the dimension of the

_mF-isotypic

_component

of

S2(ro(49))

is at

least 2 and the

_previous

_corollary

tells us that it is at most

but we do not know what it is.

In summary, the main results of this section have concerned

_{relationships}

among three spaces of

overconvergent

modular

_forms;

_{(Mk)k- 1}

and its two

subspaces

(Sk)x-i

and We have shown that

is

_isomorphic

to

_(Sk)k-1

as a Hecke module

(Theorem

7.2)

but that

may be non-zero

(Theorem 7.1)

and it is if and

_only

if there exist

non-classical forms

_Moreover,

none of the latter can be

_eigenforms

for the full Hecke

_algebra

_{(Corollary 7.2.1).}

_{We also} deduce from

_Proposition

6.6 and Theorem

_7.2;

PROPOSITION 7.4.

We can relate the

right

hand side of the above

expression

to the classical

Hecke

_polynomials

_[13].

When k &#x3E;

_2,

where

_Mk(ri(N))

denotes the space of modular forms of

weight k

on

Xi(N).

When k = 2 this formula holds with the left hand side divided

by 1 - T

8. Level

_Np

In this section we will

explain

how to

_generalize

the results of the

previ-ous sections to modular forms on

X1(Np).

We could have worked in this

generality

from the

_beginning

but

_thought

the extra

_{complications}

would

have been too

_distracting.

,

Suppose

pp C K. As

_explained

in

_§6,

the

_subgroup

1C of

_{El gives}

us an

embedding

of

_Wl

into

_{X(N; p)}

so that

E1

is the

_pullback

of the universal

elliptic

curve over

X (N; p)

with

rl (N)nTo(p)

structure. We will henceforth

regard

Z and

Wi

as

rigid

subspaces

of

X (N; p).

Let

X (N; p)

be the natural map,

Z(p) :=

g-1Z

and = Also

let

_f(p):

_El(Np)

-+

X1(Np)

the universal

generalized

elliptic

curve over

X1(Np)

and

_Ei(p)

=

E1(Np)lwi.

Then we have a commutative

_diagram

analogous

to

_(2.1).

We can define sheaves

_analogous

to w and sheaves with connection anal-ogous to

_Vk)

on

X1(Np)

_by

the same

procedures

followed in

_§2

_using

E¡(Np)

in

_place

of

_{El(N) (or}

we can

_{just pull}

back the ones we have from

Xl(N))

and we will denote them

w(p)

and

respectively.

We call sections of

Wk

on

Zi(Np)

_convergent

forms of level

_I’1(Np),

sections on _{any strict}

_neighborhood

of

_Z,(Np)

_{overconvergent}

forms of

level

_rl(Np)

and we set

_Mk(p)

:=

Mk(r1(Np)

:=

wk(W¡(P)).

Then we can define a U

_operator

on

Mk(p)

in the same _way as before.

Restriction

_gives

a natural _map from

Mk,cZ(P),

the _space of

weight k

modular forms on

X1(Np),

into

_Mk(p)

and we call the elements in the

THEOREM 8.1.

_{Every p-adic}

_{overconvergent}

_{f orm of}

_weight

k + 2 and

_slope

strictly

less that k + 1 in

_Mk+2

_(P)

is classical.

The definitions and results of Sections 2-4 _carry over without

difficulty.

For a

_rigid

_open

subspace

W of

X,

(Np)

set

let

_{]C(p) [ denote}

the union of the

_cuspidal

residue

classes,

]SS(p) [=

Z(p)

and let denote the kernel _{of the map}

The

only

results of Section 5 which

require

additional comment in this context are that

H(k,p)(W1(p»

is finite

dimensional,

the natural _map from

H(k,

p) (Wi

(p))

to

_{H(k, p) (W2 (p))}

is an

_isomorphism

and

_{p) (Wi (p))}

has a natural

non-degenerate

pairing.

We will

_again

use the main results of

~5~,

the

only

_problem

is that

_{Wi (p)}

is not

_naturally

a wide _{open in} a

complete

curve with

good

reduction to

which

_Vk(p))

extends.

_However,

we can overcome this

_difficulty

as

follows:

First let

Woo

and

Wo

be the inverse

_images

of the two

_components

of the reduction of the

_{Deligne-Rapoport}

model of

_{X1 (Np)}

over R and suppose the cusp c is a

_point

of

We.

Then C

Woo

and the

_rigid

space

Woo n Wo

is a

_disjoint

union of wide open

annuli,

Ax(p) ,

one for each

supersingular

point

x E SS. We may

glue

a disk

_Dx

to each such annulus as in

_{[8, §A2]}

to obtain a

complete

curve Y whose reduction is the

_Igusa

curve of level N. Now we will define an extension of

B7k(P»

to a sheaf

_gk

with connection on Y. Let

Bx

denote a basis of horizontal sections of

_Vx(p)

on

Ax(p)

for each x E SS.

Then Gk

is determined

_by

the data

and if

_f

E

_Maps(Bx,

0’(D,,))

The extension of is determined

_by

_requiring

the elements of

_{Maps (Bx,}

there is one

_point

(counting multiplicity)

in the

_support

of R in each disk

D,.

Now we can

_apply

[5,

Theorems 2.1 and

2.4]

to conclude that the nat-ural _maps from

_H(k,p)(Woo)

to

_H(k,p)(W1(p)

and from

_H(k,p)(W1(p)

to are

isomorphisms. Moreover,

each of these _{groups is}

isomorphic

to the first

_{hypercohomology}

group of the

complex

Thus the space

H(k, p)

:=

H(k,p)(W1(p»

is finite dimensional.

Now let

_Pk

denote Then a

_{Meyer-Vieitoris}

argu-ment

_making

use of

[3), [4]

and the

_covering

Wo}

yields

In

_particular,

the dimension over K of

_{H(k, p)(W~)}

is finite and this can be used to

_give

another

proof

of the finite

_{dimensionality}

of

_{H(k, p).}

_Moreover,

the

_self-duality

of induces a

_{perfect pairing}

( , )k

on

_Pk

and we can

define

_pairings

_{( ,}

_)~

and

_{( , )~}

on

H(k, p)(W~)

and

H(k, p)(Wo)

_by

the same

_procedure

as in the

proof

of Theorem 5.2 and show

where a and

#

are elements of

Pk.

It follows that

( , )I

is

non-degenerate.

We also conclude in the same _way as in

_§5

that

PROPOSITION 8.2. There exists an

endorrcorphisrrc

Ver

_of

H(k, p),

the quo-tient is

_naturally

_isomorphic

to

_{H(k, p)}

and the

fol-lowing diagram

commutes:

Moreover,

if

PROPOSITION 8.3. The space stable under Ver and

if a is

a rational number

One

_thing

we can now use is the action of

(Z/pZ)*,

F - for

d E

_(Z/pZ)*,

on modular forms in

Mk,cl(p)

and in

Mk(p).

It also acts on

Pk,

H(k, p)

and preserves If V is _any of these spaces, we set V"ew

_equal

to the

_subspace

of v E V such that

_¿dE(Z/PZ)*

_VI

_(d)P

= o. Let

Sk,cl(p)

denote the cusp forms in and those with trivial

residues on

]SS(p)[.

It

follows,

in

particular

that after suitable identifica-tions

and

PROPOSITION 8.4. We have

Proof.

In view of

_(8.2)

and

_(8.3)

and

_using

Lemma

_6.2,

we

_only

have to

show = On _one

hand,

the classical

Shimura

_isomorphism

tells us that

On the other

_hand,

a

_{Meyer-Vieitoris}

_argument,

as

_{above, applied}

to the

covering

of

_X1(Np)

tells us that is

_{naturally isomorphic}

to

Let w be an Atkin-Lehner

_type

automorphism

depending

on a fixed

prim-itive

_p-th

root of

_unity

as in

§5

or

[19, §6].

As w induces an

isomorphism

between these latter two

_cohomology

groups, we deduce that

This and

_(8.5)

completes

the

_{proof I}

We can _prove an

_analogue

of Lemma 6.3 in the same _way and deduce

LEMMA 8.5. The natural map an

injec-tion if a k + 1.