Families of modular forms

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

K

EVIN

B

UZZARD

Families of modular forms

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 43-52

<http://www.numdam.org/item?id=JTNB_2001__13_1_43_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

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

Families of modular forms

par KEVIN BUZZARD

RÉSUMÉ. Nous donnons une introduction terre à terre de la théo-rie des familles de formes

_modulaires,

et discutons des démonstra-tions élémentaires de résultats

_suggérant

_que les formes modu-laires _apparaissent sous forme de familles.

ABSTRACT. We

_give

a down-to-earth introduction to the

_theory

of families of modular

_forms,

and discuss

_{elementary proofs}

of

results

_suggesting

that modular forms come in families.

1. What is a

family?

Let r be the _group

_SL2(Z),

or more

_generally

a _congruence

_subgroup

of

SL2(Z).

1 be an

_integer.

A modular form of

_{weight k}

for r is a

holomorphic

function

_f

on the _upper half

_plane

_satisfying

, , ,

for all

_(~~)

E

_r,

and some boundedness

_conditions,

which ensure that for

fixed k and

_r,

the space of such forms is finite-dimensional. If r =

rl (N)

for some

_{positive integer N,}

then we _say that

_f

has

_weight

k and level N.

Any

enquiring

mind

_seeing

the

_precise

definition for the first time would

surely

wonder whether _any non-constant modular forms exist at all. But

of course

_they

do-and we shall

begin

our

study

of families with the forms

that are

frequently

the first non-trivial

_examples

of modular forms

_given

in

an

introductory

course, namely the Eisenstein series

_{Ek .}

Here l~ &#x3E; 4 is an

even

_integer,

and

_Ek

is a modular form of

_weight

and level 1 which

_has,

at least _{up to} a

_constant,

the

following

_power series

_expansion:

As is

_standard,

we _{use q to} denote

exp(27riz),

and

ad(n)

to denote the sum

of the dth _powers of the

_(positive)

divisors of n.

Note that

_although

these forms are constructed

only

for k &#x3E; 4 and _even,

the coefficients of the _{power series} above

_happen

to make sense for _any

Manuscrit requ le 2 novembre 1999.

The author would like to thank the _organisers of the Journées _{Arithmetiques} 1999 for _giving him the _opportunity to lecture in such _interesting surroundings.

non-zero k E C.

_Indeed,

if we were asked to come _up with a natural _way of

interpolating

the Fourier coefficients of the

_{Ek ,}

then the choice

_given

above would

_surely

be a natural one. Almost

_unwittingly,

we have created our

first

_family

of modular forms.

Actually,

this article is

_really

concerned with

_p-adic

families of modular

forms,

so before we

continue,

we shall

modify

the

Ek

slightly.

For

simplicity,

let us assume for the rest of this section that _{p is} odd. Recall that there

is a

p-adic analogue

of the zeta

_function,

so called because it

(essentially)

p-adically interpolates

the values of the zeta function at

_negative

_integers.

We _say

_{"essentially"}

because there are two caveats. The first is that one

has to restrict oneself to

_evaluating

the zeta function at

_negative

_integers

congruent

to 1 modulo _{p -} 1. The second is that one has to

_drop

an

Euler factor-the correct classical

_object

to

_{p-adically interpolate}

is

_{( 1}

-p-S)((s) =

q-s)-l.

Similarly

one has to

_drop

an Euler factor when

_attempting

to

_p-adically

interpolate

the Eisenstein series

Ek .

For k &#x3E; 4 an even

integer,

define

where is the sum of the dth _powers of the divisors of n that are

prime

to p. Then is an oldform of level _p, and it is these oldforms that will

interpolate beautifully,

at least if we restrict to I~ in some fixed

conjugacy

class modulo _{p -} 1.

Let us for

example

consider the set ,S’

_consisting

of

_positive

even

integers

k which are at least 4 and are

_congruent

to 0 mod _{p -} 1. Then this set

is dense in

_Zp.

It turns out that the Fourier

_expansions

for are

p-adically

continuous as k varies

_through

_S’,

where now we think of ,S’ as

having

the

_{p-adic topology.}

In

_fact,

one can show more. Consider the

functions

- - . , ,

-Then we have the

following

well-known theorem.

Theorem 1. There are

unique

p-adic analytic functions

An :

Qp

for

n &#x3E;

_0,

with

Ao

=

1,

such that

for

all k in S _we have the

following equality

of formal

sums:

D This theorem is little more than the existence of the

p-adic

zeta

function,

and the fact that it has no zeros. One

thing

which is not

immediately

for all

_{integers k &#x3E;}

_0,

the _power series is

_actually

the Fourier

expansion

of a modular form of level _p

and

_weight

k.

Motivated

_by

this

_theorem,

we shall now

_give

a definition of a

p-adic

family

of modular forms.

_Firstly,

let

_Cp

denote the

_completion

of the

algebraic

closure of

_Qp,

and fix an

_isomorphism

C ~

_Cp.

Our families will live over

p-adic

discs in

Zp,

so let us

_define,

for c E

_Zp

and r E the disc

_B(c,

_r)

to be the set of elements 1~ E

_Zp

_cl

r.

Definition. Let N be an

_{integer prime}

_{to p,} and fix a disc

B(c, r)

as above.

A

_{p-adic family of}

modular

_forms

of level N is a formal _{power series}

where each

_{Fn :}

_{B(c, r)}

-

_Cp

is a

_p-adic

_{analytic function,}

and with the

property

that for all

_{sufficiently large}

_(rational)

_{integers k}

in

_{B(c, r),}

the formal sum

¿n

is the Fourier

expansion

of a classical modular form

of

_{weight k}

and level

_NP.

Note that the

_family

of Eisenstein series that we have constructed is

indeed a

_family

in the above sense. We also remark that the restriction to p odd that we made a while _{ago is}

entirely

_unnecessary and that one can

still construct a

family

of Eisenstein series when _p = 2.

Henceforth, p

will

be an

arbitrary

prime.

2. Forms live in families.

Given what we

know,

it is now not too difficult to show that _any modular

form of level

_Np

lives in a

family

of level can

_just

_multiply

the

form in

_question

_by

the

_family

of Eisenstein series

_already

constructed

(which

fortuitously

goes

through

the constant modular form 1 when k =

0).

A much more

challenging

question

is:

(auestion.

Let c be a rational

integer,

and let

f

be a classical

eigenform

of level

_Np

and

_weight

c. Does

f

live in a

p-adic

family

of

eigenforms?

By

a

_{p-adic family}

of

eigenforms,

we

simply

mean a

family

of forms whose

_{specialisations}

to

_weight

_k,

for all

_{sufficiently large}

rational

_integers

k E

_{B(c, r),}

is an

eigenform.

We remark that the

family

of Eisenstein series

is a

family

of

eigenforms.

However,

the trick of

multiplying

an

_arbitrary

eigenform f by

this

_family

will not

_produce

a

_family

of

_eigenforms,

because

the

_product

of two

_eigenforms

is not

_necessarily

still an

eigenform

(and

indeed

_frequently

will not

_be).

Before we

_continue,

we introduce one more

_piece

of notation. If

_f

is a

classical

_eigenform

of level

_Np,

then in

_particular

it is an

_eigenvector

for

the

_Up

_operator.

We _say that the

_slope

of

_f

is the

_p-adic

valuation of the

level

_Np

_always

have finite

_slope.

We _say that a

_{p-adic family}

of

_eigenforms

has constant

_slope

s

if,

for all

_{sufficiently large}

rational

_{integers k}

E

_B(c,

_r),

the

_{corresponding weight k eigenform}

has

_slope

s.

The

_question

stated above has in fact been answered

_{affirmatiely by}

Coleman,

in the _paper

_[C].

_{Inspired by}

this

_work,

Coleman and Mazur constructed in

_[CM]

a

_geometric

object,

the so-called

_{"Eigencurve" ,}

whose

very existence

implies

that

eigenforms

come in families! Points on the

eigencurve correspond

to certain

_{p-adic eigenforms.}

So

_assuming

that the

eigencurve exists,

the

_points

in a small _open disc on this curve will

corre-spond

to a

family

of

eigenforms.

The existence of the

_eigencurve

also

_explains

more

_conceptually

what is

going

on when one

specialises

a

p-adic family

of forms at a

_point

where the

resulting

q-expansion

is not that of a classical modular

_{form-although}

we

shall not go into the details here. But one

thing

that it did not

_initially

shed

_light

on was the

_question

of how

_big

one could

_expect

the radius r of the

_family

to be-or

_{equivalently,}

how

_big

a disc could _you fit round a

point

in the

_eigencurve?

Understanding

the nature of the radius has in fact turned out to be a

tricky problem.

In

_[GM],

Gouvea and Mazur make some

_{precise conjectures}

about what r

might

be

expected

to

be,

at least when the level of the forms

in

_question

is

_ro(Np),

and the

_optimist

can

_easily

extend these

_conjectures

to cover the case of These

conjectures

were based on a lot of

numerical

_{examples computed}

_by

Mestre. What seemed to be

_going

on

in Mestre’s

_computations

was that the radius of a

family

_passing

through

an

eigenform f

of level

Np

and

slope s

was, broadly speaking,

p-’.

The

reader who wants to know the

_precise

form of the

_conjecture

_(which

is in

fact rather _easy to

_explain)

is referred to

_[GM].

As well as this

_{computational evidence,}

the _paper

[W]

_{showed, using}

Coleman’s

_work,

that an

eigenform f

of

slope s

should live in a

family

of

eigenforms

with radius where t =

0(s2).

Unfortunately,

_no-one has _so

far been able to "close the

_gap",

with the result that we still seem to be unsure whether Mestre’s

_{computational}

results are

misleading

or whether

Wan’s work can be

strengthened.

The fact that

_eigenforms

should come in families has classical

conse-quences. For

example,

the results above

imply

that there should be a lower

bound on the Newton

Polygon

of

Up

acting

on the _space of modular forms

of level

_Np

and

_weight

_k,

and that this bound should be uniform in k. This

question

was raised

explicitly by

Ulmer in

[U],

who

produced

a number

of

_interesting

results in this direction. We should

_perhaps

remark at this

stage

that Coleman’s work used a lot of

machinery

from the

theory

of

rigid-analytic

geometry,

and Ulmer’s ideas

_appealed

a lot to

_crystalline

methods.

elementary approach

could

_perhaps

be used to attack these

_problems.

In-spired

by

these

_ideas,

we shall here _present a

completely

elementary proof

of

Theorem 2. The Newton

_{Polygon of}

_Up

_acting

on the _space

of

classical modular

_{forms of weight}

k and level

_Np

is bounded below

_by

an

_explicit

quadratic

lower bound which is

_{independent of}

k.

As a

_corollary,

we

_get

_explicit

bounds for the number of forms of

slope a,

weight k

and level

_Np,

and moreover these bounds are

independent

of k.

We shall state and _prove a

_precise

version of the

theorem,

and its

corol-laries,

in the next section. As far as the author

_knows,

the first

_proof

of

this theorem was

_given

in

[W]

and made essential use of Coleman’s

rigid-analytic

methods. The

_proof

we

_present

uses

_nothing

more than _group

cohomology.

3. The

_proofs.

Some notation.

Recall that N is a

positive integer,

and _p is a

_prime

not

_dividing

N. Let R =

Zp[~]

denote the

_polynomial

_ring

in one variable over

Zp.

Let

L be a finite free

_Zp-module

of rank

_t,

_equipped

with a

_Zp-linear

endo-morphism

with non-zero determinant. Then L can be

_regarded

as a

Zp[]-module,

where e

acts via this

_{endomorphism.}

We recall the definition of the Newton

_{Polygon of ~}

_acting

on L. Write

the characteristic

_{polynomial of ~}

on L 0

_Qp

as

ct-iX 2.

Then let vp

denote the usual valuation on

_Zp,

and

_plot

the

_points

(i,

in

]R2,

for

0 i

_t,

_ignoring

the i for which ci = 0. Let Z denote the _convex hull of

these

_points.

The Newton

_Polygon

_{of ~}

on L is the lower faces of

Z,

that

is the union of the sides

_forming

the lower of the two routes from

_{(0, )}

to

(t, vp(ct))

on the

boundary

of Z. It is well-known that this

graph

encodes

the

_p-adic

valuations of the

_eigenvalues

of

_~.

In

_fact,

if the Newton

_Polygon

has a side of

_slope

a and whose

_projection

onto the x axis has

_{length n,}

then there are

precisely n

eigenvectors

of ~

with

p-adic

valuation

equal

to a.

It is our intention to say

something

about this

polygon when ~

is the

operator

U~

acting

on a certain lattice in a _space of modular forms. As a

result we shall recover results about the

p-adic

valuations of the

eigenvalues

of

_Up.

We shall first establish a

general

result,

after

_setting

_up some more

notation.

Let L be as

above,

and let K be a sub-R-module of L such that the

quotient Q

=

L / K

is _a finite

with

Now assume that there exists a

_{non-negative integer}

_n,

_greater

than or

equal

to all the _ai, and such

_{that g(K)}

C

_pnL.

_{Write bi}

=

n-ai for 1 i t

and

Lemma 1. The Newton

_{Polygon of ~}

orc L lies on or above the

_points

(.?a ~(j)) for 0 j t.

Proof.

The

_proof

is _{very easy,} and we

shall just

sketch it. First choose a

Zp-basis for L such that is a

Zp-basis

for K. With

_respect

to this

_{basis, let ~ :}

L --~ L be

_{represented by}

the matrix

_{(uj,j) .}

One sees

that the

_assumption

_~(K)

C

_{pnL implies}

that

_pbj

divides

_Then,

from the definition of the characteristic

_polynomial

as

~) = E

and from the fact that

_{the bi}

are

increasing,

we see that divides cj for

0 j

t.

_Using

once more that

the bi

are

_increasing,

the lemma follows

easily.

D

We now show how to

_apply

this in the case of classical modular forms.

Write r for and for k &#x3E; 2 an

integer,

let denote the _space

of classical _cusp forms of

_weight

k and level

_Np.

Note that if

_{Np &#x3E;}

5

then r is torsion-free and hence free. In

_fact,

if

_{Np &#x3E;}

5 then r is free on

2g(X(r))

+c-1

generators,

where is the _genus of the

_compactified

modular curve associated to

_r,

and c is the number of _cusps added to make the

_{compactification.}

For

_general

_Np,

we let

m(r)

denote the minimal

number of

_generators

of r as a _group. Hence

m(r)

=

2g(X(r))

+ c -1 if

Np &#x3E; 5,

and

_m(r)

can

_easily

be worked out

_explicitly

in the other cases.

Let g = k - 2. For a commutative

_ring

A,

let

Vg(A)

=

A9+1,

considered

as column matrices of

length

_g + 1. Let M denote the monoid

_consisting

of

two

_by

two matrices with

_integer

entries and

_positive

determinant. There

is a

unique

left action of M on with the

following

_property:

for all

y, z E

_Z,

we have

where the

_superscript

T denotes

_transpose.

In

_fact,

this action can be

con-structed

_explicitly

thus: Consider elements of

7~2

as row vectors. Then there

is a natural

_right

action of M on

7~2

and hence on its

_symmetric

gth power.

The left action we are interested in is the natural induced left

action of M on the Z-dual of Sg

(7~2) .

Define an action of M on

Vg (A)

for _any commutative

ring

A

by extending

the action on

A-linearly.

Note that r C M and hence

Vg (A)

has the

There is a well-known

isomorphism

(see

Theorem 8.4 of

(S )

due to

Eich-ler and Shimura:

ES :

_{Sk (F)}

-

where

_HP

denotes

_{parabolic cohomology. Composing}

ES with the natural inclusion

_Hp(r,

_V9(IEg))

-

Hl(r,

V9(IE8))

_gives

us an inclusion ES :

H1(r,

The Hecke

_operator

_Up

acts

_(C-linearly)

on

Sk (r),

and we shall now

recall the definition of an

_operator

on that extends

_Up.

Let A be any

ring,

and

define ~ :

Hl(T, V9(A)) ~

Hl(r,

V9(A)),

an

A-linear _map, thus.

_Firstly

note that

and write _ai for

1 Z

.

For q in F and 0

i

-

1 there exists a

unique 0 j p -1

and 7i

E IF such that ai7 =

yiaj. First we shall define

the

_{map ~}

on

1-cocycles.

If u : F -

_V9

is a

_1-cocycle,

define

by

One can check that v is a

1-cocycle

and moreover that the association u - v

induces a _map from

H1(r,

_Vg)

to

_itself,

which we define to be

_~.

Lemma 2. The

_diagram

commutes.

Proof.

This is

_Proposition

8.5 of

_[S].

D

Let

_a(X)

denote the characteristic

_polynomial

of

_Up

_acting

on the

com-plex

vector _space

_Sk(r).

It is the roots of this

_polynomial

that we

ultimately

wish to

_study.

Note first that the coefficients of a are rational. Hence

a2

is the characteristic

_polynomial

of

_UP

on

Sk (r),

considered as a real vector

space, and so if

~3(X)

is the characteristic

polynomial of ~

_acting

on the

real vector _space

_H1(r,

then we see that

a2

divides

_{3,

as ES is an

injection.

Note that

_{3

is also the characteristic

_{polynomial of ~}

_acting

on

We now _pass to the case K =

Qp.

Firstly

note

_{that, although}

the _group

may have some

torsion,

its

image

in is a

lat-tice. It is this lattice that we shall use for our L.

Let Ei E denote the column vector with 1 in its ith row and zeros

elsewhere,

and let

_{Wg (Zp)}

denote the

_Zp-subspace

of

_{generated by}

the vectors

- no

It is

_easily

checked that

_{Wg (Zp)}

is

_{preserved by}

r. Let I denote the

_quotient

Vg(Zp)/Wg(Zp)

with its induced r-action.

_Taking

the

_long

exact sequence in group

cohomology

associated to the short exact _sequence

gives

us an exact _sequence of

_{finitely-generated}

_Zp-modules

where the first and last terms in this _sequence are finite. It follows that

there is an exact sequence

where

_Q

is some

subquotient

of and the

_superscript

TF denotes

the maximal torsion-free

_quotient.

In

_particular

_Q

is finite

_(it

would be

_good

to have a more

_{explicit description}

of

Q).

Now write L for

H’ (IF

and K for

_{H1(r, Wg(Zp))TF.}

The action

_{of ~}

on induces an

action

_{of ~}

on L and hence on K. Now we have

Lemma 3.

_~(K)

C

_p9L.

Proof.

Let r, : F -

_{Wg (Zp)}

be a

_1-cocycle.

It suffices to _prove that the

cocycle

gK

is divisible

_by

_p9.

From the definition of

_6n,

this will follow if we can show that

, ,

/ ’ _B

" ,

for 0 i _{p -} 1. But in fact one can check that

for _any

( a d)

)

E M such that p divides both a and c, and we are

(

c

d )

home. D

We are now in a

position

to

apply

Lemma

1,

with n =

g. The lemma will

give

us an

_explicit

lower bound for the Newton

polygon of ~

on

L,

which

encodes the

_slopes

of the

_eigenvalues

of the

_{polynomial ,Q.}

But we remarked

above that

_{a2 divides,3}

and so we are also

getting

explicit

lower bounds

First of all we make some _easy observations. Because L is a

quotient

of

the _group of

_cocycles

c : r --~ we see that the rank t of L is at most

(g+1)m.

We do not know

_{Q explicitly,}

but we know that it is a

subquotient

of

_I),

which is itself a

quotient

of the _group of

1-cocycles

from r to I.

This _group of

_cocycles

is

_{non-canonically isomorphic}

to a

_subgroup

of

_1m,

where we recall that m =

m(r)

is the minimal number of

_generators

of r.

Moreover,

I itself is

_isomorphic

to

If x E

_R,

write for the

_largest

_integer

which is at most x, and

_fxl

for the smallest

_integer

which is at least x. The

_arguments

above show that if

we write

Q

=

with ai

decreasing,

then _r

mg.

Set ai

= 0

for r i t. Then we can deduce

that ai

Ln

+ 1 - for 1 i t

and

hence bi

= n - ai 2::

_{L(i -}

(with

notation as before Lemma

1.)

A

_simple

_application

of Lemma 1 then

_gives:

Proposition

1. The Newton

_{polygon of ~}

_acting

on L is bounded below

by

the

_function

which _goes

_through

_{(0, 0),}

has

_slope

0

_for

0 x _m,

_slope

1

for

m x

2m,

slope

2

for

2m x

3m,

and so on.

D

Note that this bound

_depends

_only

on m and in

particular

only

on N

and p. Because of the

relationship

between and

H1(r,

ex-plained

above,

we deduce

Theorem 3. The Newton

_{polygon of}

_Up

_acting

on

Sk(r)

is bounded below

by

the

_function

which _goes

_through

_{(0, 0),}

has

_slope

0

_for

0 x

m/2,

slope

1 _m, and in

_general

has

_slope

r

for mr/2 x (mr-~m)/2.

D This

_is,

of course, a more

_precise

form of the theorem that we were

_aiming

for. An easy consequence is

Corollary

1.

_{If a}

E

_Q,

the _{space of cusp}

_{forms of}

_slope

_a,

_weight

k and level

_Np

has dimension at most

_{[a + 1/2~m,}

_{independent of k.}

We remind the reader that m is the minimal number of

_generators

of

r =

r1(Np)

as a _group. As concrete

examples,

we could

_perhaps

mention

that If N = 1 and

_{p &#x3E;}

5 then m can be taken to

be 12 .

In

_general,

m is about

O((NP)2).

Moreover,

it is not difficult to show that in fact m can be taken to be any

integer

such that

rl (Np)

has a

_subgroup

of finite index

_prime

_{to p} which is

_{generated by}

m elements. We remark also that

our

_analysis

can

easily

be

generalised

to other _congruence

subgroups

F of

the form

_rOnr1(p)

or

ro

fl

_ro(p),

where

ro

is any congruence

subgroup

of level

_prime

to p.

Finally,

we should mention that for

explicit

values of N

and _p

_(typically

N = 1

by

Coleman,

Stevens,

Teitelbaum,

and Emerton and Smithline

_using

rigid-analytic

methods. It would be

_interesting

to know whether there were more

down-to-earth

_techniques

that could establish their results. References

[C] R. COLEMAN, p-adic Banach spaces and families of modular forms. Invent. Math. 127

(1997), 417-479.

[CM] R. COLEMAN, B. MAZUR, The _eigencurve. In Galois _{representations} in arithmetic algebraic

geometry (Durham, 1996), CUP _1998, 1-113.

[GM] F. GOUVÊA, B. MAZUR, Families of modular _eigenforms. Math. _Comp. 58 no. 198 (1992),

793-805.

[S] G. _SHIMURA, Introduction to the arithmetic _{theory of automorphic functions.} Princeton

University Press, 1994.

[T] R. _TAYLOR, Princeton PhD thesis.

[U] D. _{ULMER, Slopes of} modular _{forms. Contemp.} Math. 174 _(1994), 167-183.

[W] D. WAN, Dimension variation _of classical and _p-adic modular _forms. Invent. Math. 133

(1998), 449-463. Kevin BUZZAR.D Department of Maths Huxley Building Imperial College London, SW7 2BZ England