A computational study of the asymptotic behaviour of coefficient fields of modular forms

by

Mar elMohyla& GaborWiese

Abstra t.  The arti le motivates,presentsand des ribeslarge omputer al ulations

on- erningtheasymptoti behaviourofarithmeti propertiesof oe ienteldsofmodularforms.

Theobservationssuggest ertainpatterns,whi hdeservefurtherstudy.

Résumé.  Le but de et arti le est de motiver, présenter et dé rirede nombreux al uls

menés sur ordinateur on ernant le omportement asymptotique de propriétés arithmétiques

des orpsdes oe ientsdeformesmodulaires.Lesobservationssuggèrent plusieursquestions

quiméritentd'êtreétudiéesultérieurement.

1. Introdu tion

A re ent breakthrough in Arithmeti Geometry is the proof of the Sato-Tate onje ture by

Barnet-Lamb, Clozel, Geraghty, Harris, Shepherd-Barron and Taylor ([BLGHT℄, [CHT℄,

[HSHT℄, [T℄). It states that the normalised He ke eigenvalues

a

p

(f )

2p

(k−1)/2

on a holomorphi newform

f

ofweight

k ≥ 2

(andtrivialDiri hlet hara ter

(1)

)areequidistributedwithrespe t

toa ertainmeasure(theso- alledSato-Tatemeasure),when

p

runsthroughthesetofprime

numbers. Thename horizontalSato-Tateis sometimesusedfor this situation.

The reversed situation, to be referred to as verti al horizontal Sato-Tate, was su essfully

treated by Serre in [S℄. He xes a prime

p

and allows any sequen e of positive integers

(Nn, kn)

witheven

kn

and

p ∤ Nn

su hthat

Nn

+ kn

tendstoinnityandprovesthat

a

p

(f )

2p

(k−1)/2

2000 Mathemati sSubje tClassi ation.  11F33(primary);11F11,11Y40.

Key words and phrases.  Modularforms, oe ient elds,asymptoti behaviour, ongruen es, He ke

algebras.

G.W.wouldliketothankGabrielChênevertforinspiringdis ussionsofvariantsofSato-Tate. Morethanksare

duetoPierreParent,ChristopheRitzenthaler,RenéS hoofandMarkWatkinsforinterestingdis ussionsand

suggestions. G.W.a knowledgespartialsupportbytheEuropeanResear hTrainingNetworkGaloisTheory

and Expli it Methods MRTN-CT-2006-035495 and by the Sonderfors hungsberei h Transregio 45 Periods,

modulispa esandarithmeti ofalgebrai varieties oftheDeuts heFors hungsgemeins haft.

(1)

isequidistributed with respe t to a ertain measuredepending on

p

(whi h is related to the

Sato-Tatemeasure),when

f

runsthroughallthenewforms inanyofthespa es of uspforms

oflevel

Γ

0

(N

n

)

and weight

k

n

. A orollaryisthat forxed positive even weight

k

,theset

(1.1)

{[Q

f

: Q] | f

newformof level

Γ

0

(N

n

)

and weight

k}

is unbounded for any sequen e

N

n

tending to innity. Here,

Q

f

denotes the number eld

obtainedfrom

Q

by adjoining allHe ke eigenvalues on

f

.

Inthisarti leweperformarst omputationalstudytowardsa(weak)arithmeti analogueof

verti alSato-Tate,wherethenamearithmeti referstotakinganitepla eof

Q

,asopposedto

theinnitepla eusedinusualSato-Tate(theassertionofusualSato-Tate on ernstheHe ke

eigenvalues as real numbers). An arithmeti analog of horizontal Sato-Tate is Chebotarev's

density theorem. Consider, for example, a normalised uspidal He ke eigenform

f

with

at-ta hed Galois representation

ρ

f

: Gal(Q/Q) → GL

2

(Z

ℓ

)

.

(2)

Fix some

x ∈ Z

ℓ

and let

n ∈ N

.

Let

G

be the image of the omposite representation

Gal(Q/Q)

ρ

f

−→ GL

2

(Z

ℓ) ։ GL2

(Z/ℓ

n

Z)

andlet

d(x)

bethenumberofelements in

G

withtra eequal to

x

modulo

ℓ

n

. Thenthe

den-sity of the set

{p | |a

p

− x|

ℓ

≤ ℓ

−n

_}

isequal to

d(x)

|G|

byChebotarev'sdensity theorem; hen e, thesituation is ompletely lear. Whereasat theinnite pla e horizontal Sato-Tateseemsto

be more di ultthan verti al Sato-Tate, the situation appearsto be reversed for arithmeti

analogs. We are not goingto proposesu h an analogue. But, we aregoing to study related

questions by means of omputer al ulations. For instan e, as a motivation let us onsider

theset

(1.2)

{[F

ℓ,f

: F

ℓ

] | f ∈ S

k

(N

n

; F

ℓ

)

normalisedHe ke eigenform

}

inanalogyto Equation1.1 . Here,

S

k

(N

n

; F

ℓ

)

denotesthe

F

ℓ

-ve torspa eof uspidalmodular

forms over

F

ℓ

(see Se tion 2 for denitions) and

F

ℓ,f

is dened by adjoining to

F

ℓ

all He ke

eigenvalues on

f

. It is easy to onstru t sequen es

(N

n

, k

n

)

for whi h theset inquestion is

innite(seee.g. [DiWi℄and[W℄), butitdoesnot seemsimple toobtain allnaturalnumbers

asdegrees. Mostimportantly,itseemsto beunknownwhetherthis setis innitewhen

(Nn)

isthesequen eof prime numbers,

k = 2

and

ℓ > 2

.

Con erningproperties of modularforms inpositive hara teristi , thereis other, mu h more

subtleinformation thanjustthedegrees of

F

ℓ,f

tobestudied,e.g. ongruen esbetween

mod-ularforms. Inorder to take thefull information into a ount, inthis arti le we examine the

properties of the

F

ℓ

-He ke algebras

T

k

(Nn)

on

S

k

(Nn; F

ℓ

)

asymptoti ally for xed weight

k

(mostly

2

) and running level

N

n

(mostly the set ofprime numbers) by means of

experimen-tation. Morepre isely,weinvestigate three quantities:

(a) The deviation of

T

k

(N

n

)

from being semisimple. In Se tion 3 , we in lude a proposition

relating nonsemisimpli ity to ongruen es, rami ation and ertain indi es. Our

exper-iments suggest that for odd primes

ℓ

, the He ke algebra

T

k

(N

n

)

tends to be lose to

(2)

semisimple, whereas the situation seems to be ompletely dierent for

p = 2

(see

Se -tion 4.1).

(b) The average residue degree of

T

k

(Nn)

. That is the arithmeti mean of the degrees of

F

ℓ,f

for all

f

in

S

k

(N

n

; F

ℓ

)

. Our omputations (see Se tion 4.2 ) strongly suggest that this quantity is unbounded. More pre isely, we seem to observe a ertain asymptoti

behaviour,whi h we formulate asaquestion.

( ) The maximum residue degree of

T

k

(N

n

)

. That is the maximum of the degrees of

F

ℓ,f

for all

f

in

S

k

(N

n

; F

ℓ

)

. Our experiments suggest that this quantity is 'asymptoti ally'

proportional to the dimension of

S

k

(N

n

; F

ℓ

)

.

InSe tion4wedes ribeour omputationsandderive ertainquestionsfromourobservations.

However, we do not attempt to propose any heuristi explanations in this arti le. This will

havetobethe subje tofsubsequent studies,buildingonrenedandextended omputations.

We see (b ) and ( ) as strong eviden e for the innity of the set in Equation 1.2, when

N

n

runs through the primes. Generally speaking, there appears to be some regularity in the

otherwise quite errati behaviour of the examined quantities, lending some support to the

hopeof ndinga formulationof an arithmeti analogue ofverti alSato-Tate.

2. Ba kground and notation

We start by introdu ing the ne essary notation and explaining the ba kground. For fa ts

on modular forms, we refer to [DI℄. Let us x an interger

k

and a ongruen e subgroup

Γ ⊆ SL

2

(Z)

. Denoteby

S

k

(Γ)

the omplexve torspa eofholomorphi uspformsofweight

k

for

Γ

. Dene

T := T

k

(Γ)

to be the

Z

-He ke algebra of weight

k

for

Γ

, i.e. the subring of

EndC(S

k

(Γ))

spanned by all He ke operators

Tn

for

n ∈ N

. If

Γ = Γ0(N )

we simply write

T

k

(N )

. Weuse similarnotation in other ontexts, too. It isan important theoremthat

T

is freeof niterankasa

Z

-module, hen ehasKrulldimension one asa ring, andthatthemap

(2.3)

HomZ(T, C) → S

k

(Γ), φ 7→

∞

X

n=1

φ(T

n

)q

n

with

q = q(z) = e

2πiz

denes an isomorphism of

C

-ve tor spa es, whi h is ompatible with

thenatural He ke a tion. For anyring

A

,dene

S

k

(Γ; A) := HomZ(T, A)

equipped withthe

natural He kea tion (i.e.

T

-a tion), sothatwehave

S

k

(Γ) ∼

= Sk

(Γ; C)

. Wethink ofelements

in

S

k

(Γ; A)

intermsofformal

q

-expansions, i.e.asformalpowerseriesin

A[[q]]

,byananalog

of Eq. 2.3 . Note that normalised He ke eigenforms, i.e. those

f =

P∞

n=1

a

n

q

n

∈ S

k

(Γ; A)

that satisfy

a

1

= 1

and

T

n

f = a

n

f

,pre isely orrespond to ring homomorphisms

φ : T → A

with

φ(Tn) = an

. When

A

is an integral domain, a normalised eigenfun tion gives rise to

a prime ideal

p

of

T

, namely the kernel of

φ

. We may think of

T/p

as thesmallest subring

of

A

generated bythe

a

n

for

n ∈ N

: the oe ient ring of

f

in

A

. Note that

Aut(A)

a tson

S

k

(Γ; A)

by omposing

φ : T → A

with

σ ∈ Aut(A)

. Obviously,this a tion doesnot hange

Wexaprimenumber

p

. Weput

T := b

b

T

k

(Γ) := T

k

(Γ) ⊗Z

Z

p

,

T

b

η

:= T

k

(Γ) ⊗Z

Q

p

= b

T ⊗Z

p

Q

p

and

T := T

k

(Γ) := T

k

(Γ)⊗

Z

F

p

= b

T⊗

Z

p

F

p

. Notetheisomorphisms

S

k

(Γ; Z

p

) ∼

= HomZ

p

(b

T, Z

p

)

,

S

k

(Γ; Q

p

) ∼

= HomZ

p

(b

T, Q

p

)

and

S

k

(Γ; Fp) ∼

= HomZ

p

(b

T, Fp) ∼

= HomF

p

(T, Fp)

.

The

Gal(Q

p

/Qp)

- onjuga y lassesofnormalisedeigenformsin

S

k

(Γ; Q

p

)

(bywhi hwe mean

the lasses for the

Gal(Q

p

/Q

p

)

-a tion des ribed above) arein bije tion withtheprime (and automati allymaximal)idealsof

T

b

η

andalsoinbije tionwiththeminimalprimeidealsof

T

b

,

whosesetisdenotedby

MinSpec(b

T)

. These ondbije tionisexpli itlygivenbytaking

preim-agesfortheinje tion

T ֒→ b

b

Tη

. Notethat

T

b

hasKrulldimensionone,meaningthatanyprime

ideal is either minimal (i.e. not ontaining any smaller prime ideal) or maximal. Moreover,

the

Gal(F

p

/F

p

)

- onjuga y lasses of normalisedeigenforms in

S

k

(Γ; F

p

)

areinbije tion with

Spec(T) = MaxSpec(T)

. Furthermore,

Spec(T)

is innatural bije tion with

MaxSpec(b

T)

un-der takingpreimages for the natural proje tion

T ։ T

b

. By a resultin ommutative algebra,

wehave dire tprodu tde ompositions

b

T =

Y

m∈MaxSpec(b

T)

b

Tm, T =

Y

m∈MaxSpec(b

T)

Tm

and

T

b

η

=

Y

p∈Spec(b

T

η

)

b

T

η,p

,

where thefa torsarethe lo alisations at theprime idealsindi atedbythesubs ripts.

Denition 2.1.  We say that two

p

1

, p

2

∈ MinSpec(b

T)

are ongruent if they lie in the

same maximalideal

m

∈ MaxSpec(b

T)

. For

p

∈ MinSpec(b

T)

, we all

T/p

b

thelo al oe ient

ring and

L

p

:= Frac(b

T/p)

the lo al oe ient eld. We say that

p

∈ MinSpec(b

T)

is ramied

if

L

p

isa ramied extension of

Q

p

. We denote by

i

p

the index of

T/p

b

in thering of integers

of

Lp

. The residue eld

T/m ∼

b

= T/m

will be denoted by

Fm

and will be alled the residual

oe ient eld.

Wenowestablishthe onne tionwiththe usualunderstanding ofthetermsinthedenition.

Let

Z ⊂ Q ⊂ C

be the algebrai integers andthe algebrai numbers, respe tively. As

T

isof

nite

Z

-rank, theset of normalisedeigenforms in

S

k

(Γ)

isthe same astheset of normalised

eigenforms in

S

k

(Γ; Z)

. Fixhomorphisms

Z

 

ι

//

π

 =

=

=

=

=

=

=

=

=

Zp

π



F

p

giving rise to

S

k(Γ; Z)

 

ι

_//

π

%% %%K

K

K

K

K

K

K

K

K

K

K

S

k(Γ; Zp

)

π



S

k

(Γ; F

p

)

.

From this perspe tive, a holomorphi normalised He ke eigenform

f =

P∞

n=1

a

n

q

n

∈ S

k

(Γ)

gives rise to an eigenform in

S

k

(Γ; F

p

)

, alled the redu tion of

f

modulo

p

, whose formal

q

-expansionis

π(f ) :=

P∞

n=1

π(an)q

n

∈ Fp[[q]]

. Theredu tion orrespondsto

m

∈ MaxSpec(b

T)

and to

m

∈ Spec(T)

(we use the same symbol due to the natural bije tion between the

two sets). The form

f

also gives rise to an eigenform in

S

k

(Γ; Q

p

)

, whi h orresponds to

p

_f

∈ MinSpec(b

T)

and to

p

f

∈ Spec(b

T

η

)

(the same symbolis used again due to the natural

Let

g =

P∞

n=1

b

n

q

n

beanotherholomorphi normalisedHe keeigenform. If

π(f ) = π(g)

,then learly

p

f

⊂ m

and

p

g

⊂ m

,i.e.

p

f

and

p

g

are ongruent. Conversely,let

p

f

, p

g

∈ MinSpec(b

T)

su h that

p

f

⊂ m

and

p

g

⊂ m

for some

m

∈ MaxSpec(b

T)

, so that

p

f

and

p

g

are ongruent.

The ideals

p

f

and

p

g

orrespond to

Gal(Q

p

/Q

p

)

- onjuga y lasses in

S

k

(Γ; Q

p

)

and we an

hoose

f, g ∈ S

k

(Γ; Z

p

)

orresponding to

p

f

and

p

g

with

π(f ) = π(g)

. We illustrate the

situation by thediagram

b

T/p

f

(( ((R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

R

 _



b

T

88 88q

q

q

q

q

q

q

q

q

q

q

q

q

q

q

&& &&M

M

M

M

M

M

M

M

M

M

M

M

M

M

M

f

,,

g

22

Z

p

π

// //

F

p

oo

? _

T/m = Fm.

b

T/p

g

66 66

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

 ?

OO

Notethat

f, g

an alreadybefoundin

S

k

(Γ; Z) ⊂ S

k

(Γ)

. Thisjustiesour usageof theterm

ongruen e.

Moreover, the lo al oe ient ring

T/p

b

an be identied with

Z

p,f

:= Z

p

[ι(a

n

)|n ∈ N]

and

its fra tion eld

Lp

with

Q

p,f

:= Qp(ι(an)|n ∈ N)

, when e

ip

is the index of

Z

p,f

in its

normalisation. Furthermore, the residual oe ient eld, i.e.

Fm

= T/m

, an be interpreted

as

F

p

[π(a

n

)|n ∈ N]

. The relation to thearithmeti of the oe ient eld

Q

f

:= Q(a

n

|n ∈ N)

and the oe ient ring

Z

f

:= Z[an|n ∈ N]

is apparent.

In order to on lude this ba kground se tion, we point out that in the ase

k = 2

, the

oe ient ring

Z

f

is theendomorphism ring of theabelian variety

A

f

atta hed to

f

. From

thatpointof view,thefollowing analysis analso be interpretedasastudy ofthearithmeti

of the endomorphismalgebrasof

GL

2

-abelian varieties.

3. Semisimpli ityof He ke algebras

We re allthata nitedimensional ommutative

K

-algebra,where

K

is aeld, issemisimple

if and only ifit is isomorphi to a dire t produ tof elds (whi h are ne essarily nite eld

extensionsof

K

).

In this se tion we rst study the semisimpli ity of the He ke algebra

T

b

η

. In the ase when

it is semisimple, we relate the non-semisimpli ity of the mod

p

He ke algebra

T

to three

phonomena: ongruen es between

Gal(Q

p

/Qp)

- onjuga y lasses of newforms, rami ation

at

p

of the oe ient elds of newforms and the

p

-index of the lo al oe ient ring in the

ringof integersof thelo al oe ient eld.

Let

f =

P∞

n=1

a

n

(f )q

n

∈ S

k

(Γ

1

(M ))

new

be a normalised He ke eigenform and let

m

be any

positiveinteger. Wedene the

C

-ve torspa e

V

f

(m)

tobe thespanof

{f (q

d

_{) | d | m}}

d

runs throughall positive divisorsof

m

(in luding

1

and

m

). Newform theorystatesthat

S

k

(Γ

1

(N )) ∼

=

M

m|N

M

f ∈S

k

(Γ

1

(N/m))

new

V

f

(m).

Thisis anisomorphism of He ke modules. TheHe ke operators

Tn

for

(n, m) = 1

restri ted

to

V

f

(m)

are s alar matri es with

a

n

(f )

as diagonal entries. We now des ribe the He ke

operator

T

ℓ

on

V

f

(m)

for a prime

ℓ

. Suppose that

ℓ

r

_{|| m}

. Let

ǫ

be theDiri hlet hara ter

of

f

. Consider the

(r + 1) × (r + 1)

-matrix

A := A

f

(m, ℓ) :=





















a

ℓ

(f )

1

0 0 . . . 0

−δǫ(ℓ)ℓ

k−1

0

1 0 . . . 0

0

0

0 1 . . . 0

. . . . . .

0

. . . 0 0

0

1

0

. . . 0 0

0

0





















,

where

δ = 0

if

ℓ | (N/m)

and

δ = 1

otherwise. TheHe ke operator

T

ℓ

isthengiven on

V

f

(m)

(fora ertainnatural basis)byadiagonal blo kmatrix having onlyblo ks equalto

A

onthe

diagonal, where ea h blo k on thediagonal orresponds to a divisor of

m/ℓ

r

. Let

T

be the

He kealgebraof

S

k

(Γ

1

(N ))

(asinSe tion 2). Thealgebra

TQ

:= T ⊗Z

Q

issemisimpleifand

onlyif

TC

:= T ⊗Z

C

issemisimple(ifandonlyif

b

T

η

issemisimple). Bytheabovedis ussion,

T

C

is semisimpleifand onlyifall thematri es

A

f

(m, ℓ)

thatappeararediagonalisable. Proposition 3.1.  Assume the notationabove,

M = N/m

and

k ≥ 2

. Moreover, if

k ≥ 3

assumeTate's onje ture (see [CE ℄, Se tion 1).

(a) Assume

ℓ ∤ M

. Then

A

f

(m, ℓ)

isdiagonalisable if andonlyif

r ≤ 2

.

(b) Assume that

ℓ | M

and that either

ℓ || M

or that

ǫ

annot be dened mod

M/ℓ

. Then

A

f

(m, ℓ)

is diagonalisableif and onlyif

r ≤ 1

.

( ) Assumethat

ℓ

2

_{| M}

andthat

ǫ

anbedenedmodulo

M/ℓ

. Then

A

f

(m, ℓ)

isdiagonalisable

ifand onlyif

r = 0

.

Proof.  (a)Assume

r ≥ 1

(otherwise theresultistrivial)and all

B

thetopleft

2 × 2

-blo k

of

A = A

f

(m, ℓ)

. The hara teristi polynomial of

B

is

g(X) = X

2

_{− a}

ℓ

(f )X + ǫ(ℓ)ℓ

k−1

.

We have

g(0) 6= 0

and

g(X)

has dis riminant

a

ℓ

(f )

2

_{− 4ǫ(ℓ)ℓ}

k−1

, whi h is non-zero, sin e

|a

ℓ

(f )| = 2|ℓ|

(k−1)/2

would ontradi t [CE℄, Theorem 4.1. Consequently,

A

is diagonalisable

ifand only ifapartfrom

B

there isat most one morerow and olumn.

In ases (b)and( ), notethat

A

isinJordanform. Theresultisnowimmediate, sin e

a

ℓ

(f )

isnon-zero for (b)andzero for ( )(see [DS℄,1.8).

Wehave theimmediate orollary (whi h isTheorem 4.2in[CE℄).

Corollary 3.2.  Let

N

be ubefree and

k ≥ 2

. If

k ≥ 3

assumeTate's onje ture (see[CE℄,

Se tion 1). Then the He ke algebras

T

k

(Γ

1

(N )) ⊗ Q

and

T

b

k

(Γ

1

(N ))

η

as well as

T

k

(N ) ⊗ Q

Theprin ipalresultofthisse tionisthefollowing propositiononthestru tureoftheresidual

He ke algebra. We assume thenotation laid out in Se tion 2, inparti ular, we work with a

general ongruen e subgroup

Γ

.

Proposition 3.3.  Assume that

T

b

η

is semisimple (see, e.g., Corollary 3.2), i.e.

b

T

η

∼

=

Y

p∈MinSpec(b

T)

Lp.

Then the residual He ke algebra

T

issemisimple if and onlyif all of the followingthree

on-ditionsare satised:

(i) No two

p

1, p2

∈ MinSpec(b

T)

are ongruent.

(ii) None of the

p

∈ MinSpec(b

T)

is ramied.

(iii) Forall

p

∈ MinSpec(b

T)

, the index

ip

= 1

.

Proof.  We rst prove that(i),(ii)and (iii)imply the semisimpli ityof

T

.

Thefa t thatthere isno ongruen e means thatinevery

m

∈ MaxSpec(b

T)

there isa unique

p

∈ MinSpec(b

T

m

)

. As

T

b

m

⊗

Z

p

Q

p

∼

= Lp

,itfollows that

T

b

m

isasubring of

L

p

. Dueto (iii),

T

b

m

isthe ringofintegersof

Lp

. Sin eby(ii)

Lp

isunramied, we getthat

Tm

istheresidueeld

of the integersof

Lp

. Thisshows that

T

isa produ tofnite elds,i.e.semisimple.

Nowweprovethe onversedire tion andassumethat

T

issemisimple. Let

m

∈ MaxSpec(b

T)

.

Let

p

1

, p

2

, . . . , p

m

∈ MinSpec(b

T)

bethedistin tminimalprimes ontainedin

m

. Then

T

b

m

⊗

Z

p

Qp

∼

= Lp

1

× · · · × Lp

m

. Dueto thenon-degeneration

Tm

∼

= b

Tm⊗Z

p

Fp

∼

= Fp

n

forsome

n

. Sin e

dimQ

_p

Tm

b

⊗Z

_p

Q

p

= n

,we have

[Lp

i

: Q

p

] ≤ n

for

i = 1, . . . , m

.

Let

O

i

be the ring of integers of

L

p

i

for

i = 1, . . . , m

. It ontains

T

b

m

/p

i

with index

i

p

i

. Tensoring theexa t sequen eof

Z

p

-modules

0 → b

Tm/p

i

→ O

i

→ O

i

/(b

Tm/p

i

) → 0

with

F

p

over

Z

p

we obtaintheexa t sequen eof

F

p

-ve tor spa es:

F

p

n

→ O

_i

⊗Z

_p

F

_p

→ (O

_i

/(b

Tm/p

_i

)) ⊗Z

_p

F

_p

→ 0.

Sin e the mapon theleft isa ring homomorphism,itis inje tive. Now

dimF

p

O

i

⊗Z

p

F

p

≤ n

implies that

O

i

is unramied and that

i

p

i

= 1

. Thus

[L

p

i

: Q

p

] = n

for

i = 1, . . . , m

and,

hen e,

m = 1

, on ludingtheproof.

4. Observations and Questions

In this se tion, we explain and expose our omputer experiments and we ask some

ques-tions suggestedbyour studies. All omputer al ulationswereperformedusingMagma (see

4.1. Semisimpli ityof the residualHe ke algebra.  Alo alnite-dimensional

om-mutative

F

p

-algebra

A

issemisimpleifand onlyifitissimple,whi hisequivalentto

A

being

eld. Wetakethedimensionofthe maximalideal

m

of

A

asameasureforthedeviationof

A

frombeing semisimple. Inparti ular,

A

isa eldifand onlyif

m

hasdimension

0

.

Forgiven prime

p

,level

N

and weight

k

we study thesumof the residue degrees ofall prime

ideals:

a

(p)

_N,k

=

X

m∈Spec(T

k

(N ))

[Fm

: F

p

].

Clearly,

a

(p)

N,k

islessthanorequaltothe

F

p

-dimensionof

S

k

(N ; F

p

)

. Hen e,

T

k

(N )

is semisim-pleifandonly if

a

(p)

N,k

isequal to this dimension.

Weintendtostudytheasymptoti behaviourofthefun tion

a

(p)

N,k

foraxedprime

p

andxed weight

k

asafun tionofthelevel

N

. Forsimpli ity,welet

N

runthroughtheprimenumbers

only in order to avoid ontributions from lower levels via the degenera y maps. We should

point outthatthere anbe ontributions fromlowerweights: aneigenform in

S

k

(N ; F

p

)

also

livesin

S

k+n(p−1)

(N ; F

p

)

forall

n ≥ 0

bymultipli ation bytheHasseinvariant. Notethatfor

p > 2

and

k = 2

,aswell asfor

p > 3

and

k = 4

there isno su h ontribution.

Our omputational ndings are best illustrated by plotting graphs. In ea h of the

follow-ing plots, the prime

p

and the weight

k

are xed and on the

x

-axis we plot

d(N ) :=

dim

_F

p

S

k

(N ; F

p

)

and on the

y

-axis the fun tion

a

(p)

N,k

as a fun tion of

N

, i.e. ea h

N

gives riseto a dot at theappropriate pla e. Thestraight line inthegraphs wasdetermined asthe

linearfun tion

α · d(N )

whi h bestts thedata (a ordingto gnuplot and theleastsquares

method).

In the weights that we onsidered we observed a behaviour for

p = 2

whi h seems to be

ompletelydierentfromthebehaviouratallotherprimes. Wemadeplotsforalloddprimes

lessthan

100

and weight

2

and present a sele tion here. The graphs that we leave out look

verysimilar. Figure1

0

20

40

60

80

100

120

140

160

0

20

40

60

80

100

120

140

160

Sum of Residue Degrees

Dimension k = 2, p = 3

x * 0.990097

Figure2

0

20

40

60

80

100

120

140

160

0

20

40

60

80

100

120

140

160

Sum of Residue Degrees

Dimension k = 2, p = 5

Figure3

0

20

40

60

80

100

120

140

160

0

20

40

60

80

100

120

140

160

Sum of Residue Degrees

Dimension k = 2, p = 7

x * 0.995265

Figure4

0

20

40

60

80

100

0

20

40

60

80

100

Sum of Residue Degrees

Dimension k = 2, p = 19

x * 0.997344

Figure5

0

20

40

60

80

100

120

140

160

0

20

40

60

80

100

120

140

160

Sum of Residue Degrees

Dimension k = 2, p = 31

x * 0.999248

Figure6

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

Sum of Residue Degrees

Dimension k = 2, p = 41

x * 0.998996

Figure7

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

Sum of Residue Degrees

Dimension k = 2, p = 53

x * 0.999397

Figure8

0

10

20

30

40

50

60

0

10

20

30

40

50

60

Sum of Residue Degrees

Dimension k = 2, p = 73

Figure9

0

10

20

30

40

50

60

0

10

20

30

40

50

60

Sum of Residue Degrees

Dimension k = 2, p = 83

x * 0.999396

Figure10

0

20

40

60

80

100

120

140

0

20

40

60

80

100

120

140

Sum of Residue Degrees

Dimension k = 2, p = 97

x * 0.999889

InFigures110 thelevelsrangeoverallprimesup toa ertainbound(whi h isnot thesame

for all

p

). We observe that non-semisimpli ity seemsto be a rather rare phenomenon whi h

be omes rarer for growing

p

,asone might have guessed. In thenext gures, we analyse the

ases

p = 3, 5

still for weight

2

more losely by letting the levels range through all primes

between

3000

and

10009

subdivided into four onse utive intervals.

Figure11

240

260

280

300

320

340

360

380

240

260

280

300

320

340

360

380

Sum of Residue Degrees

Dimension k = 2, p = 3

x * 0.995086

Figure12

380

400

420

440

460

480

500

520

540

380

400

420

440

460

480

500

520

540

Sum of Residue Degrees

Dimension k = 2, p = 3

x * 0.996909

Figure13

540

560

580

600

620

640

660

680

540

560

580

600

620

640

660

680

Sum of Residue Degrees

Dimension k = 2, p = 3

x * 0.997720

Figure14

680

700

720

740

760

780

800

820

840

680

700

720

740

760

780

800

820

840

Sum of Residue Degrees

Dimension k = 2, p = 3

Figure15

240

260

280

300

320

340

360

380

240

260

280

300

320

340

360

380

Sum of Residue Degrees

Dimension k = 2, p = 5

x * 0.997053

Figure16

380

400

420

440

460

480

500

520

540

380

400

420

440

460

480

500

520

540

Sum of Residue Degrees

Dimension k = 2, p = 5

x * 0.997917

Figure17

520

540

560

580

600

620

640

660

680

520

540

560

580

600

620

640

660

680

Sum of Residue Degrees

Dimension k = 2, p = 5

x * 0.998676

Figure18

680

700

720

740

760

780

800

820

840

680

700

720

740

760

780

800

820

840

Sum of Residue Degrees

Dimension k = 2, p = 5

x * 0.998808

One anobservethattheslopeofthebestttinglinethroughtheoriginseemstobein reasing

with growing dimension. Although we only omputed relatively litte data, we in lude two

examplesfor weight

4

. Theydo not suggestanysigni ant dieren e to theweight

2

ase.

Figure19

0

50

100

150

200

0

50

100

150

200

Sum of Residue Degrees

Dimension k = 4, p = 5

x * 0.993821

Figure20

0

50

100

150

200

0

50

100

150

200

Sum of Residue Degrees

Dimension k = 4, p = 7

x * 0.994612

Question 4.1.  Fix an odd prime

p

and an even

k ≥ 2

. Let

a(N ) := a

(p)

k,N

and

d(N ) :=

dim

_F

_p

S

k

(N ; Fp

)

. Does the following statement hold?

For all

ǫ > 0

there is

C

ǫ

> 0

su h that for all primes

N

the inequality

a(N ) > (1 − ǫ)d(N ) − C

ǫ

holds.

We ontrast the situation, whi h seems very similar for every odd prime, with the one for

p = 2

and

k = 2

. We do not onsider any higher weights due to the ontributions from

weight

2

,whi hwould`disturb'thesituation. Thefollowingplotstakeprimenumbers

N

into

a ount thatlie insixdierent intervals upto

12000

.

Figure21

0

20

40

60

80

100

0

20

40

60

80

100

120

140

160

Sum of Residue Degrees

Dimension k = 2, p = 2

x * 0.521382

Figure22

200

220

240

260

280

420

430

440

450

460

470

480

490

500

510

Sum of Residue Degrees

Dimension k = 2, p = 2

x * 0.510937

Figure23

240

260

280

300

320

520

540

560

580

600

Sum of Residue Degrees

Dimension k = 2, p = 2

x * 0.509345

Figure24

280

300

320

340

360

380

400

620

640

660

680

700

720

Sum of Residue Degrees

Dimension k = 2, p = 2

Figure25

320

340

360

380

400

420

440

460

480

720

740

760

780

800

820

Sum of Residue Degrees

Dimension k = 2, p = 2

x * 0.508675

Figure26

440

460

480

500

520

540

560

930

940

950

960

970

980

990

1000

Sum of Residue Degrees

Dimension k = 2, p = 2

x * 0.508181

Inspite oftheveryirregular behaviour, itisremarkablethattheslope ofthebesttting line

throughthe origin isalways justa little biggerthan

1

2

.

At the moment we annot fullyexplain this behaviour. Contributions from weight one play

somerole. However,probablymoreimportant are ongruen es offormshavingAtkin-Lehner

eigenvalue

+1

with forms having eigenvalue

−1

. As Johan Bosman observes in a remark

in[B℄,itfollows fromthe onne tednessof thespe trumoftheHe ke algebra([M ℄,10.6) for

k = 2

andprimelevels thatthere isalwaysat leastone su h ongruen e for

p = 2

,whenever

the

+1

- andthe

−1

-eigenspa e are nonempty.

We areledto ask thefollowing question.

Question 4.2.  Fixanevenweight

k ≥ 2

. Let

a(N ) := a

(2)

k,N

and

d(N ) := dim

F

2

S

k

(N ; F

2

)

.

Are there

1 > α ≥ β > 0

and onstants

C, D > 0

su h that the inequality

α · d(N ) + C > a(N ) > β · d(N ) − D

holds?

4.2. Average Residue Degree.  We now study the average residue degree, whi h we

dene for givenlevel

N

,weight

k

and prime

p

as

b

(p)

_N,k

=

1

# Spec(T

k

(N ))

X

m∈Spec(T

k

(N ))

[Fm

: F

p

] =

a

(p)

_N,k

# Spec(T

k

(N ))

.

We made omputations for weight

2

and all primes

p

lessthan

100

, where

N

runs through

the same ranges as previously. We again plot the dimension

d(N )

on the

x

-axis and the

fun tion

b

(p)

N,k

on the

y

-axis and the straight line is again the best tting fun tion

α · d(N )

, although we believe thatthis isnot the right fun tion totake (seebelow). Weagain present

a sele tion of prime numbers as before, however, in luding

p = 2

from the beginning. The

Figure27

0

5

10

15

20

25

30

35

0

20

40

60

80

100

120

140

160

Average Residue Degree

Dimension k = 2, p = 2

x * 0.098129

Figure28

0

10

20

30

40

50

0

20

40

60

80

100

120

140

160

Average Residue Degree

Dimension k = 2, p = 3

x * 0.129624

Figure29

0

5

10

15

20

25

30

35

40

45

0

20

40

60

80

100

120

140

160

Average Residue Degree

Dimension k = 2, p = 5

x * 0.114418

Figure30

0

5

10

15

20

25

30

35

40

0

20

40

60

80

100

120

140

160

Average Residue Degree

Dimension k = 2, p = 7

x * 0.113730

Figure31

0

5

10

15

20

0

20

40

60

80

100

Average Residue Degree

Dimension k = 2, p = 19

x * 0.128238

Figure32

0

5

10

15

20

25

30

0

20

40

60

80

100

120

140

160

Average Residue Degree

Dimension k = 2, p = 31

Figure33

0

2

4

6

8

10

12

14

16

18

0

10

20

30

40

50

60

70

80

Average Residue Degree

Dimension k = 2, p = 53

x * 0.127727

Figure34

0

2

4

6

8

10

12

14

0

10

20

30

40

50

60

Average Residue Degree

Dimension k = 2, p = 73

x * 0.134275

Figure35

0

5

10

15

20

0

10

20

30

40

50

60

Average Residue Degree

Dimension k = 2, p = 83

x * 0.141422

Figure36

0

5

10

15

20

25

30

35

40

0

20

40

60

80

100

120

140

Average Residue Degree

Dimension k = 2, p = 97

x * 0.119130

Hereare againtwo examplesfor weight

4

.

Figure 37

0

5

10

15

20

25

0

50

100

150

200

Average Residue Degree

Dimension k = 4, p = 5

x * 0.110425

Figure38

5

10

15

20

25

30

35

0

20

40

60

80

100

120

140

160

180

Average Residue Degree

Dimension k = 4, p = 11

x * 0.106811

Very roughly speaking the data suggest that the average residue degree grows with the

di-mension, asis ertainlyto be expe ted. Wealso ondu teda loseranalysisfortheprimes

2

,

3

and

5

. For

p = 2

we used all primes in dierent intervals up to

12000

and obtained these

Figure39

15

20

25

30

35

40

45

50

55

60

420

430

440

450

460

470

480

490

500

510

Average Residue Degree

Dimension k = 2, p = 2

x * 0.062388

Figure40

20

30

40

50

60

520

540

560

580

600

Average Residue Degree

Dimension k = 2, p = 2

x * 0.057421

Figure41

20

40

60

80

100

120

620

640

660

680

700

720

Average Residue Degree

Dimension k = 2, p = 2

x * 0.057158

Figure42

20

40

60

80

100

120

140

160

180

200

720

740

760

780

800

820

Average Residue Degree

Dimension k = 2, p = 2

x * 0.055027

Figure43

20

40

60

80

100

120

140

160

180

930

940

950

960

970

980

990

1000

Average Residue Degree

Dimension k = 2, p = 2

x * 0.055893

Hereare theplots for

p = 3, 5

and the primes between

3000

and

10009

subdivided into four

Figure44

20

40

60

80

100

120

240

260

280

300

320

340

360

380

Average Residue Degree

Dimension k = 2, p = 3

x * 0.096554

Figure45

20

40

60

80

100

120

140

160

180

380

400

420

440

460

480

500

520

540

Average Residue Degree

Dimension k = 2, p = 3

x * 0.089212

Figure46

30

40

50

60

70

80

540

560

580

600

620

640

660

680

Average Residue Degree

Dimension k = 2, p = 3

x * 0.081910

Figure47

20

40

60

80

100

120

140

680

700

720

740

760

780

800

820

840

Average Residue Degree

Dimension k = 2, p = 3

x * 0.078536

Figure48

20

30

40

50

60

70

80

240

260

280

300

320

340

360

380

Average Residue Degree

Dimension k = 2, p = 5

x * 0.090116

Figure49

30

40

50

60

70

380

400

420

440

460

480

500

520

540

Average Residue Degree

Dimension k = 2, p = 5

Figure50

30

40

50

60

70

80

90

520

540

560

580

600

620

640

660

680

Average Residue Degree

Dimension k = 2, p = 5

x * 0.080942

Figure51

40

50

60

70

80

90

680

700

720

740

760

780

800

820

840

Average Residue Degree

Dimension k = 2, p = 5

x * 0.077091

We observe that the slope of the best tting line goes slightly down with the dimension.

Thisstrongly suggeststhat takingastraight line doesnot seem to bequite orre t. Wealso

madelogarithmi plots, whi h we do not reprodu e here; they seemed to suggest to us that

abehaviour oftheform

b

(p)

N,k

∼ const · d(N )

α

isnot quite orre teither (thebest hoi eof

α

seemsto be lose to

1

ina ordan e withthe previous dis ussion).

These omputationssuggestthe following question.

Question 4.3.  Fix a prime

p

and an even weight

k ≥ 2

. Let

b(N ) := b

(p)

k,N

and

d(N ) :=

dim

_F

_p

S

k

(N ; F

p

)

. Do there exist onstants

C

1

, C

2

and

0 < α ≤ β < 1

su h that the inequality

C

1

+ α

d(N )

log(d(N ))

≤ b(N ) ≤ C

2

+ β · d(N )

holds? Weremarkthatif

a

(p)

N,k

behaveslike

d(N )

,assuggested byQuestion4.1, thenQuestion 4.3is equivalenttoaskingthat

# Spec(T

k

(N ))

doesnotgrowfasterthana onstanttimes

log(d(N ))

.

The phenomenon that for odd primes

p

most of the dots in the diagrams seem to lie on or

very lose to ertaindistinguished linesthroughthe originis naturalinviewof Question4.1:

theslope oftheline on or loseto whi h adot lies isjust

1

# Spec(T

k

(N ))

.

4.3. MaximumResidue Degree.  Nowweturnour attention tothemaximum residue

degree, whi h wedene for given level

N

,weight

k

andprime

p

as

c

(p)

_N,k

= max{[Fm

: F

p

] | m ∈ Spec(T

k

(N ))}.

We made omputations for weight

2

and all primes

p

less than

100

, where

N

runs through

the same ranges as previously. We again plot the dimension

d(N )

on the

x

-axis and the

fun tion

c

(p)

N,k

on the

y

-axis and the straight line is again the best tting fun tion

α · d(N )

. Thistimewe believe thatthis might be the right fun tion to take. Here isagain a sele tion

Figure52

0

10

20

30

40

50

60

70

0

20

40

60

80

100

120

140

160

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.277970

Figure53

0

10

20

30

40

50

60

70

80

0

20

40

60

80

100

120

140

160

Maximum Residue Degree

Dimension k = 2, p = 3

x * 0.390147

Figure54

0

10

20

30

40

50

60

70

80

90

0

20

40

60

80

100

120

140

160

Maximum Residue Degree

Dimension k = 2, p = 5

x * 0.377419

Figure55

0

10

20

30

40

50

60

70

80

90

0

20

40

60

80

100

120

140

160

Maximum Residue Degree

Dimension k = 2, p = 7

x * 0.387289

Figure56

0

10

20

30

40

50

0

20

40

60

80

100

Maximum Residue Degree

Dimension k = 2, p = 19

x * 0.391898

Figure57

0

10

20

30

40

50

60

70

80

0

20

40

60

80

100

120

140

160

Maximum Residue Degree

Dimension k = 2, p = 31

Figure58

0

10

20

30

40

50

0

10

20

30

40

50

60

70

80

Maximum Residue Degree

Dimension k = 2, p = 53

x * 0.384707

Figure59

0

5

10

15

20

25

30

35

0

10

20

30

40

50

60

Maximum Residue Degree

Dimension k = 2, p = 73

x * 0.416753

Figure60

0

5

10

15

20

25

30

35

0

10

20

30

40

50

60

Maximum Residue Degree

Dimension k = 2, p = 83

x * 0.405110

Figure61

0

10

20

30

40

50

60

70

80

0

20

40

60

80

100

120

140

Maximum Residue Degree

Dimension k = 2, p = 97

x * 0.393488

Hereareagaintwo examplesfor weight

4

.

Figure62

0

10

20

30

40

50

60

70

80

90

0

50

100

150

200

Maximum Residue Degree

Dimension k = 4, p = 5

x * 0.362600

Figure63

10

20

30

40

50

60

70

80

0

20

40

60

80

100

120

140

160

180

Maximum Residue Degree

Dimension k = 4, p = 11

x * 0.377393

Thedata ertainly suggestthatthemaximumresiduedegreegrowswiththedimension. It is

remarkabletoseethattheslopesofthebestttinglinesallseemtobevery losetoea hother

withthesingleex eptionofthe ase

p = 2

,whi hmightbe ausedbythesamephenomenon

as earlier. Also in this ase we ondu ted a loser analysis for the primes

2

,

3

and

5

. For

Figure64

60

80

100

120

140

160

180

200

220

240

420

430

440

450

460

470

480

490

500

510

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.282934

Figure65

100

150

200

250

520

540

560

580

600

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.266686

Figure66

100

150

200

250

300

620

640

660

680

700

720

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.289400

Figure67

100

150

200

250

300

350

720

740

760

780

800

820

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.274635

Figure68

150

200

250

300

350

400

450

930

940

950

960

970

980

990

1000

Maximum Residue Degree

Dimension k = 2, p = 2

x * 0.276529

Here arethe plots for

p = 3, 5

and the primes between

3000

and

10009

subdivided into four

Figure69

40

60

80

100

120

140

160

180

200

240

260

280

300

320

340

360

380

Maximum Residue Degree

Dimension k = 2, p = 3

x * 0.378322

Figure70

100

150

200

250

300

380

400

420

440

460

480

500

520

540

Maximum Residue Degree

Dimension k = 2, p = 3

x * 0.374745

Figure71

100

150

200

250

300

350

540

560

580

600

620

640

660

680

Maximum Residue Degree

Dimension k = 2, p = 3

x * 0.366550

Figure72

150

200

250

300

350

400

680

700

720

740

760

780

800

820

840

Maximum Residue Degree

Dimension k = 2, p = 3

x * 0.374045

Figure73

60

80

100

120

140

160

180

200

220

240

260

280

300

320

340

360

380

Maximum Residue Degree

Dimension k = 2, p = 5

x * 0.372593

Figure74

100

150

200

250

300

380

400

420

440

460

480

500

520

540

Maximum Residue Degree

Dimension k = 2, p = 5

Figure75

100

150

200

250

300

350

520

540

560

580

600

620

640

660

680

Maximum Residue Degree

Dimension k = 2, p = 5

x * 0.370869

Figure76

150

200

250

300

350

400

680

700

720

740

760

780

800

820

840

Maximum Residue Degree

Dimension k = 2, p = 5

x * 0.366508

We observethat,although thebestttinglineswere omputedusingdierentintervals,their

slopesarevery loseto ea h other. These omputationssuggestthefollowing question.

Question 4.4.  Fix a prime

p

and an even weight

k ≥ 2

. Let

c(N ) := c

(p)

k,N

and

d(N ) :=

dim

_F

_p

S

k

(N ; F

p

)

. Dothere exist onstants

C

1

, C

2

and

0 < α ≤ β < 1

su h that the inequality

C

1

+ α · d(N ) ≤ c(N ) ≤ C

2

+ β · d(N )

holds?

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[S℄ J.-P.Serre.Répartitionasymptotiquedesvaleurspropres del'opérateurde He ke

T

[T℄ R.Taylor.Automorphy forsomel-adi liftsofautomorphi modlrepresentations.IIPub.Math.

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23o tobre 2010

Mar el Mohyla, Universität Duisburg-Essen,Institutfür ExperimentelleMathematik,Ellernstraÿe 29,

45326Essen,Germany

GaborWiese,UniversitätDuisburg-Essen,InstitutfürExperimentelleMathematik,Ellernstraÿe 29,45326