Computing coefficients of modular forms

(1)

by

Peter Bruin

Abstrat. Weprovethat oeients of

q

-expansionsof modularforms anbeomputed in polynomial timeunder ertain assumptions, the most important of whih is the Riemann

hypothesisfor

ζ

^-funtions^of^number^elds. ^We^giveappliationstoomputingHekeoperators, ountingpointsonmodularurvesoverniteelds,and omputingthenumberofrepresenta-

tionsofanintegerasasumofagivennumberofsquares.

Résumé (Sur lealul desoeients desformes modulaires). Ondémontreque

les oeients des

q

-développements des formes modulaires peuvent être alulés en temps polynomialsousertainesonditions,dontlaplusimportanteestl'hypothèsedeRiemannpour

lesfontions

ζ

^des^orps^de^nombres.^On^donne^desappliationsauxproblèmessuivants:aluler desopérateursdeHeke;ompterlenombredepointsd'uneourbemodulairesurunorpsni;

alulerlenombredereprésentationsd'unentierommesommed'unnombredonnédearrés.

1. Introdution

Let

n

^and

k

^be ^positive ⁱⁿ^tegers, ^and ^let

M _k (Γ ₁ (n))

^be ^the^omplex ^vetor ^spae^of ^modular

formsof weight

k

^for ^the^group

Γ ₁ (n)

^. ^A^modular^form

f ∈ M _k (Γ ₁ (n))

^is^determined^by

n

^,

k

and its

q

^-expansion ^oeients

a m (f)

^for

0 ≤ m ≤ k · d(Γ 1 (n))

^,^where

d(Γ 1 (n))

^is^a ^funtion

growing roughlyquadratially in

n

^.

Anaturalquestiontoaskiswhether,given

a _m (f )

^for

0 ≤ m ≤ k · d(Γ ₁ (n))

^,ôneânêiently

ompute

a _m (f )

^for ^large

m

^. ^In ^the ^ase

n = 1

^, Couveignes, Edixhoven et al. [2℄ desribed a deterministi algorithm that aomplishes this in time polynomial in

log m

^for ^xed

k

^.

Under the generalised Riemann hypothesis, their algorithm runs in time polynomial in

k

and

log m

^. ^Earlieralgorithms, basedonmodularsymbols,requiretimepolynomialin

m

^. ^The

2000 MathematisSubjetClassiation. 11E25,11F11,11F30,11F80,11Y16.

Key wordsand phrases. Algorithms,Hekealgebras,modularforms,sumsofsquares.

The results of this artile are based on those of my thesis [1 ℄. I am muh indebted to my advisors Bas

Edixhovenand RobindeJong for theirsupport. I wouldalso like tothanktheorganisers ofthe onferene

Théorie des nombres et appliations for theopportunity to speak about thissubjet, whihhas ledto this

artile.

TheresearhforthisartilewassupportedbytheNetherlandsOrganisationforSientiResearh.

(2)

method of [2℄ is, very briey, to ompute two-dimensional Galois representations assoiated

toeigenforms of level

1

^over^nite ^elds.

Inthisartile,resultsfromtheauthor'sthesis[1℄onomputingGaloisrepresentationsassoi-

atedto eigenforms ofhigherlevels areusedto generalisetheresultofCouveignes,Edixhoven

et al., be it that we an urrently only give a probabilisti algorithm. The preise result

from[1℄ thatweneed isTheorem3.1below. We willusethisto proveour mainresult,whih

readsasfollows.

Theorem 1.1. Let

n ₀

^be â ^positive înteger. ^There êxists â probabilisti algorithm that, given

a positive integer

k

^,

a squarefree positive integer

n ₁

^oprime ^to

n ₀

^,

a number eld

K

^,

a modular form

f

^of ^weight

k

^for

Γ ₁ (n)

^over

K

^,^where

n = n ₀ n ₁

^, ^and

a positive integer

m

ⁱⁿ ^fatored ^form,

omputes

a _m (f )

^, ând ^whose êxpeted ^running ^time îs ^bounded ^by â ^polynomial ⁱⁿ ^the ^length

of the inputunder the Riemann hypothesis for

ζ

^-funtions^of ^number ^elds.

Letusmakepreisehowthenumbereld

K

^and^the ^form

f

^should^be^given^to^the^algorithm

and how it returns

a _m (f)

^. ^W^e ^represent

K

^by ^its multipliation table with respet to some

Q

^-basis

(b ₁ , . . . , b _r )

^of

K

^. ^By ^this ^we ^mean ^the ^rational ^numbers

c _i,j,k

^with

1 ≤ i, j, k ≤ r

suhthat

b _i b _j = X r k=1

c _i,j,k b _k .

We represent elements of

K

^as

Q

^-linear ombinations of

(b ₁ , . . . , b _r )

^. ^W^e ^represent

f

^by ^its

oeients

a 0 (f )

^,^.^.^.^,

a _k _· _d(Γ ₁ _(n)) (f )

^;^these,âs^wellâs^the ôutput

a m (f )

^,âreêlementsôf

K

^.

Weshouldalsomakepreisewhattheword`probabilisti'inTheorem1.1means. Theorret

interpretationisthattheresultisguaranteedtobeorret,butthattherunningtimedepends

on random hoies made during exeution. Probabilisti algorithms with this property are

ommonlyalled Las Vegas algorithms. These areto be ontrasted with Monte Carlo algo-

rithms,where therandomnessinuenesthe orretnessoftheoutputinsteadoftherunning

time. It is worth emphasising that the expeted running time is dened by averaging only

over the randomhoies made duringexeution, not over thepossible inputs. For any input

x

^, ^the âtual ^running ^time ôf ^the âlgorithm ^given ^this înput ân ^be ^modelled âs â ^random

variable

T _x

^. ^The ^laim ^that ^the êxpeted ^running ^time îs ^polynomial ⁱⁿ ^the ^length ôf ^the

inputmeansthatthereexistsapolynomial

P

^suh^that^for^any^input

x

^,^the^expetation^of

T _x

isat most

P (

^length^of

x)

^. ^W^e ^refer ^to ^Lenstra ^and^Pomerane ^[10,

§

^12℄ ^for ^an enlightening disussionofprobabilisti algorithms.

Remark 1.2. The length of theinput depends not only on

k

^,

n ₀

^,

n ₁

^,

log m

^and

K

^,^but

alsoon theomplexityofthe given oeientsof the modularform

f

^. ^Fôr êxample,îf

f

^is^a

primitiveform

f 0

^multiplied^by^an^integer

A

^,^then^for^xed

f 0

^and

A

^tending^to

∞

^,^the^length

(3)

of the input inreases approximately by a multiple of

log A

^,^and ^the ^running ^time ^inreases

approximately byapolynomialin

log A

^.

Remark 1.3. Without the generalised Riemann hypothesis, we are only able to prove

thattherunningtimeofouralgorithm ispolynomialin

exp(n ₁ )

^,

exp(k)

^and^the^length^of^the

input. Inother words, we arestill ableto prove unonditionally that ifnot only

n ₀

^,^but ^also

n ₁

^and

k

âre ^xed,^then^theêxpeted^running ^timeîs^polynomialⁱⁿ^the^length ôf ^theînput.

Remark 1.4. Omitting the onditionthat

m

^be ^given ⁱⁿ ^fatored ^form ^would ^be ^equiv-

alent to laiming that integers that are produts of two prime numbers an be fatored in

polynomialtime. Namely,suppose thatthe theoremholds without this ondition. Applying

thehypothetial strongerversionof the theoremwith

k

â^xedêven înteger^greater^than

2

^,

n 0 = n 1 = 1

^,

K = Q

^,

f = E _k

^,^the ^lassial ^Eisenstein^series

E _k

^of^weight

k

^for

Γ ₁ (1) = SL ₂ (Z)

^,^and

m = pq

^,^where

p

^and

q

^are^two ^distint ^prime ^numbers,

we onlude that there existsa probabilisti algorithm that omputes

a _m (E _k )

ⁱⁿ^time ^poly-

nomialin

log m

^. ^F^rom ^the ^formula

a _m (E _k ) = X

d | m

d ^k−1

= 1 + p ^k ⁻ ¹ + q ^k ⁻ ¹ + m ^k ⁻ ¹ ,

it follows that

{ p ^k ⁻ ¹ , q ^k ⁻ ¹ }

ân ^be ômputed ^quikly âs ^the ^set ôf ^roots ôf ^the ^polynomial

x ² − (a _m (E _k ) − m ^k−1 − 1)x + m ^k−1 ∈ Z[x]

^. ^Hene^we ^would^be^able^to^ompute

{ p, q }

^from

m

intimepolynomialin

log m

^,^whih îsâ ^laim^we êrtainly^do ^not ^wish^to ^make.

Remark 1.5. The reason why our algorithm is probabilisti is that this is the urrent

state of aairs for the algorithm to whih Theorem 3.1 refers. This algorithm an perhaps

be turned into a deterministi one by replaing the arithmeti over nite elds that is used

in[1℄byapproximatearithmeti overtheomplex numbers. Thelatterapproah istaken by

Couveignes,Edixhovenetal.[2,Chapter12℄formodularformsoflevel1. Thereareurrently

still some diulties with this approah for modular forms of higher level. We refer to [1,

Introdution℄for adisussion ofthese.

Remark 1.6. It wouldbemore satisfatoryifwe ouldprove thetheoremwiththelevel

ranging over all positive integers

n

^. ^Wê ûrrently ânnot ^do ^this ^for ^the ^following ^reason.

The modular urve

X 1 (n)

^has ^a ^regular ^and semi-stable model over thering of integers

Z L

of asuitable numbereld

L

^,^but ⁱⁿ^general ^we ^do ^not ^knowâ ^good ^bound ôn ^the^numberôf

irreduibleomponentsofthegeometribresofsuhamodelatprimesof

Z _L

^that^divide

n

^.

Ifweouldprovethetheoreminthismoregeneralform, thentherestritiontomodularforms

forongruenesubgroupsoftheform

Γ ₁ (n)

ôuldâlso^be^removed. ^The^reason^for^thisîs^that

(4)

thespaeofmodular formsofweight

k

^for^the^prinipal^ongruene ^subgroup

Γ(n) ⊆ SL ₂ (Z)

anbe embedded into

M _k (Γ ₁ (n ² ))

^by^a^map^that ^on

q

-expansions isgivenby

q 7→ q ⁿ

^.

Wenowturnto someappliationsof Theorem1.1. Wewill prove thatthereexistprobabilis-

ti algorithms that solve the following problems in expeted polynomial time in the input,

assumingtheRiemann hypothesisfor

ζ

^-funtions^of ^number^elds:

Given a positive integer

k

^, ^a ^squarefree ^positive ^integer

n

ând â ^positive înteger

m

ⁱⁿ

fatored form, ompute the matrix of the Heke operator

T m

ⁱⁿ

T(M _k (Γ 1 (n)))

^with

respetto a xed

Z

^-basis^of

T(M _k (Γ ₁ (n)))

^.

Given a squarefree positive integer

n

^and ^a ^prime ^number

p ∤ n

^, ^ompute ^the ^zeta

funtion of the modularurve

X ₁ (n)

^over

F _p

^.

Givenanevenpositive integer

k

ândâ^positiveînteger

m

ⁱⁿ^fatored ^form,^ompute ^the

number ofways inwhih

m

ân ^be ^written âsâ ^sumôf

k

^squares^of ^integers.

Atually, we do not prove our results in exatly the same order as presented above. We

rst prove Theorem 1.1 in the speial ase where

f

îs ân Êisenstein ^series ôr â ^primitive

usp form. This sues to solve (a slightly more general version of) the above problem of

omputingHekeoperators. Wethenprove Theorem1.1ingeneral. Finally,weshowhowto

solvetheproblems ofomputing zetafuntionsofmodularurves andndingthenumberof

representationsof aninteger asa sumof squares.

To onlude this introdution, we remark that in order to keep this artile at a reasonable

length, we have omitted, or only briey touhed upon, muh material that an be found in

[1℄and [2℄. Thismeans thatthe ontents of this artile arelargely disjoint from thoseof [1℄

and[2℄.

2. Bakground

We begin by olletingthe neessary preliminaries and introduing our notation. For deni-

tionsand more bakground, we referto the manytextson modular forms, suh asDiamond

andIm [5℄ or Diamondand Shurman [6℄.

2.1. Modular forms. Let

n

^and

k

^be ^positive ^integers. ^Let

M _k (Γ ₁ (n))

^denote ^the

C

^-vetor^spae ^of^modular ^forms ^of^weight

k

^for ^the ^group

Γ ₁ (n) = a

c b d

∈ SL ₂ (Z)

a ≡ d ≡ 1 (mod n), c ≡ 0 (mod n)

.

For every

f ∈ M _k (Γ ₁ (n))

^and ^every

m ≥ 0

^, ^we ^write

a _m (f)

^for ^the ^oeient ^of

q ^m

ⁱⁿ ^the

q

^-expansion ^of

f

^, ^so ^the

q

^-expansion ^of

f

^is ^the ^power ^series

P _∞

m=0 a m (f)q ^m

ⁱⁿ

K [[q]]

^. ^F^or

everydivisor

d

^of

n

^and ^every ^divisor

e

^of

n/d

^,^thereêxistsân înjetive

C

^-linear ^map

b ^d,n _e : M _k (Γ ₁ (d)) ֌ M _k (Γ ₁ (n))

that,on

q

-expansions, hasthe eet ofsending

q

^to

q ^e

^.

(5)

We dene

(1)

d(Γ ₁ (n)) = 1

12 [SL ₂ (Z) : {± 1 } Γ ₁ (n)].

This

d(Γ ₁ (n))

^grows ^roughlyquadratially in

n

^. Â ^basi ^fat ^that ^we ^will ^need ôftenîs ^the

following.

Lemma 2.1. Any

f ∈ M _k (Γ ₁ (n))

^is ^determined ^by

n

^,

k

^and ^the ^oeients

a _m (f)

^for

0 ≤ m ≤ k · d(Γ ₁ (n))

^.

Proof. If

n ≥ 5

^, ^then

d(Γ 1 (n))

^is ^the ^degree ^of ^the ^line ^bundle

ω

^of ^modular ^forms ^of

weight

1

^on ^the ^modular ^urve

X ₁ (n)

^. În ^that âse, ^we ân ^view ^modular ^forms âs ^global

setions of

ω ^⊗k

^. ^If

f, g ∈ M _k (Γ ₁ (n))

^are^suh ^that

a _m (f ) = a _m (g)

^for

0 ≤ m ≤ k · d(Γ ₁ (n))

^,

then

f − g

^hasâ^zeroôfôrderât ^least

k · d(Γ ₁ (n)) + 1

^at^the^usp

∞

^of

X ₁ (n)

^,^and^we^onlude

that

f = g

^. Ôneân ^prove ^the ^lemmaⁱⁿ^general ^by^reduing ^to ^theâse

n ≥ 5

^. ^W^e ^refer ^to

Sturm [14℄for afull proof.

2.2. Heke algebras. Let

T(M _k (Γ ₁ (n)))

^be ^the ^Heke ^algebra ^on

M _k (Γ ₁ (n))

^. ^This ^is

a ommutative ring, free of nite rank as a

Z

^-module ând ^generated âs â

Z

^-algebra ^by ^the

Heke operators

T m

^for

m ∈ { 1, 2, . . . }

^and ^the^diamond ^operators

h d i

^for

d ∈ (Z/nZ) ^×

^. ^It

atson the

C

^-vetor ^spae

M _k (Γ ₁ (n))

^of ^modular^forms.

Letus give someuseful formulae. We have

(2)

T _m ₁ _m ₂ = T _m ₁ T _m ₂

^if

gcd(m ₁ , m ₂ ) = 1

and

(3)

T _p ⁱ⁺² = T p T _p ⁱ⁺¹ − p ^k ⁻ ¹ h p i T _p ⁱ (p

^prime^and

i ≥ 0),

where

h p i

^is ^to ^be interpretedas

0

^if

p

^divides

n

^. ^F^or ^all

f ∈ M _k (Γ ₁ (n))

^,^we ^have

(4)

a _m (T _p (f )) = a _pm (f ) + p ^k ⁻ ¹ a _m/p ( h p i f ) (p

^prime^and

m ≥ 1),

where theseond termis

0

^if

p

^divides

n

^or ^if

p

^does^not ^divide

m

^,^and

(5)

a ₁ (T _m f ) = a _m (f ) (m ≥ 1).

There existsaanonial bilinear map

T(M _k (Γ ₁ (n))) × M _k (Γ ₁ (n)) −→ C (t, f ) 7−→ a ₁ (tf ),

induing an isomorphism

(6)

M _k (Γ ₁ (n)) −→ ^∼ Hom _Z-modules (T(M _k (Γ ₁ (n))), C)

of

C ⊗ Z T(M _k (Γ 1 (n)))

^-modules.

An eigenform of weight

k

^for

Γ ₁ (n)

îs ân êlement ôf

M _k (Γ ₁ (n))

^spanning ^a one-dimensional eigenspae for the ation of

T(M _k (Γ ₁ (n)))

^. ^Let

f

^be ^suh ^a ^form. ^Then

a ₁ (f ) 6 = 0

^, ^and ^we

maysale

f

^suh^that

a ₁ (f ) = 1

^. ^Now ⁽⁵⁾^implies^that

T _m f = a _m (f )f

^for^all

m ≥ 1.

(6)

Furthermore,there existsaunique grouphomomorphism

ǫ: (Z/nZ) ^× → C ^× ,

alledtheharater of

f

^,^suh^that

h d i f = ǫ(d)f

^for ^all

d ∈ (Z/nZ) ^× .

Underthe isomorphism (6), the eigenforms

f ∈ M _k (Γ ₁ (n))

^with

a ₁ (f) = 1

^orrespond ^to ^the

ring homomorphisms

T(M _k (Γ ₁ (n))) → C

^.

The

C

^-vetor ^spae

M _k (Γ 1 (n))

ân ^be ^written âsâ^diret ^sum

M _k (Γ ₁ (n)) = E _k (Γ ₁ (n)) ⊕ S _k (Γ ₁ (n)).

Here

S _k (Γ ₁ (n))

^denotes ^the ^subspae ôf ûsp ^forms ând

E _k (Γ ₁ (n))

^denotes ^the ^subspae

of Eisenstein series. The ation of

T(M _k (Γ 1 (n)))

^respets ^these ^subspaes, ^and ^we ^get ^a

orrespondingdeomposition

T(M _k (Γ ₁ (n))) = T(E _k (Γ ₁ (n))) × T(S _k (Γ ₁ (n)))

of

Z

^-algebras.

2.3. Eisenstein series. Let

d ₁

^and

d ₂

^be^positive^integers^suh^that

d ₁ d ₂

^divides

n

^,^and

onsiderprimitive haraters

ǫ ₁ : (Z/d ₁ Z) ^× → C ^× , ǫ ₂ : (Z/d ₂ Z) ^× → C ^× .

(Aharater

ǫ : (Z/dZ) ^× → C ^×

^, ^with

d

â ^positive înteger, îs âlled ^primitive îf^there îs ^no

stritdivisor

e | d

^suh ^that

ǫ

^fators^through^the ^quotient

(Z/dZ) ^× → (Z/eZ) ^×

^.) ^W^e ^dene

theformal powerseries

(7)

E _k ^ǫ ¹ ^,ǫ ² (q) = − δ _d ₁ _,1 B _k ^ǫ ² 2k +

X ∞ m=1

X

d|m

ǫ 1 (m/d)ǫ 2 (d)d ^k ⁻ ¹

!

q ^m ∈ C[[q]].

Here

B _k ^ǫ ²

^is ^a generalised Bernoullinumber and

δ _d ₁ _,1

^is

1

^or

0

^depending ^on ^whether

d ₁ = 1

or

d ₁ > 1

^.

If

k 6 = 2

^, ^or ^if

k = 2

ând ât ^least ône ôf

ǫ 1

^and

ǫ 2

^is non-trivial, then

E _k ^ǫ ¹ ^,ǫ ² (q)

^is ^the

q

^-

expansionofan eigenformin

E _k (Γ ₁ (d ₁ d ₂ ))

^with^harater

ǫ ₁ ǫ ₂

^. ^For^any^divisor

e

^of

n/(d ₁ d ₂ )

^,

themap

b ^d e ¹ ^d ² ^,n : M _k (Γ ₁ (d ₁ d ₂ )) → M _k (Γ ₁ (n))

^sends^this^form^toânêlementôf

M _k (Γ ₁ (n))

^with

q

^-expansion

E _k ^ǫ ¹ ^,ǫ ² (q ^e )

^. ^As^for ^the^ase ^where

k = 2

^and ^both

ǫ ₁

^and

ǫ ₂

^are ^trivial, ^for ^every

divisor

e | n

^with

e > 1

^there îs ân êlement ôf

E ₂ (Γ ₁ (n))

^with

q

^-expansion

E ₂ (q) − eE ₂ (q ^e )

^,

where

E ₂ (q)

^is^the^power^series

(8)

E ₂ (q) = − 1

24 + X ∞ m=1

X

d | m

d

! q ^m .

For

k 6 = 2

^,^the^nite^set

(9)

F _k (Γ ₁ (n)) = G

d 1 d 2 |n

G

e|(n/d 1 d 2 )

E _k ^ǫ ¹ ^,ǫ ² (q ^e ) | ǫ _i : (Z/d _i Z) ^× → C ^×

^primitive

(7)

isa

C

^-basis^of

E _k (Γ ₁ (n))

^. ^F^or

k = 2

^,^we^take^all

E _k ^ǫ ¹ ^ǫ ² (q ^e )

^for

ǫ ₁

^,

ǫ ₂

^not ^both ^trivial,^together

withthe

E ₂ (q) − eE ₂ (q ^e )

^for ^all

e | n

^with

e > 1

^.

2.4. Cusp forms. We write

S ^new _k (Γ 1 (n))

^for ^the ^orthogonal ^omplement, ^with ^respet

to the Petersson inner produt, of the subspae of

S _k (Γ ₁ (n))

^spanned ^by ^the ^images ^of

all the

b ^d,n e

^with

d

^stritly ^dividing

n

^. ^The ^spae

S ^new _k (Γ ₁ (n))

^is ^preserved ^by ^the ^ation

of

T(S _k (Γ ₁ (n)))

^. ^The^unique ^quotient ^of

T(S _k (Γ ₁ (n)))

^that^ats ^faithfully^on

S ^new _k (Γ ₁ (n))

^is

denotedby

T(S ^new _k (Γ 1 (n)))

^.

An eigenform

f ∈ S ^new _k (Γ ₁ (n))

^with

a ₁ (f ) = 1

îsâlledâ ^primitive ûsp^form. ^The ^nite^set

(10)

B _k (Γ 1 (n)) = G

d|n

G

e|(n/d)

b ^d,n _e

primitive uspforms in

S ^new _k (Γ 1 (n))

isa

C

^-basis^for

S _k (Γ ₁ (n))

^.

2.5. Modular forms over other rings. We dene

M ^int _k (Γ ₁ (n)) = {

^forms ⁱⁿ

M _k (Γ ₁ (n))

^with

q

^-expansion ⁱⁿ

Z[[q]] } .

Thisisa

T(M _k (Γ ₁ (n)))

^-module^thatîs^freeôf^nite^rankâsâ

Z

^-module. ^Fôrânyômmutative

Z[1/n]

^-algebra

R

^, ^we ^dene ^the

R

^-module ^of ^modular ^forms ^of ^weight

k

^for

Γ ₁ (n)

^with

oeientsin

R

^as

M _k (Γ ₁ (n), R) = R ⊗ Z M ^int _k (Γ ₁ (n)).

Apartfromthe omplexnumbers, the important examplesforus arenumberelds andnite

elds of harateristi not dividing

n

^. ^If

R

îs âny êld ôf harateristi not dividing

n

^, ^we

dene eigenforms over

R

ⁱⁿ^the^same ^way^asⁱⁿ ^the^ase

R = C

^.

If

R

^is^a^sub-

Z[1/n]

^-algebra^of

C

^,^we^identify

M _k (Γ ₁ (n), R)

^with^the^submodule^of

M _k (Γ ₁ (n))

onsistingof forms with

q

^-expansion ⁱⁿ

R[[q]]

^.

3. Modular Galois representations

Let

n

^and

k

^be ^positive ^integers, ^let

F

^be â ^nite êldôf harateristi not dividing

n

^, ^and

let

f ∈ M _k (Γ ₁ (n), F)

^beân êigenformôver

F

^.

Itfollows fromwork ofEihler,Shimura, Igusa,Deligne andSerre thatthere existsa ontin-

uoussemi-simple representation

ρ _f : Gal(Q/Q) → Aut F V _f ,

where

V _f

^is ^atwo-dimensional

F

^-vetor ^spae,^with ^the^following properties:

ρ _f

îsûnramied ât âll^prime^numbers

p

^not ^dividing

nl

^;

if

p

^is^suh ^a ^prime ^number, ^then^theharateristi polynomialof theFrobenius onju- gay lassat

p

^equals

t ² − a _p (f )t + ǫ(p)p ^k−1

^,^where

ǫ

^is^the^harater ^of

f

^.

(8)

This

ρ _f

îsûnique ûp^to isomorphism.

The end produt of [1℄ is a probabilisti algorithm for omputing representations of the

form

ρ _f

^, ^where

f

îs ân êigenform ôver â ^nite êld

F

^. ^This ^allows ^us ^to ^state ^the^following

theorem.

Theorem 3.1. Let

n ₀

a positive integer

k

^,

n ₁

^oprime ^to

n ₀

^,

a niteeld

F

^of harateristi greater than

k

^, ^and

an eigenform

f ∈ M _k (Γ 1 (n))

^, ^given ^by ^its ^oeients

a m (f )

^for

0 ≤ m ≤ k · d(Γ 1 (n))

^,

omputes

ρ _f

ⁱⁿ ^the ^form ^of ^the ^following^data:

the nite Galois extension

K _f

^of

Q

^suh ^that

ρ _f

^fators ^as

Gal(Q/Q) ։ Gal(K _f /Q) ֌ Aut F V _f ,

given by the multipliationtable of some

Q

^-basis

(b ₁ , . . . , b _r )

^of

K _f

^;

for every

σ ∈ Gal(K _f /Q)

^, ^the ^matrix ^of

σ

^with ^respet ^to ^the ^basis

(b 1 , . . . , b r )

^and^the

matrix of

ρ _f (σ)

^with^respet ^to^some ^xed

F

^-basis ^of

V _f

^,

andthat runs in expeted time polynomial in

k

^,

n ₁

^and

#F

^.

Moreover, one

ρ _f

^has ^been ômputed, ône ân ômpute

ρ _f (Frob p )

^using ^a deterministi al- gorithmin time polynomialin

k

^,

n ₁

^,

#F

^and

log p

^.

Remark 3.2. Thisrunningtimeisoptimalfromaertainperspetive,giventhefatthat

thelengthof theinput andoutput ofsuh an algorithm isneessarily at leastpolynomial in

k

^,

n 1

^and

#F

^(and

log p

^for ^the^seond^part).

4. Some bounds

Inthis setionwe olletsome boundsthatwe will need in

§

⁵^below^to ^prove ^Theorem ^1.1.

4.1. The disriminant of the new quotient of the Heke algebra. Let

n

^and

k

be positive integers. The

Z

^-algebra

T(S ^new _k (Γ ₁ (n)))

îs ^redued, ^beause ^there îs â ^basis ôf

eigenformsfor itsation on

S ^new _k (Γ ₁ (n))

^. ^Furthermore, itis freeof niterankasa

Z

^-module.

Inpartiular,it hasanon-zero disriminant

disc T(S ^new _k (Γ 1 (n)))

^.

Lemma 4.1. The logarithm of

| disc T(S ^new _k (Γ 1 (n))) |

^is ^bounded ^by ^a ^polynomial ⁱⁿ

n

and

k

^.

Proof. ThemethodofUllmo[15℄,whoonsidereduspformsofweight 2for

Γ ₀ (n)

^with

n

squarefree,extends without diulty to our situation. For ompleteness, let us give a proof

inthis more general setting.

Weabbreviate

T = T(S ^new _k (Γ ₁ (n)))

(9)

and

r = dim _C S ^new _k (Γ ₁ (n))

= rank _Z T.

ItfollowsfromLemma2.1and(6 )thatthe

Q

^-vetor^spae

Q ⊗ Z T

^is^spanned^by^the^elements

T ₁

^,^.^.^.^,

T _k _· _d(Γ ₁ _(n))

^,^with

d(Γ ₁ (n))

âsⁱⁿ^(1). ^Weân ^therefore^hooseîntegers

1 ≤ m 1 ≤ · · · ≤ m r ≤ k · d(Γ 1 (n))

suh thattheelements

T _m ₁

^,^.^.^.^,

T _m _r

^of

T

^are

Z

^-linearly independent. Welet

T ^′

^denote^the

subgroup of

T

^spanned^by

T _m ₁

^,^.^.^.^,

T _m _r

^. ^This

T ^′

^is ^free ^of ^rank

r

^as^a

Z

^-module, ^so ^it^has

nite index

(T : T ^′ )

ⁱⁿ

T

^,^and

disc T = disc T ^′ (T ^′ : T) ² .

Inpartiular, this implies

| disc T | ≤ disc T ^′ .

We next usethe denitionof thedisriminant:

disc T ^′ = det tr(T _m _u T _m _v ) ^r _u,v=1 ,

where

tr(e)

^denotes ^the ^trae ^of ^the

Z

^-linear ^map

T ^′ → T ^′

^sending

t

^to

et

^. ^Now ^the ^trae

ofan endomorphism

e

^of

T ^′

^equals^the ^trae ^of^theendomorphism dualto

e

^on ^the

C

^-vetor

spae

Hom _Z-modules (T ^′ , C) ∼ = S ^new _k (Γ ₁ (n)).

We let

f ₁

^,^.^.^.^,

f _r

^be ^the ^primitive ^usp^forms ⁱⁿ

S ^new _k (Γ ₁ (n))

^,^and^we^abbreviate

α _t,u = a _m _t (f _u ).

Thenweget

tr(T _m _u T _m _v ) = X r t=1

α _t,u α _t,v .

We thenompute

disc T ^′

^as^follows:

disc T ^′ = det X r

t=1

α _t,u α _t,v r

u,v=1

= det



 

 



 

 

α 1,1 α 2,1 . . . α r,1

α _1,2 α _2,2 . . . α _r,2

.

. .

.

. .

.

. .

.

α _1,r α _2,r . . . α _r,r



 

 



 

 

α 1,1 α 1,2 . . . α 1,r

α _2,1 α _2,2 . . . α _2,r

.

. .

.

. .

.

. .

.

α _r,1 α _r,2 . . . α _r,r



 

 



 

 

= det (α _t,u ) ^r _t,u=1 2

.

(10)

Deligne'sboundfortheoeientsofeigenforms,provedin[3℄ and[4℄,impliestheinequality

| α t,u | = | a m t (f u ) |

≤ σ ₀ (m _t )m ^(k _t ⁻ ^1)/2 .

Here

σ ₀ (m)

^denotes ^the ^number ^of ^positive ^divisors ^of

m

^. ^Elementary ^estimates ^now ^show

that

log | disc T |

^is ^bounded ^by^a^polynomialⁱⁿ

n

^and

k

^.

4.2. Primes of small norm in number elds. The Riemann hypothesis for the

ζ

^-

funtionof anumbereld

K

^has^the ^following ^well-knownîmpliation ôf^for ^theêxisteneôf

primeidealsofsmall normin thering ofintegers of

K

^.

Lemma 4.2. Let

ǫ

^and

δ

^be ^positive ^real ^numbêrs. ^There êxist ^positive ^rêal ^numbêrs

A

^and

B

^suh ^that ^the ^following ^holds. ^Let

K

^be ^an ^number ^eld ^suh ^that ^the ^Riemann

hypothesis is true for the

ζ

^-funtion ^of

K

^. ^Let

Z _K

^denote ^the ^ring ôf întegers ôf

K

^, ^and^for

everyprimenumber

p

^let

λ _K (p)

^denote^the ^numberôf ^primeîdealsôf

Z _K

^of ^norm^equal ^to

p

^.

Thenfor all real numbers

x ≥ 2

^suh^that

x ^δ

(log x) ² ≥ A[K : Q]

^and

x ^δ

log x ≥ B log | disc Z _K |

wehave

X

p ≤ x

^prime

λ _K (p) log p − x

≤ ǫx ^1/2+δ .

Proof. For every prime number

p

ândêvery^positiveînteger

m

^,^we^dene

Λ K (p ^m ) = X

t|m

t · # {

^prime ^ideals^of ^norm

p ^t

ⁱⁿ

Z K } · log p.

Inpartiular,thisimplies

Λ _K (p) = λ _K (p) log p

^for^every^prime^number

p

^. ^W^e^dene

Λ _K (n) = 0

^if

n

^is ^not ^a^prime^power. ^The^relation ^between

ζ _K

^and

Λ _K

^is^the^Dirihlet ^series

− ζ _K ^′ (s) ζ _K (s) =

X ∞ n=1

Λ _K (n)n ⁻ ^s .

Wedene

ψ K : [1, ∞ ) −→ R x 7→ X

n≤x

Λ _K (n).

Now there existsa positive real number

c

^, independent of

K

^, ^suh ^that^the generalisedRie- mannhypothesis for

ζ _K

^implies^the^estimate

| ψ _K (x) − x | ≤ c √

x log(x) log x ^[K:Q] | disc Z _K |

for all

x ≥ 2;

(11)

seeIwanieand Kowalski [9, Theorem 5.15℄. Byelementary arguments, it follows thatthere

existsapositive realnumber

c ^′

^,^alsoindependent of

K

^,^suh ^that

X

p ≤ x

^prime

λ _K (p) log p − x ≤ c ^′ √

x log(x) log x ^[K:Q] | disc Z _K |

forall

x ≥ 2.

It isnowstraightforward to hekthattaking

A = B = 2c ^′ /ǫ

^works.

5. Proof of Theorem 1.1

Asalready mentioned brieyintheintrodution, Theorem1.1 willbe proved asfollows. We

rstprove thefollowing basi ases:

f

îs ânêlement ôf ^the^form

E _k ^ǫ ¹ ^,ǫ ²

ⁱⁿ

E _k (Γ ₁ (d ₁ d ₂ ))

^,^where

ǫ ₁ : (Z/d ₁ Z) ^× → C ^×

^and

ǫ ₂ : (Z/d ₂ Z) ^× → C ^×

^are^primitive^haraters;

f

îs â^primitive ûsp^formⁱⁿ

S _k (Γ ₁ (n))

^.

In eah ase, we take

K

^to ^be ^the ^number êld ^generated ^by ^the ôeients ôf

f

^, ^and ^we

assume that

m

îs â ^prime ^number. Âfter ^proving ^these ^speial âses, ^we ^show ^that ^we ân

ompute theHeke algebra

T(M _k (Γ 1 (n)))

ⁱⁿ^a ^sense ^that^will^be ^explainedⁱⁿ

§

^5.3^below. ^It

isthenstraightforward to dedueTheorem 1.1ingeneral.

5.1. Eisenstein series. We start by onsidering the Eisenstein series

E _k ^ǫ ¹ ^,ǫ ²

^, ^where

ǫ ₁ : (Z/d ₁ Z) ^× → C ^×

^and

ǫ ₂ : (Z/d ₂ Z) ^× → C ^×

^are ^primitive ^haraters ^and

e

^is ^a ^divisor

of

n/(d ₁ d ₂ )

^. ^Fôr ônveniene, ^we âlsoâllow ^theâse ôf ^the`pseudo-Eisenstein series'

E ₂

^de-

nedby(8 ). Let

K

^be^the^ylotomi^extension^of

Q

^generated^by^the^images^of

ǫ ₁

^and

ǫ ₂

^. ^The

formula(7)showsthatforeveryprimenumber

p

^,^weânômpute^theêlement

a _p (E _k ^ǫ ¹ ^,ǫ ² ) ∈ K

intimepolynomialin

n

^,

k

^and

log p

^.

5.2. Primitive forms. We ontinue with the ase where

f

îs â ^primitive ûsp ^form

in

S ^new _k (Γ ₁ (n))

^and

K

îs^the ^number êld ^generated ^by^the ôeientsôf

f

^. ^Let

Z _K

^denote

thering ofintegersof

K

^. ^There êxistsâûnique ^ringhomomorphism

e _f : T(S ^new _k (Γ ₁ (n))) → Z _K

sending eah Heke operator to its eigenvalue on

f

^. ^Let

A

^denote ^the ^image ^of

e _f

^. Ît îs ôf

nite index

(Z _K : A)

ⁱⁿ

Z _K

^,^and ^we ^have

disc A = (Z _K : A) ² disc Z _K .

Furthermore, we have

| disc A | ≤ | disc T(S ^new _k (Γ ₁ (n))) |

^and

[K : Q] ≤ rank _Z T(S ^new _k (Γ ₁ (n))).

Lemma 4.1 now implies that

log | disc A |

^, ^and ^hene ^also

log | disc Z _K |

^and

log(Z _K : A)

^, ^are

boundedbya polynomial in

n

^and

k

^. ^The ^same ^learly ^holds^for

[K : Q]

^.

(12)

Nowlet

p

^beâ^prime^number. ^Wê^have^to^show^that^weânômpute

a _p (f )

ⁱⁿ^time^polynomial

in

n

^,

k

^and

log p

^. ^In Couveignes, Edixhoven et al. [2,

§

^15.2℄ ît îs êxplained ⁱⁿ^detail ^how ^to

dothis. We only givea sketh.

Wemayassumethat

p

^does^not ^divide

n

^;^namely,^if

p

^does^divide

n

^,^then^we ^an ^spend^time

polynomial in

p

^,^soûsing^modular ^symbolsîs^fast ênough;^see

§

^5.3^below.

ByLemma 4.2 applied to

K

^and ^the^fat^that

log(Z _K : A)

^is^bounded ^by^a ^polynomial ⁱⁿ

n

and

k

^,^we^an ^hoose

x

^suiently ^large,^but^bounded^by^a^polynomial ⁱⁿ

n

^and

k

^,^suh ^that

if

M

îs ^the ^set ôf ^maximal îdeals ôf

A

^whose ^norm îs â ^prime ^number ^lying ⁱⁿ ^the înterval

(k, x]

^and ^dierent ^from

p

^,^we ^have

(11)

Y

m ∈ M

Norm(m) ≥

2 ([K:Q]+1)/2 · 2p ^(k ⁻ ^1)/2 [K:Q]

.

An explanation for the right-hand side will be given below. We ompute

a _p (f )

^using ^the

following algorithm.

1. Compute a

Z

^-basis^for

A

^.

2. Compute abound

x

^and ^the ^set

M

ôf^maximal îdealsôf

A

^suh^that ^the^set

M

^dened

above satises(11 ).

3. For all

m ∈ M

^,^ompute ^the ^Galois representation

ρ _f _mod _m : Gal(Q/Q) → GL ₂ (A/m)

using Theorem 3.1 .

4. For all

m ∈ M

^,^ompute

(a p (f ) mod m) = tr(ρ _f _mod _m (Frob p )) ∈ A/m,

again using Theorem3.1.

5. Compute an LLL-redued

Z

^-basis^for ^the^ideal

a = Q

m∈M m

^of

A

^.

6. Fromthe

a _p (f ) mod m

^,ômpute^the îmage ôf

a _p (f )

ⁱⁿ

A/a

^.

7. Using the LLL algorithm, reonstrut

a _p (f )

^as ^the ^shortest representative in

A

^of ^the

image of

a p (f )

ⁱⁿ

A/a

^. ^This^works^beause ^of^the ^inequality ^{(11) .}

5.3. ComputingHeke operators. Werepresent

T(M _k (Γ ₁ (n)))

ⁱⁿ^the^following^form:

we speify its multipliation table with respet to a suitable

Z

^-basis

(b 1 , . . . , b r )

^, ^together

with the Heke operators

T _m

^for

1 ≤ m ≤ k · d(Γ ₁ (n))

^and ^the ^diamond ^operators

h d i

^for

all

d ∈ (Z/nZ) ^×

^as

Z

^-linear ombinations of

(b ₁ , . . . , b _r )

^. ^These ^data ^speify

T(M _k (Γ ₁ (n)))

uniquely beause the above operators generate

T(M _k (Γ ₁ (n)))

^. În ôther ^words, îf ^the ^same

data are given with respet to a dierent basis of

T(M _k (Γ ₁ (n)))

^, ^there êxists êxatly ône

hange of

Z

^-basis ^ompatible ^with^the^given

T m

^and

h d i

^.

Theorem 5.1. Let

n ₀

a positive integer

k

^,

n ₁

^oprime ^to

n ₀

^, ^and

a positive integer

m

ⁱⁿ ^fatored ^form,

omputes

(13)

the Heke algebra

T(M _k (Γ ₁ (n)))

âs âbôve, ^where

n = n ₀ n ₁

^, ^and

the element

T _m

^on ^the ^basis

(b ₁ , . . . , b _r )

^,

and that runs in expeted time polynomial in

k

^,

n ₁

^and

log m

^under ^the ^Riemann ^hypothesis

for

ζ

^-funtions ôf ^numbêr êlds.

Proof. We need some more information about the ation of Heke operators on

q

^-

expansions. Asa basisfor

M _k (Γ ₁ (n))

^we ^take ^the ^union ^of^the ^basis

F _k (Γ ₁ (n))

^of

E _k (Γ ₁ (n))

dened by (9 )and the basis

B _k (Γ ₁ (n))

^of

S _k (Γ ₁ (n))

^dened ^by^{(10 ).}

Let

f

^be êither ân Êisenstein ^series

E _k ^ǫ ¹ ^,ǫ ² ∈ E _k (Γ ₁ (d ₁ d ₂ ))

âs âbove ôr â ^primitive ^form

in

S _k (Γ ₁ (d))

^. ^In ^the ^rst ^ase, ^we ^put

d = d ₁ d ₂

^. ^The^formula⁽⁴⁾ ^for ^the^ation ^of^the^Heke

operator

T _p

^shows ^that^the^relation^between

T _p

^and^the ^maps

b ^d,n e : M _k (Γ ₁ (d)) → M _k (Γ ₁ (n))

^,

where

e

^runs^through ^the^divisors^of

n/d

^,^is ^as^follows:

(12)

T _p (b ^d,n _e f ) =

 

 

 

 



a _p · b ^d,n _e f

^if

p ∤ n;

b ^d,n _e/p f

^if

p | e;

a p · b ^d,n e f − p ^k ⁻ ¹ ǫ(p)b ^d,n pe f

^if

p ∤ d

^,

p ∤ e

^and

p | n;

a _p · b ^d,n _e f

^if

p | d

^and

p ∤ e.

Thisformulagivesthematrix of

T p

^with^respet^to^the^basis

F _k (Γ 1 (n))

^of

E _k (Γ 1 (n))

^and^the

basis

B _k (Γ ₁ (n))

^of

S _k (Γ ₁ (n))

^.

We rst ompute the

q

-expansions of the Eisenstein series

E _k ^ǫ ¹ ^,ǫ ² ∈ E _k (Γ ₁ (d ₁ d ₂ ))

^, ^with

ǫ _i : (Z/d _i Z) ^× → C ^×

^primitive ^haraters ^suh ^that

d ₁ d ₂ | n

^, ^as ⁱⁿ

§

^{2.3 .} ^F^rom ^these

q

^-

expansionsand(12 )wethenompute theHekealgebra

T(E _k (Γ ₁ (n)))

ⁱⁿ^the^form ^desribed

above intime polynomialin

n

^and

k

^.

Given a prime number

p

^, ^we ^ompute ^all ^the

a _p (E _k ^ǫ ¹ ^,ǫ ² )

^as ⁱⁿ

§

^{5.1 ,} ^and ^we ^nd ^the ^matrix

of

T _p

^using ^(12). ^We^then^express

T _p

^on ^the^basis^of

T(E _k (Γ ₁ (n)))

^that^we^omputed ^earlier.

Inthis way,we anompute theHeke operator

T p ∈ T(E _k (Γ 1 (n)))

ⁱⁿ^time^polynomialⁱⁿ

n

^,

k

^and

log p

^.

For uspforms, the

q

-expansions areomputed fromtheHeke algebra insteadofvieversa.

We ompute theHekealgebras

T(S _k (Γ ₁ (d)))

^,^where

d

^runs^through ^the^divisors^of

n

^,ⁱⁿ^the

form desribed above. These data an be omputed in time polynomial in

n

^and

k

^using

deterministialgorithmsbasedonmodularsymbolsandtheLLLlattiebasisredutionalgo-

rithm;seeStein[13,Chapter8℄andtheauthor'sthesis[1,

§

^IV.4.1℄. ^F^rom^eah

T(S _k (Γ ₁ (d)))

^,

we ompute the

q

-expansionsof the primitiveusp formsin

S _k (Γ ₁ (d))

^.

Sofar,wehaveonlyusedexistingmethods. ToomputetheHekeoperator

T p ∈ T(S _k (Γ 1 (n)))

for a primenumber

p

ⁱⁿ ^time^polynomial ⁱⁿ

log p

^,^we ^need ôur ^new ^tools. ^Fôr êvery ^divisor

d

^of

n

^and ^every ^primitive ^form

f ∈ S _k (Γ ₁ (d))

^,^we^ompute

a _p (f )

^as ⁱⁿ

§

^{5.2 .} ^Using ^(12),^we

obtain the matrix of

T _p

^with ^respet ^to ^the ^basis

B _k (Γ ₁ (n))

^. ^W^e ^nally ^express

T _p

^on ^the

basisof

T(S _k (Γ ₁ (n)))

^that^we^omputed ^earlier.

(14)

Nowlet

m

^beânârbitrary^positive înteger,ând^suppose^that^we ^know^thefatorisation of

m

^.

Thenwe an ompute the element

T _m ∈ T(M _k (Γ ₁ (n))) = T(E _k (Γ ₁ (n))) × T(S _k (Γ ₁ (n)))

fromthe

T _p

^for^for

p | m

^primeⁱⁿ^time^polynomialⁱⁿ

log m

ûsing^theîdentities^{(2 )}ând^(3).

5.4. Proof of Theorem 1.1 in general. Given

n

^,

k

^,

K

^,

f

^and

m

^as ⁱⁿ ^the^theorem,

weompute

a m (f )

âs^follows. ^Wê ^rstômpute

T(M _k (Γ 1 (n)))

^using^modular^symbols. ^F^rom

f

^we ^then^determine ^the ^unique

Z

^-linear ^map

e _f : T(M _k (Γ ₁ (n))) → K

sending

T i

^to

a i (f )

^for ^all

i

^with

1 ≤ i ≤ k · d(Γ 1 (n))

^. ^Using ^Theorem^{5.1 ,} ^we ^then^ompute

theHeke operator

T _m

^. ^Finally,^we ^ompute

a _m (f )

^as

a _m (f ) = e _f (T _m ).

Itis straightforward to hekthat allthese omputationsan be done intimepolynomial in

thelength of theinput.

Remark 5.2. The proof shows that the Riemann hypothesis only needs to be assumed

forthe

ζ

^-funtionsôf^numberêlds ^thatâriseâsêldsôf ôeients ôf^primitive ûsp^forms.

6. Appliations

6.1. Counting points on modular urves. Thease

k = 2

ôf^Theorem ^5.1împliesâ

newresulton ountingpointson modularurvesovernite elds.

Theorem 6.1. There exists a probabilisti algorithm that, given a squarefree positive in-

teger

n

^and ^a ^prime ^number

p ∤ n

^, ômputes ^the ^zeta ^funtion ôf ^the ^mo^dular ûrve

X 1 (n)

over

F _p

^, ^and^that ^runs ⁱⁿ ^time ^polynomial ⁱⁿ

n

^and

log p

^under ^the ^Riemann ^hypothesis^for

ζ

^-funtions^of ^number ^elds.

Proof. Let

J ₁ (n) _F _p

^denote^the^Jaobian^of

X ₁ (n) _F _p

^. ^Let

χ

^be^theharateristipolynomial of the Frobenius endomorphism of the

l

^-adi ^Tate ^module

T _l J 1 (n) F p

^, ^where

l

^is ^any ^prime

number dierent from

p

^; ^then

χ

^hasîntegral ôeients ând ^does^not ^depend ôn ^the ^hoie

of

l

^. ^Beause^of ^the^well-known^identity

Z _X ₁ _(n)/F _p (t) = χ ^∗ (t) (1 − t)(1 − pt) ,

where

χ ^∗ (t) = t ^deg ^χ χ(1/t)

^is^the ^reiproal ^polynomial ^of

χ

^,^it^sues^to ^ompute

χ

^.

Let

T ₁ (n)

^denote ^the ^Heke âlgebra âting ôn

J ₁ (n) _F _p

^. ^Then

Q _l ⊗ Z l T _l J ₁ (n) _F _p

^is ^a ^free

Q _l ⊗ Z T ₁ (n)

^-module^of^rank^2. ^By^theEihlerShimurarelation,theharateristipolynomial of

Frob p

ônîtêquals

x ² − T p x+p h p i ∈ T 1 (n)[x]

^. ^This^implies^that^theharateristipolynomial of

Frob _p

^viewed ^as^a

Q _l

^-linear ^map^equals

Computing coefficients of modular forms

q

ζ

q

ζ

n

k

M k (Γ 1 (n))

k

Γ 1 (n)

f ∈ M k (Γ 1 (n))

n

k

q

a m (f)

0 ≤ m ≤ k · d(Γ 1 (n))

d(Γ 1 (n))

n

a m (f )

0 ≤ m ≤ k · d(Γ 1 (n))

a m (f )

m

n = 1

log m

k

k

log m

m

1

n 0

k

n 1

n 0

K

f

k

Γ 1 (n)

K

n = n 0 n 1

m

a m (f )

ζ

K

f

a m (f)

K

Q

(b 1 , . . . , b r )

K

c i,j,k

1 ≤ i, j, k ≤ r

b i b j = X r k=1

c i,j,k b k .

K

Q

(b 1 , . . . , b r )

f

a 0 (f )

a k · d(Γ 1 (n)) (f )

a m (f )

K

x

T x

P

x

T x

P (

x)

§

k

n 0

n 1

log m

K

f

f

f 0

A

f 0

A

M _k (Γ ₁ (n))

Γ ₁ (n)

f ∈ M _k (Γ ₁ (n))

a _m (f )

0 ≤ m ≤ k · d(Γ ₁ (n))

a _m (f )

n ₀

n ₁

n ₀

Γ ₁ (n)

n = n ₀ n ₁

a _m (f )

a _m (f)

(b ₁ , . . . , b _r )

c _i,j,k

b _i b _j = X r k=1

c _i,j,k b _k .

(b ₁ , . . . , b _r )

a _k _· _d(Γ ₁ _(n)) (f )

T _x

T _x

n ₀

n ₁

exp(n ₁ )

n ₀

n ₁

f = E _k

E _k

Γ ₁ (1) = SL ₂ (Z)

a _m (E _k )

a _m (E _k ) = X

d ^k−1

= 1 + p ^k ⁻ ¹ + q ^k ⁻ ¹ + m ^k ⁻ ¹ ,

{ p ^k ⁻ ¹ , q ^k ⁻ ¹ }

x ² − (a _m (E _k ) − m ^k−1 − 1)x + m ^k−1 ∈ Z[x]

Z _L

Γ ₁ (n)

Γ(n) ⊆ SL ₂ (Z)

M _k (Γ ₁ (n ² ))

q 7→ q ⁿ

T(M _k (Γ 1 (n)))

T(M _k (Γ ₁ (n)))

X ₁ (n)

F _p

M _k (Γ ₁ (n))

Γ ₁ (n) = a

∈ SL ₂ (Z)