Computing modular degrees using $L$-functions

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

C

HRISTOPHE

D

ELAUNAY

Computing modular degrees using L-functions

Journal de Théorie des Nombres de Bordeaux, tome 15, n

o

3 (2003),

p. 673-682.

<http://www.numdam.org/item?id=JTNB_2003__15_3_673_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Computing

modular

_{degrees using}

L-functions

par CHRISTOPHE DELAUNAY

RÉSUMÉ. Nous donnons un

_algorithme

_pour calculer le

degré

mo-dulaire d’une courbe

_elliptique

définie sur

Q.

Notre méthode

est basée sur le calcul de la valeur

spéciale

en s = 2 du carré

symétrique

de la fonction L attachée à la courbe

_elliptique.

Cette

méthode est assez efficace et facile à

_{implémenter.}

ABSTRACT. We

_give

an

_algorithm

to _compute the modular

_degree

of an

elliptic

curve

defined

over

Q.

Our method is based on the

computation

of the

special

value at s = 2 of the

symmetric

_square of the L-function attached to the

_elliptic

curve. This method is

quite

efficient and _easy to

_implement.

1. Introduction

From the recent and difficult work of

_[14],

_[11]

and

_[3],

it is now known

that _every

_elliptic

curves

E/Q

is modular. If N denotes its

conductor,

this

implies

that there exists a

_covering

_{map cp} from

Xo (N)

to E. The

pull-back

_by

_cp of the

_unique

_(up

to

_{multiplication)}

invariant differential form

w on E is

2i’lrcf(r)dr,

where

f (T)

is a normalized newform of level N and

weight

2 on

ro(N)

and where the ’Manin’s constant’ c is rational and can

be assumed

_positive.

_Furthermore,

the L-function associated to

_f

coincides with the Hasse-Weil L-function of E.

The

_question

of

_computing

the

_degree

is natural and

_interesting

because of

_{important conjectures}

related to this number

_deg(cp).

It is well known

_(cf.

_[15])

that there exists a

simple

relation between

deg(cp)

and

where

_11-IIN

denotes the Petersson norm. In

[15],

D.

_{Zagier explains}

how to

_compute

_explicitly

_{Ilf JIN}

in the

_general

case of a _congruence

sub-group r. J.

Cremona,

in

[7]

interprets Zagier’s

method in the

_langage

of

"M-symbols"

and

_computes

_deg(cp)

for many

elliptic

curves

(large

tables of

elliptic

curves are

_given

in

_[6]).

Both methods are

_geometric

and efficient

but tend to be

_quite

slow when the conductor is

_large.

The purpose of this paper is to

_give

an alternative _way of

_computing

Ilf JIN.

This is an

_analytic

method based on _{well-known results which relate the}

_special

_{value of the}

L-function associated to the

_symmetric

_square of E with

_Ilf

2. The

_imprimitive

_symmetric

_{square of E}

Let E be an

elliptic

curve defined

over Q

of conductor N. The normalized

newform attached to E is

_{f (T) _}

_En

_anqn

_(q

= We have

~p* (w) _

2i1rcf(r)dr,

and a

_conjecture

of Manin asserts that c =1 whenever E is a

strong

Weil curve

(there

is

_exactly

one such curve in an

_isogeny

class).

We

then have

_([15]):

- -

-where

_vol(E)

is the volume of a minimal

period

lattice A with

C/A.

Now,

the Hasse-Weil function

_{L(E, s)}

is

_equal

to

_{Fln a,n-3}

and can be

expanded

as an Euler

product:

f

-1 if E has

_{non-split multiplicative}

reduction at p

ap = 1 if E has

split

multiplicative

reduction at p

0

if E has additive reduction at _p

_V

_{I N).}

We define the

_{imprimitive symmetric}

square L-function of

f

to be:

The

_subscript

N means that we have omitted the Euler factors at the

_primes

dividing

N.

It can be shown that

L(Syml2

f ,

s)

has a

holomorphic

continuation to the whole

_complex

_plane,

and

_by

Rankin’s method that

_(cf.

_[10]):

-1.

This formula allows us to

_{study quadratic}

twists of an

elliptic

curve.

In-deed,

assume that E is the

_quadratic

twist of an

_elliptic

curve E’ with

conductor N’ such that

_ordp(N)

for all

prime

p. We denote

by

X the

underlying

quadratic

character and

by

cond(x)

its conductor. From

classical results about twists of newforms

_(cf.

_[2])

and from the fact that for an odd

_prime

_p the

_p-adic

valuation of

cond(x)

is

_{1 ,}

we can obtain

the

_following.

Let

_{P &#x3E;}

3 be a

_prime

number with _p

cond(x);

Thus,

we can write N = and N’ =

MD22 A

where

D¡

(resp. D2)

is the

_product

of the odd

_primes

p such that p

cond(x)

and p

~’

N’

(resp.

p

cond(x) and pIIN’),

A =

ord2(N’),

k =

ord2(N)

_so that A k and

M,

D1,

D2

are odd. We can now state :

Theorem 1. Assume that E is the

_quadratic

twist

_of

E’ with

conduc-tor N’ such that

_ordp(N)

_for

all _p. Write

_{f’ =}

_{E n ann-S}

=

aP s)-1(1 -

for

the

newform

attached to

E’.

Let

N = and N’ =

MD22~

_as

explained

above. Then:

REMARK : From this

_theorem,

it is _easy to relate

_deg(cp)

with

PROOF: We observe that we have _aP = = and that the Euler

product

(2)

for

_f

and

_f’

are

clearly

related since

;~2

is the trivial

char-acter modulo

_cond(X).

_Furthermore,

this Euler

_product

allows us to

_give

a "local"

proof

of the theorem.

So,

_suppose that E is the twist of E’

by

a character of

_prime

conductor

_{p &#x3E;}

3 with

ordp(N’)

ordp(N)

(if

ordp (N’)

=

ordp(N)

then we have

L(Sym;2

f’, s)

=

f, s)).

We have

N =

lcm(N’,p2)

=

N’p2

(resp.,

=

N’p)

if

(N’, p)

= 1

(resp., (N’, p)

=

p).

p the Euler factor at q of both

f,

s)

and

f’,

s)

are

the same. Since

I

N we have _ap = 0

(~1~),

and there is _no Euler factor

at p in

f, s).

When

_{(N~, P)}

= 1

(i.e.

p

Di)

we have:

The case _p = 2 follows

by

the same

_argument

_except

that there is no

character of conductor 2 and so we have to deal with

_cond(x)

= 4 _or 8.

This also

_explains

_why

some cases cannot

_(and

do

_not)

occur in list of cases

relating

II f II N

and 0

This theorem asserts that we

_only

have to consider

_elliptic

curves E

which are not twists of another curve E’

_having

a lower conductor.

3. The

_primitive

_symmetric

square of E

The

_{imprimitive symmetric}

_square

_L(Syml2

_f,

_s)

does not have a

"tra-ditional" functional

_equation

and there is no

simple

method to

_compute

L(Sym2

f,

2)

directly. Thus,

we consider the

_{primitive symmetric}

_square

L-function of

_E,

_L(Sym2

_f,

_s):

where the

_product

is over the finite set S of bad

primes

where E has bad

but

_{potentially good reduction,}

in other words

_{primes p}

such that

_{p I N}

and

_{ordp(j(E)) &#x3E;}

_0,

_j(E)

_being

the

_j-invariant

of E. The

_properties

of the

primitive symmetric

square function are studied in

[4].

In

_particular,

the

following

is

_proved:

Theorem 2

_{(Coates-Schmidt).}

The

_function

_L(Sym2

_f,

_s)

has a

holomor-phic

continuation to the whole

_{complex plane}

and there exists B E Z such that the

_{completed function:}

is entire and admits the

_{functional equation:}

REMARKS: 1. the Euler factor at _p of the

_primitive

and

imprimi-tive

_symmetric

_square functions of E are the same and we have

ordp(B) =

ordp (N) .

In

_particular,

if N is

_squarefree,

then

_L(Sym2

_f,

_s)

=

L(Syml2

f,

s)

and B = N.

2. The function

_L(Sym2

_f,

_s)

is invariant if we twist E

_by

a

quadratic

character of

_Q.

This is not true in

_general

for the

_{imprimitive symmetric}

square function.

In order to write down the correct Euler factor _{at P ~}

_I

_N2,

we assume

that E is not the

_quadratic

twist of a curve E’ of lower conductor. For the cases _p = 2 and _p =

3,

we have the

_following

tables

_coming

from

_[4]

_(we

should mention that two cases have been

_{initially forgotten}

in

[4]

whenever

2$ ~

_{1 N,}

and that

_[13]

corrects this

_mistake).

When several

_{possibilities}

occur in theses

_tables,

the correct Euler factor is

given by

certain

_properties

of the fields The cases

2411N

and

2611 N

never _appear since we assumed that E is minimal _among its

quadratic

twists.

_{If p ~}

_{2, 3}

then

_ordp(B)

= 1 and

_Lp(Sym2

_f,

_X)

_=1-pX

or 1

_{+ pX}

depending

on whether or not

_Qp(El)/Qp

is abelian.

_{Nevertheless,}

for each

ambiguous

case, one can find in

~13~ 1

the correct Euler factor: first assume

that

_{p &#x3E;}

5. Then we have

Lp(Sym2

f , X )

= 1 -

pX

if and

only

if one of

the

_following

conditions

holds,

where c6 and c4 are the classical invariants

attached to E:

~ _{p =1}

(mod 12);

p ’ 5

(mod 12),

p2

I c6

and

p2 f

C4;

w _p = 7

(mod 12)

and either _c6, or

p2

I C6

and

p2

04.

For _p =

2,

2811N

is the

only

ambiguous

case and:

. if

29

I C6

then

LP(SYM2

f,

X)

=

_1;

~ if

29 f

_c6 and

_{c4 - ~ 32}

(mod 128)

then 1 +

epX,

where c = ±1.

For _p =

3,

3411N

is the

only ambiguous

case and we have

Lp (Sym2

f,

X)

1-

_pX

when one of the two

_following

holds:

o _{c4 -} 27

(mod 81);

. _C4 == 9

(mod 27)

and _{c6 m} ~1Q8

(mod

243).

4.

_Computation

of

_L(Sym2

_{f , s)}

For

_simplicity,

we write:

where C = and

L(Sym2f,s) =

Note that the

coeffi-cients

_bn

are

easily computable

from the definitions.

Furthermore,

it

fol-lows from

_Deligne’s

bounds and the Euler

_product

for

_L(Sym2

_{f ,}

_s)

that

n2.

Classical estimates

_coming

from the functional

_equation

of

1 During

the _preparation of this paper, we were informed of the preprint [13] of M. Watkins where a similar _(but not so _detailed) method of _{computing deg(W)} is described and used to

A(Sym2

f,

s)

give:

This formula

_implies

that the series

_{En bn/n2}

_converges to

L(Sym2 f, 2).

Of course, this is not an efficient method to

_compute

because the convergence is very slow.

However,

it

easily

gives

us a first

approximation

of

Fortunately,

a classical method for

_computing

Dirichlet series with

func-tional

_equation

can be

applied

to our case

(cf.

[5],

Chapter

10):

Proposition

3. We have:

where

,forallb&#x3E;0.

This is a

_rapidly

_convergent

series since we have:

Proposition

4. Let s = (J + it and A

_{= 21/4C}

then:

I

PROOF: We have:

With an _easy but tedious

calculation,

we obtain:

Thus,

we have:

/ 2/3

The

_proposition

is then

_{proved by}

_{taking J =}

’ 0

This

_proposition

allows us to estimate the tail of the series in

(4) (we

have

_~,2).

In order to

_compute

_F(s,

_x),

we

push

the line of

integration

to the left

_catching

all the residues of

_t-’-t(z):

Proposition

5.

with

It is clear that the terms in this

_expression

can be

_{recursively computed.}

We then

_{compute io}

terms in the series of

_{proposition 5,}

where

_io

is the smallest

_integer

such that

_(cf.

_[12]):

We thus obtain

_A(Sym2

_{f ,}

₂₎

with a

sufficiently

_high

_accuracy and we

de-duce from it the value of hence of

_deg(p).

_Using

this

_method,

we can

quickly

_compute

modular

_degrees

of

_strong

Weil curves. As a check on

the

_{computations,}

we use the fact that

_deg(p)

is an

integer.

REMARK. In

_fact,

what is

_really

obtained here is an

algorithm

to

com-pute

L(Sym2

f ,

2)

from which

_{lif 11}

and then

_deg(p)

can be

easily

recovered.

Nevertheless,

the

_quantity

_{ilf 11}

makes sense and is also

interesting

in

_greater

generality,

namely

for _any

_holomorphic

form

_f

of

_integral

_weight

k &#x3E; 2 and level

_N,

not

_necessarily

related to an

_elliptic

curve. In this

general

_case, one can also define

L(Sym2

f ,

s)

the

primitive symmetric

_square L-function

related to the L-function of

_f

and we have:

This L-function does have a traditional functional

_equation

and the

adap-tation of the method above is

_{possible. However,}

in our case

( f

is related

to an

_elliptic

curve),

_computing

the Euler factors involves

_looking

at the

elliptic

curve whenever the reduction is additive

N);

in

general,

such a

study

is not

_possible

and the case of

_{non-squarefree}

N seems not to be _easy. When N is

_squarefree

the

_adaptation

of the method is _very

_simple

since the Euler factors of

_L(Sym2

_f,

_s)

are all

given by

(2)

and the functional

equation

is :

-

-Furthermore,

when N is

_squarefree,

one can

_adapt

the method to

_compute

(conjecturally)

special

values of

_general

_symmetric

_powers

_L(Sym~

_{f ,}

_k)

since

_they

also

_satisfy

a traditional

(and conjectural)

functional

_equation.

5. Some estimates

From the functional

_equation

of

_L(Sym2

_{f , s),}

one can show that

I I f 112

~ë

Nl+c.

In

_fact,

Nt: can be

_{replaced by}

a suitable _power of

log(N).

Thus,

estimates for

_vol(E)

provide

upper bounds on

deg(p) (modulo

Manin’s

Proposition

6. Let C be a

_nonnegative

real number. There exist a E 1I~ and A E R

_depending

on C such that:

where

_{Ami,, is}

the discriminant

_of

the minimal model

_of

E.

PROOF: This

_proposition

comes from a

straightforward

estimate for the

fundamental

_periods

W1 and w2 of

E,

since we have

vol(E)

=

0

Assuming

Manin ’s

_conjecture,

we see that

proposition

6

gives

the

up-per bound « for

_elliptic

curves with bounded

j-invaxiant.

Proposition

7. Let 6 be an

infinite family of elliptic

curves

defined

over

Q

such that:

PROOF: The upper bound comes from the classical estimate

L(Sym2

f,

2)

K

log(N)3

and from the fact that the Manin’s constant is bounded whenever the conductor is

_squaxefree.

The last estimate comes from the lower bound

L(Sym2

f,

2)

»

_{1/ log(N)}

for N

_squarefree

_(cf.

_[8]).

11

The curves

E~

defined

by y2

_{+ xy}

= x3

+ k

_(with

432k2

+ k

_squarefree)

give

a infinite

_family

of

_elliptic

curves for which the conditions in the

propo-sition hold.

The lower bound of the

_proposition

also holds in the more

_general

set-ting

where the condition is

_squarefree"

is

_{replaced by}

"E is semi-stable

_(i.e.

N is

_{squarefree)".}

We wrote a GP-PARI

([9])

_{program for}

_computing

the modular

_degrees

using

the method

_explained

above. In the

_following

table we

_give

three

examples

for which the modular

_degree

is very

large.

In each cases,

deg(cp)

was

computed

in a few minutes. The column

j(an)

indicates the number

The last curve is in fact the

quadratic

twist of the curve

E’

with

coef-ficients

_{[1, 0,1,120229952, -3351306510322]}

of conductor 1290. We need 5000 coefficients an to

compute

deg(p(E’))

= 1068480.

References

[1] A. _ATKIN, J. _LEHNER, Hecke _operators on 03930(m). Math. Ann. 185 (1970), 134-160.

[2] A. _ATKIN, W. LI, Twists of newforms and _{pseudo-eigenvalues of} W-operators. Invent. Math. 48 _(1978), 221-243.

[3] C. BREUIL, B. CONRAD, F. DIAMOND, R. TAYLOR, On the _{modularity of elliptic}

curves over _Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 no. 4 _(2001), 843-939.

(electronic).

[4] J. COATES, C.-G. SCHMIDT, Iwasawa theory for the symmetric square of an _elliptic curve. _{J. reine angew. Math. 375 (1987),} 104-156.

[5] H. COHEN, Advanced _topics in computational algebraic number theory. Graduate Texts in Mathematics, 193, Springer-Verlag, New-York, 2000.

[6] J. CREMONA, Algorithms for modular elliptic curves. Second _edition, Cambridge

Uni-versity Press, 1997.

[7] J. _{CREMONA, Computing} the _{degree of} the modular parametrization of a modular elliptic curve. Math. Comp. 64 _(1995), 1235-1250.

[8] D. GOLDFELD, J. HOFFSTEIN, D. LIEMAN, An _{effective zero-free region.} Ann. of Math.

(2) 140 no. 1 _(1994), 177-181.

[9] C. _BATUT, K. _BELABAS, D. _BERNARDI, H. COHEN, M. _{OLIVIER, pari-gp,} available

by anonymous ftp.

[10] G. _SHIMURA, The _special values _of the zeta _functions associated with _cusp forms.

Com. Pure _Appl. Math. 29 _(1976), 783-804.

[11] R. _TAYLOR, A. _{WILES, Ring-theoretic properties of} certain Hecke algebras. Ann. of Math. ₍₂₎ 141 no. 3 _(1995), 553-572.

[12] E. TOLLIS, Zeros _of Dedekind zeta functions in the critical strip. Math. Comp. 66

(1997), 1295-1321.

[13] M. WATKINS, Computing the modular _{degree of an elliptic} curve. _Experimental Maths

11 no. 4 _(2003), 487-502.

[14] A. _WILES, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. ₍₂₎ 141

no. 3 _(1995), 443-551.

[15] D. ZAGIER, Modular parametrizations of elliptic curves. Canad. Math. Bull. 28

no. 3 _(1985), 372-384.

Christophe DELAUNAY

Universite Bordeaux I, Laboratoire A2X 351 Cours de la Liberation

33 405 Talence, France

E-mail : _{delaunayGmath.u-bordeaux.fr}

et Ecole _{Polytechnique} Fédérale de Lausanne CH- 1015 Lausanne, Suisse