Classical and overconvergent modular forms of higher level

R

OBERT

F. C

OLEMAN

Classical and overconvergent modular forms of higher level

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{2 (1997),}

p. 395-403

<http://www.numdam.org/item?id=JTNB_1997__9_2_395_0>

© Université Bordeaux 1, 1997, tous droits réservés.

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

Classical and

_{Overconvergent}

Modular

Forms

of

_Higher

Level

par ROBERT F. COLEMAN

RÉSUMÉ. Nous définissons la notion de forme modulaire _{surconvergente}

pour

0393(Npn)

où p est un nombre _premier, N et n sont des entiers et N est

premier à _p. Nous démontrons _que toute forme _{primitive surconvergente} pour 03931(NPn), de poids k et dont la valeur propre de Up associée est de valuation strictement inférieure à k - 1 est une forme modulaire au sens

classique.

ABSTRACT. We define the notion _{overconvergent} modular forms on ₀₃₉₃₁

_(Npn)

where p is a _prime, N and n are _{positive integers} and N is _prime to _p. We show that an _{overconvergent eigenform} on

_{03931 (Npn)}

of weight k whose

Up-eigenvalue has valuation _strictly less than k - 1 is classical.

In this note we define a notion of

_{overconvergent}

modular form of level

where lV is a

_{positive integer,}

_{p is} a

_prime,

_{(N, p)}

= 1 1

and

_generalize

the main result of

_[2].

That

_is,

we show that

_{overconvergent}

forms of level

_weight

l~ and

_{slope strictly}

less that k - 1 are

classical.

This allows one to recover most of the finite dimensional space of classical

forms in the immense

_(of

uncountable

_dimension)

_space of

_{overconvergent}

forms.

_Using

the extra freedom one has with

_{overconvergent}

modular forms one can use this to prove

important

results about classical forms

_{(see below).}

The

_{only essentially}

new wrinkle in this _{paper is} the

_proof

of the

_equality

of the dimensions:

which is a

_{generalization}

of

_[2,

_Prop.

_8.4].

The terms on the

_right

are

defined in terms of

_{overconvergent}

forms of level in

_§l

and the term on the left is defined in terms of modular forms on in

§2

(and

in

_[6]~.

The

_following

observation of

_Ogus

is crucial for our

proof.

OBSERVATION

_(Ogus).

S is a

locally free

rank m coherent

sheaf

a curve X over a

_field

K and S2 is an invertible

sheaf. Suppose

S’:

is a complex.

Thenx(S’)

=:

=-mdegf2.

Proof. This follows from Riemann-Roch and the

_magic

of Euler

charac-teristics.

_Namely,

_using

the

_"Hodge

to de Rham"

_spectral

sequence we

deduce

Now

_{using using}

the fact that the

_alternating

sum of the Euler

character-istics of the terms in an exact sequence of

locally

free sheaves is zero and

Riemann-Roch we see

X(S ~

S2)

= ₊

m deg

S2.

(Of

course we have connections in

_mind.)

A _consequence of this are the results in

~10~.

THEOREM.

_Suppose

a E

_Q

and _{X is} a character on

(ZjNpnz)*.

Then the

dimension

_of

the space

of

modular

forms

on

Xl (N pn)

of

weight

k &#x3E; a +

_1,

and

_slope

a is

_locally

constant as a

_{function of}

k where k is

considered as an element

_of

(Z/(p - I)Z)*

x

_Zp.

1.

_{Overconvergent}

forms of level

_Np’~.

In this section we will _suppress the level N from the notation when

conve-nient.

Recall from

_[2~,

we have a commutative

_diagram

in level

_Np.

Here

_Wi(p)

is the connected

_component

of the

_rigid

sub-space of

Xl(Np)

fv(Ep-,(x))

p2-i/(p

+

_1)}

_containing

the cusp 00

and

_Ei(p)

is the

_pullback

of the universal

_elliptic

curve over

Xl(Np)

to

Wi(p).

Also, 16 and o

are the

_{Tate-Deligne morphisms.}

We let for n &#x3E;

_1,

denote the inverse

_image

in of with

_respect

to the

_morphism

from

_X1(Np)

which takes the

_point

corre-sponding

to

_(E,

a: _ppm -~

E)

_to the

_point

corresponding

_to

(E’,

a’)

where

E’ = and a’ is the

map induced

by

a from up to

E’.

_Better,

_Wi(pn)

consists of

_points

_{corresponding}

to

_pairs

_E)

such that the

_point

_{corresponding}

to

_{(E, alJLp)}

is in and

Also let

_Z(p)

be the inverse

_image

of

_Z(p),

where

_Z(p)

is the

_ordinary

locus in

_W2(p).

_(and

also

_Wl(p)).

Then

_Z(p’)

is the minimal

_underlying

affinoid of

_Wi(pn)

_containing

the _{cusps in} this wide open for i = 1 _or 2

(the

requirement

about the cusps is

only

necessary when the genus of the

Igusa

curve

(see

below)

is 0 and the number of

supersingular

points

on is less than 3

_(see

_[1]).

We have a natural

_{lifting of 0}

to a _map from

W2(p-)

to

_Wl(p")

which takes the

_point

_{corresponding}

to a

_pair

(E,

a:

E)

to the

_point

corre-sponding

to

_a’)

where

_{a’(~) =}

if

_Q

E and

_pQ

=

a(~’).

Then we have a commutative

_{diagram analogous}

to

_(1.1).

Let

_{f n:}

_{Xl (Np")}

be the universal

_{generalized elliptic}

curve over

_Xl

_{(Npn) (with}

the

_singular

_points

above the _cusps

removed)

and let w = * For k E Z we call sections of

Wk on

convergent

forms of level sections on _{any strict}

_neighborhood

of

_{overconvergent}

forms of level

_]F,(Npn).

We note that _any

overconvergent

form of finite

_slope

extends to an section of

wk on

W,

(pn).

Therefore,

we set

_Mk(p")

=

Mk(ri(Npn))

=: The _space

of modular forms of

_{weight k}

on

Xl(Np"),

which we denote

naturally

injects

into and we _say that elements in the

_image

are

classical. We

_identify

_{Mk,,, (pn)}

with the _space of classical modular forms. Then we can define an

_operator

U on

Mk(p")

in the same _way as in

[2,§2].

On

_Mk,n(p")

this acts as the Hecke

_operator

_Up

and so it follows that the

slope

of any classical form of

weight k

+ 2 is at most k + 1 if it is finite. THEOREM 1.1.

_{Every p-adic}

_{overconvergent}

_{form of weight}

k + 2 and

_slope

strictly

less that k + 1 in

_Mk+2

_(pn)

is classical.

The

_proof

of this theorem follows the same lines as the

proof

of Theorem

8.1 of

_[2].

The definitions and results of Sections 2-4 of

_[2]

carry over to this situation without

_difficulty.

In

_particular,

there is a linear

_{map 0}

from

M-k(pn)

to

_Mk+,(pn)

which on

_q-expansion

s is

(qdldq)’+’

and which

satisfies

We will use the notation

Symk

to denote the

_{kth symmetric}

power of

the relative de Rham

_cohomology

_complex

of

_{El(Npn) (minus}

its

_singular

points)

over

_X,(Np).

Let denote the union of the

_cuspidal

residue

classes,

and let denote the kernel of the map

We may deduce in the way in

[2]

that there exists an

endomor-phism

Ver of the

_quotient

is

naturally isomorphic

to this space and the

following

diagram

commutes:

For k &#x3E; 0,

we set

_Sk+2(p")

_equal

to the

_subspace

of

_Mk(pn)

consist-ing

of _cusp forms with trivial residues

_(in

the sense of

[11])

on

This is also

_equal

to the inverse

_image

with

_respect

to the above map of in

2. Dimensions: Modular forms and

_Cohomology

Let X denote the curve

Xl(Npn)

over

Cp.

We let denote

the

_subspace

of

_primitive

forms in

_Recall

this is the space

spanned by

the

_images

of the

_weight

k forms of level

_rl(NpT)

of

_primitive

nebentypus

at p of

0 ~

r n and

is the

subspace

of

primitive

cusp forms.

(We

point

out that two

copies

of the level N forms are contained

in

_{when n &#x3E;}

1 via the two

_degeneracy

_maps.)

It follows from the work of

_Ogg

and Li that the

_subspace

of on which

(Z/p"Z)*

acts

_{non-trivially}

is the same as the

subspace

of on which

(Z/p"Z)*

acts

_{non-trivially}

and

_rlsUpMk(p")

=

M~~"’’(p’’~)

₊

We set

_{r(N; p’’)}

= =

X(N;pT)

= _etc., in the notation of Mazur-Wiles

[MW].

For a

representable

moduli

problem P

over

Ell/R

we let

In(P)

denote

the

_{Igusa problem}

of level

_0,

over P

_(so

that

_{Io(P) = P).}

If P is a

representable

moduli

problem

over

Ell/k

we let

X(P)

denote the

completion

of

_{representation}

space

yep), C(P)

the

cuspidal

divisor on

X(P)

and

_Sl(P)

= We also set

etc. We will also use to denote the k-th

_symmetric

_product

of the first relative de Rham

_{cohomology complex}

of the universal

_elliptic

curve over

X(P)

and V will denote the relevant connection. The group

is defined

_similarly

to

Mk (p

-We let X be stable model of

_{(X, C)}

over R

(we

only

need to worry

about C when X has genus 0 or

1) .

Then let be the

"good" Igusa

whose closed

_points

reduce to

_points

in

_{In(N) (this}

is the

_generic

fiber of the tube of

_In(N)

in

_X)

and I’

=

I’(N)

be the

rigid

_space

corresponding

to the other

_good

_Igusa

_component

_(which

equals

w~I

for _any

_primitive

_p-the

root

_unity

_~).

We also let U =

Ul(Npn)

denote the reduction inverse of the union of all the

_components

of the stable model of X besides the

_good

ones. Let SS denote the

_{supersingular}

divisor on or its

degree

when no confusion will arise. Then the spaces I n U and are each the union

of SS annuli.

For a

rigid subspace

W of X we let denote the kernel of

the restriction map from to

_{H’ (W}

n THEOREM 2.1. The natural map

from

is an

_isomorphism.

Note: The _map from

_{HI (X,}

_Symk)prim

to

HI

_(I,

_Symk)

factors

_through

Symk)*

because

_(Z/pnZ)*

acts

_trivially

on the

cohomology

of the

complex

Sym k

when restricted to the

_{supersingular}

annuli. Proof.

_First,

_using

_[5,

Thm.

_10.13.12~,

we see that when n &#x3E; 2

and

_similarly

Hence,

as

H1 (X,

modulo

HI (Xl

is

isomor-phic to

when n &#x3E; 1

_(all

the _maps involved here are

injections)

we see

_using

the

ob-servation of

_Ogus

and

_[5,

Cor.

_12.9.4],

that the dimension of

Now we have to

_compute

We do this

_{by adding}

disks to I to make a

_complete

smooth

_lifting

of the

_Igusa

curve

In(N),

i,

and

_extending

the connection

_SyTnk

to a

connection

on

I

which

_only

has

_log

_poles

at the _cusps C in I

_{(see [2,}

_§8]).

Then the observation of

Ogus

mentioned in the introduction

_implies

that

X(t)

=

deg

91

(log C) =

Since

H°(tf)

= 0 if k _&#x3E; 2 this

_gives

us what we want.

Similarly,

we can work on the reduction inverse

_image

I(N, p")

of the

good

component

containing

o0 on the stable reduction of the curve

X(N, p")

define and _{prove its} dimension is

We also let denote the reduction inverse of the bad

components

of the stable model of

To _prove the

_isomorphism

in the theorem we use

_{Meyer-Vietoris}

and

induction on n. It is true when n = 1

by

[2].

We have exact

sequences

Where Y is either

Xl

(Npn)

or

X(N,pn),

V is either

_{Ui (Np")}

or

_{U(N; p"),}

J is either

_I(N)

or etc.

_accordingly.

When k &#x3E; 0 the first _maps

are

_injections

and the last _maps are

_surjections.

In _any case we deduce that

(it

is easy to see that the _map from the second of these _spaces to the first is an

_injection,

and we

_identify

it with its

_image)

is

_isomorphic

to the direct

sum of

But,

using

the

_triviality

of the action of

_(Z/eZ)*

on the

_cohomology

over

the

_{supersingular annuli,}

either of the first two spaces is

isomorphic

under

the natural _map to

and it follows from the above that this space has dimension

Hence the _map to the sum of the first two spaces above is an

_isomorphism,

which proves the

theorem,

and it follows that the dimension of the third is zero. I

COROLLARY 2.2.

In

_particular,

_every

_primitive

modular

_{form of}

_weight

k + 2 when considered

as a section

_of

_{Symk 0!1l(Npn)}

and restricted to

_Ul(Npn)

is

_equal

to B7G

for

some section G

of

Symk

on

For a

_subspace

W of X we set

_Symk)

_equal

to

Then

COROLLARY 2.3. We have which

equals

where 6 = 2

if

k = 0 and 6 = 0 otherwise.

Proof.

_By

Riemann-Roch and

_[5,

Thm.

_10.13.12]

when k &#x3E; 0. It follows

_that,

in this case,

which

_implies

the result

_using

_Ogus’

observation and

_[5,

Cor.

_12.9.4]..

This and the

_isomorphism

between

and

_(p’~),

_Symk)

enables us to prove

(0.1).

REMARK. It

_{follows from}

the results in

_{~8,§~ ~ §3]}

that the stable

reduc-tion

_of

_X(N;pn)

is the same as that

of

Xl (Np")

except

that the two

_good

components in the latter are each

replaced by

a _copy

_of

_In-l(N).

On the other hand we

_point

out that the stable reduction

of

Xl

(Npn)

is not known

3. Conclusion

As in

_{[2, §5]}

we can

_put

a

_pairings

( , )1

and

( , ) I,

on

Syrrvk)*

and on We can conclude in the same _way as in

[2, §5]

that

Moreover,

if

_{( , )}

is the natural

_pairing

on

( , )

restricts

to a

_{perfect pairing}

on and if ac and b are elements of

H

This and Theorem 2.1

_implies

that

_{( , )1}

is

_{non-degenerate}

which allows us

to

_generalize

_[2,

_Prop.

_{8.3~ .}

We can conclude that the _map from the

slope

a

subspace

of

P72 M

to the

slope

a

_subspace

of

Symk)*

is an

isomorphism

if a k + 1

exactly

as in

[2, §8].

The

proof

of Theorem 1.1 may be

completed by

arguments

similar to those used in

_[2].

REFERENCES

[1]

Coleman R., Reciprocity Laws on Curves, Compositio 72

(1989),

205-235.

[2]

Coleman R., Classical and _{Overconvergent} modular _forms, Invent. Math. 124 _(1996), 215-241.

[3]

Edixhoven B., Stable models _of modular curves and applications, Thesis, University

of Utrecht _{(unpublished).}

[4]

Katz _N., P-adic _{properties of} modular schemes and modular forms Modular Func-tions _of one Variable III, Springer Lecture Notes 350

(197),

69-190.

[5]

Katz N. and B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Math. Stud.

108, Princeton _{University Press,} 1985.

[6]

Mazur B. and A. _Wiles, "Class _fields and abelian extensions _{of Q",} Invent. Math. 76 _(1984), 179-330.

[7]

Li _{W., "Newforms} and _{functional equations, ",} Math. Ann. 212 _{(1975), 285-315.}

[8]

Mazur B. and A. _Wiles, "On p-adic analytic families of Galois _{representations",} Compositio Math. 59

_(1986),

231-264.

[9]

Ogg A., "On the _{eigenvalues of} Hecke _operators", Math. Ann. 179 _{(1969), 101-108.}

[10]

Coleman _{R., p-adic} Banach spaces and families of modular forms, Invent. math? 127 _(1992), 917-979.

[11]

Coleman R., p-adic Shimura Isomorphism and p-adic Periods _of modular forms, Contemp. Math. 165

_(1997),

21-51.

Robert F. COLEMAN Department of Mathematics

Berkeley, California 94720-3840 USA email: coleman~math.berkel.ey.edu