Modularity of an odd icosahedral representation

A

RNAUD

J

EHANNE

M

ICHAEL

M

ÜLLER

Modularity of an odd icosahedral representation

Journal de Théorie des Nombres de Bordeaux, tome

12, n

o

_{2 (2000),}

p. 475-482

<http://www.numdam.org/item?id=JTNB_2000__12_2_475_0>

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

Modularity

of

an

odd icosahedral

representation

par ARNAUD JEHANNE et MICHAEL

MÜLLER

d Jacques Martinet

RÉSUMÉ. Dans cet _article, nour démontrons _que la

_{représentation}

03C1 de

GQ

dans

d’image A5

dans

PGL2(A5)

correspondant

à

_l’exemple

16 dans

_[B-K]

est modulaire. Cette

_{représentation}

est

de conducteur 5203 et de déterminant _~-43. La modularité de

cette

_{représentation}

n’était _pas encore

_{prouvée ;}

en

_effet,

elle ne

vérifie _pas les

_hypothèses

des théorèmes de

_[B-D-SB-T]

et

_[Tay2].

ABSTRACT. In this paper, we _prove that the _{representation p}

from

_GQ

in with

_{image A5}

in

_{PGL2(A5) corresponding}

to the

_example

16 in

_[B-K]

is modular. This _{representation} has conductor 5203 and determinant _~-43; its

_modularity

was not _yet

proved,

since this _{representation} does not

_satisfy

the

_hypothesis

of the theorems of

_[B-D-SB-T]

and

_[Tay2].

1. Introduction Consider an irreducible Galois

_{representation}

and the

_{corresponding}

induced

_{projective representation}

where we denote

_{by Q}

an

_algebraic

closure

_{of Q}

and

_by

_{GQ =Gal(Q/Q)}

the absolute Galois _group over

Q.

Artin has

_conjectured

_(for

_any nontrival irreducible

_{representation}

_{p :}

GLn((C)),

that the

_analytic

continuation of the L-series associated to the

_{representation}

_p is an entire function. It is known

_by

results of

_Hecke,

Langlands

and Tunnel

_{([Lan], [Tun])}

that p satisfies Artin’s

conjecture,

if

Im(p)

is solvable. If

_Im(p)

is

_isomorphic

to

_A5,

this

_conjecture

is not

_yet

proved

in

_general,

but it has been

_proved

for

_particular

cases

_by

Buhler

([Buh])

in

_1978,

and

_{by Kiming}

and

_Wang

_([K-W])

in 1994. In

_1999,

Buz-zard,

Dickinson, Shepherd-Baron

and

_Taylor

_proved

the

_conjecture

for

eight

examples

[B-S]

and

_Taylor

gave another result that proves the

conjecture

for

infinitely

many icosahedral

representations

([Tay2]).

In all these

_examples

and theorems one has _{to assume,} that the

repre-sentation p

is

_odd,

that

_is,

_{det(p)(c) =}

_-1,

where c denotes the

complex

conjugation.

And for odd

_{representations,}

Artin’s

_conjecture

is

equiva-lent to the fact that the L-series of this

_{representation}

coincides with the L-series

_{L( f,}

_s)

of a

weight

one modular form

f

(see (D-S)).

We

_give

one more

_example

for the Artin

_conjecture

with

det(p)

= X-43 and conductor 5203. With this method one can

verify

Artin’s

_conjecture

for

each odd

_{representation}

with

_quadratic

determinant and conductor lower than 10000. We _prove in fact the

_modularity

of this

_{representation}

_by

_using

the

_computation

of its L-function

_([Jeh])

and a basis of the C-vector _space

S2(5203,1)

of cusp forms of

weight

2,

conductor 5203 and

nebentype

1. Note that this

_{representation}

does not

_satisfy

the

_hypothesis

of the main

theorem of

_[B-D-SB-T].

_Indeed,

if A

_{and /i}

are the

_eigenvalues

of the

Frobe-nius element

_Frobp,2

in

_2,

then

_A/p

has order 5. We check too that the

representation

does not

_satisfy

the

_hypothesis

of the theorems of

_[Tay2].

Acknowledgement.

It is a

pleasure

for the authors to thank Gerhard

_Frey

for fruitful discussions and Kevin Buzzard for useful comments and

ques-tions.

2.

_Modularity

of _p We consider the L-function

of our irreducible two-dimensional

representation

_p. We want to _prove that

the series deduced

_formally

from

_L(p,

_s)

(where

q =

e2i1rz)

is a newform of

weight

one and level

_equal

to the

con-ductor of _p. We make use of the

following

_proposition

(see

[Frel],

proposition

1. 1).

Proposition

2.1. Assume that

_belongs

to the _space

Mk(N, X)

of

modular

_{forms of}

level N and

_nebentype

_X, where N is an

inte-ger and X :

(Z/NZ) * ---+

C* a Dirichlet character. Let _J-l = N

+ ~)

(where

p runs

through

the set

of

prime

numbers

dividing

N).

If

bn

= 0

for

Let N =

8(p)

be the conductor and

_{let X}

be the determinant of _p. Via class field

_theory,

we can

consider X,

as a Dirichlet character

(7G/N7G)* -&#x3E;

C* .

Proposition

2.2. be an elements

of Ml (N, X)

and

If

there are two elements _g, and _{92 in}

_S2(N,

_X2)

such that

and

then

_belongs

to

_{81(N, X).}

Proof.

Since _gl

_belongs

to

_{S2(N, X)}

and 0 to

_{Ml(N, X),}

it is

_enough

to prove that the

meromorphic

function

gi /0

is

holomorphic.

Moreover we

have

_92g2

mod

_q¡.t/3.

. Furthermore

gi

and

92g2

are both elements of

S4(N,

X4),

so we can

apply

proposition

2.1. This shows that

gi

= and

therefore that is

_holomorphic.

D

Now we consider a

projective representation

_p

given

by

the

polynomial

P(X)

=

X5

₊

5X2 -

6X ₊ 27. It is the

example

16 of

_(B-K~.

We consider the

_{lifting p}

of

_p

with conductor 5203 and determinant _X-43

_(the

character associated to

_{Q(V/’--4-3))}

of

_[Jeh],

_example

6.

Theorem 2.3. The

_{representations}

_{p is} modular.

Proof.

Let

_Q

be the

_quadratic

form

_{Q(x, y)}

= x2

+ xy +

lly2;

it is known that the associated theta series

/ B

belongs

to

_{Ml(43, X_43).}

We recall that

_f

is the series deduced

_formally

from

_{L(p, s).}

For N =

5203,

the element of

_proposition

2.2 is

_equal

to 1936. We shall _prove in section 3 that there are two elements _gl and _g2 such that

----and

By

proposition 2.2,

we conclude that

f

is

_congruent

to an

_{element 9}

=

gl/8Q

in

_{81(5203,X-43)}

modulo As the series

_{L(p, s)}

has an Euler

qlooo

eigenvector

the associated

_eigenvalue

is

_equal

to

i 1-2v’s.

We obtain that

and then

_by

_proposition

2.1:

It _proves that there exists an

eigenform h

in

81

(5203,

X-43)

with

eigen-value for

_T2

_equal

to

il-2v’5.

_By

_[D-S],

theorem

_4,

we obtain a Galois 2 *

representation

associated to h. The conductor of this

_{representation}

divides 5203 = 11 ~ .43 and its determinant is

_equal

_{to X_43·} After

_[D-S]

the

_eigenvalue

of

_T2

is the trace of a Frobenius

endomorphism

at 2. The trace

i 1 2

can be obtained

neither

_by

a tetrahedral

_{representation}

(Im(ph) - A4)

nor

_by

an octahedral one

S4).

Assume that _{ph is} a dihedral

representation.

We know

that there is a

quadratic

field and a

character p

of the absolute

Galois _group of such that _ph is the induction of _cp. The conductor

8(ph)

is

_given

_by

the formula

where

_Od

is the discriminant of and

_fcp

the conductor of _cp.

_Here,

d E

_{{-11,-43,473}.}

In

_{~( -43)}

and in

_{Q(v/’--11),}

the

_prime

number 2 is inert and then if

d = -43

or -11,

the trace of the

image

of _a Frobenius at 2 is zero.

_Hence,

we cannot have

_(where

_Frobp,2

is the Frobenius

element of _{ph in}

_2).

Because of the formula

_(1),

the

_only

ramified

_primes

for _~p are those above 11. We denote

by p

the

only

ideal above 11 in the

ring

of

_integers

of

_{~( 473).}

_By

_[PARI],

we

_compute

that the _ray class

group

of p

in

~( 473)

has order 15.

Hence,

the

eigenvalues

of the

image

of a Frobenius

endomorphism

at 2 are 15th roots of

unity.

Therefore the

trace of a Frobenius at 2

_belongs

to the

_cyclotomic

field which does not contain

21-.

We conclude that if d =

_473,

_{we cannot} have

2 *

Tr(FrobPh?2)

=

2 -

We have _now

proved

that the

representation

ph is icosahedral.

Let

_Kh

be a

_quintic

subfield of . The

only possibilities

for the

valuations of the discriminant

_dKh

of

_Kh

for

_prime

numbers not

_dividing

60 are 2 and 4

(see

[Ser2],

chapter IV, §

_{1, 2,}

3).

Since the determinant

of _ph is _X-43, the

_{only possibility}

for the valuation of

_dKh

in 43 is 2. For the valuation of

_dKh

in

_11,

we can have the valuations 4

(if

the order of

ph(hl)

is

_equal

to

_5,

where

_III

is the inertia in

_11),

2 or 0. We check then

in table 1 of

_[B-K]

that the

_{only possibilities}

are

_{dKh =}

112.432,

and then

ph is

isomorphic

to p or

dKh

=

114.432,

that is too

large

for this table.

Now,

we assume that

dKh

=

114.432.

In this

case, the order

Ofph(Ill)

is

equal

to 5. Since _{ph is}

_modular,

all the

_liftings

_{(where X}

runs

though

the Dirichlet

_characters)

are modular.

_Moreover,

there exists such a

_lifting

p’

with

_quadratic

nontrivial determinant and such that the order of

_{p’ (I5 )}

is 5

_(see

_[Serl],

theorem 5 _p.

_228).

Since

_p’

is

_modular,

we can assume that

Ph =

p’ (but

then,

the trace of the Frobenius in 2 of _ph is _no more 1

but

1-2v’5,

_multiplied

_by

an element of

f 1,

_{-1, i,}

-i}).

We can _{view Ph} as a 2 1

_ _

representation

with

_image

in

_{GL2(Z) ,}

where Z is the

_integral

closure of Z in

_Q.

We choose a

_{place v}

in Z above 11 and we reduce _ph modulo v. We

obtain a

_{representation}

Consider the Eisenstein series

E1o

normalized such that its constant term

is

_equal

to 1. Then the coefhcients of the

_q-expansion

of

Elo

are

11-integers

and

_{Elo -}

1 mod 11.

_Moreover,

the reduction 8

_{= E}

_bnqn

of modulo v is a _cusp form with coefhcients in

IF11

of

weight

11,

level 5203 and

determinant

_jij-43

_(the

reduction of _X-43 modulo

_v),

that is an

_eigenform

for the Hecke

_operators

and such that

for all

_prime

number

_{p ~ {11,43}}

_(see

_[D-S]

_p. 520 and

_521).

_{Then, ;5h}

satisfies Serre’s

_conjecture

_(3.2.3?)

of

_[Ser3],

that

_is,

_Ph

arises from a _cusp

form with coefhcients in

_IF’11.

The level of this _cusp form is

_equal

to

112.43.

In this _case, we know that we can "take off" the

_11-part

of this level. This

representation

arises from a _cusp form with level 43

(see

theorem

(2.1)

p. 643 of

[Rib]).

Finally, by

theorem 4.5 p. 572 of

[Edi],

we obtain that

;5h

arises from a _cusp form with the

weight predicted by

Serre. We conclude

that

_ph

satisfies Serre’s

_conjecture

_(3.2.4?),

that

_is,

with the level and

_weight

predicted by

Serre. Then this

_{representation}

arises from a _cusp form with

coefhcients in

_Fn

of level

_43,

character

_jij-43

and

_weight

k as in

[Ser3],

_§

2.

To

_compute

this

_weight,

we consider the restriction of

_ph

to the inertia group

7n.

We denote

by 0

the

cyclotomic

character. Since has order 5 and has a trivial

_determinant,

then

l is

conjugated

to

, , , , - ,

and we obtain k = 51 _or

31, using

the

recipe

of

[Ser3].

By

theorem 3.4 of

_[Edi],

that we recall below

(theorem 2.4),

we see that we

only

have

to deal with the _spaces of _cusp forms with coefficients in

_Fii

of level

_43,

character

_{X_43}

and

_weight

3

_{(resp. 7).}

We

_compute

a basis for each of

eigenform

to

-Indeed,

we know that the values of the trace of Frobenius span

Q(i,

J5).

And the values of the reductions modulo 11 _span

_IF121.

Now if we look

at

_{S3(43, X-43),}

we see that the coefficients of an

eigenform

of this _space

span the root field of the

polynomial

X6

+

20X4

+

121X2

+

_214,

that is

congruent

modulo 11 to

_(X

_{+ 3) (X}

_{+ 8) (X~ + 7X}

_{+ 8).}

Then we don’t have an

_eigenform

in

S3(43,

k-43)

modulo 11 such that its

_eigenvalues

span

F121 .

For

_{S7(43, X-43),}

the Hecke

_operator

_T2

has the

_polynomial

characteristic

The reduction of this

_polynomial

modulo 11 has two factors of

_degree

2:

X 2

+ X - 3 and

X 2 -

X - 3. The roots of

X 2

+ X - 3 are

6 (-1

+ and

6(-1 -

If the derivation

_application

acts twice on the

_{corresponding}

eigenforms

(see

theorem 2.4

_below),

we obtain the

eigenvalues

2(-1

+ and

_{2(201312013B/2).}

These values cannot

_correspond

with an

eigenvalue -’+V’5

2

multiply by

an element of

f 1, -1,

_i,

2013z}.

We solve the case of the

_polynomial

X 2 -

X - 3

_by

the same _way. D

In order to write theorem

_2.4,

we have to introduce the derivation 0. If

_f

_{= E}

_anqn

is a formal

series,

we define the series

Of by

_setting

Of

=

E

If

_f

is a modular form with coefficients in

_Fp,

then

Of

is such

a modular form too.

_Moreover,

the

_{application 0}

does neither

_change

the level nor the

character,

but it increases the

weight by

_p + 1

_(see

_[Edi]).

Theorem 2.4

_(Edixhoven).

Let

_f

be an

eigenform

in a space

Sk(N,

X)

of

modular

_forms

with

_coefficients

in

_IE’p.

Then there exists

_integers

n and k’

with

_{0 n p - 1,}

_{k’ p -f- 1,}

and an

eigenform g

in

Sk’

(N, X)

such that

f

and have the same

eigenvalues

for

all

p).

3. Existence of _gl and _g2

In this section we describe _very

briefly

(since

we use

only

standard

tech-niques

in

_{computational}

number

_theory)

how one can

_show,

that there are

two cusp forms _{gl, g2 E}

_{S’2(5203,1)}

which satisfies

and n E N as in section 2.

The method is as follows: We

_compute

a basis

(hi , ... ,

of

S2(5203,1)

with

-

’

(It

is

_enough

for us to

_compute

the first 2000 coefficients of the

_q-expansion

of the

_{hm ) .}

Here _g = dim

S2 (5203,1 )

= 473 is the dimension of the _space of _cusp forms of level 5203. Such a basis exists due to results of

_[Shi],

see also

[D-D-T].

To

_compute

the basis we use the modular

_Symbol

method

(introduced

by

Birch)

and the

_{explicit knowledge}

of the action of the Hecke

algebra

on the modular

symbols.

For a

description

of this method see for

example

[Cre]

or

[Mer].

The rest is linear

_algebra.

After

_computing

such a basis we have to

_check,

that there exists two

_cuspforms

_{gl, g2} which satisfies 2. Let

be the matrix of the coefhcients of the basis

_{~hl, ... ,}

and the coefhcients of

the t2

and the Then one has to show that

rank(A)

= dim

s2(5203,1)

= 473.

(We

multiply

by

2 to

_get

a

_q-expansion

with

_integral

coefficients for

r~

2000.)

To

_compute

the rank we reduce the matrix A modulo

differ-ent

_prime

numbers _p and check that

_{rank(A mod p)}

= 473. Hadamard’s

inequality

gives

us a bound for the number of

_primes

_p that

_guarantee

that

rank(A)

= 473

over Q

(we

need 380

_prime

numbers _p with _p &#x3E;

2 ’ 10~;

it is

clear that these bounds are not

good,

but the

_computation

takes

_only

20

hours on a PII 450MHz

Linux-PC,

so we did not

_try

to

_improve

_them).

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Arnaud JEHANNE

Laboratoire de theorie des nombres

et _{d’Algorithmique Arithmetique}

Universite Bordeaux I

351 cours de la Liberation

33405 Talence cedex

France

E-mail : j ehanne(dmath . u-borde aux . f r

Michael MULLER

Institut fur _{Experimentelle} Mathematik Universitat Gesamthochschule Essen Ellernstralie 29

45326 Essen

Germany