Arithmetic of elliptic curves and diophantine equations

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

L

OÏC

M

EREL

Arithmetic of elliptic curves and diophantine equations

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{1 (1999),}

p. 173-200

<http://www.numdam.org/item?id=JTNB_1999__11_1_173_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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

Arithmetic of

_elliptic

curves

and

diophantine

equations

par Loïc MEREL

RÉSUMÉ. Nous décrivons un _panorama des méthodes reliant l’étu-de l’étu-des _équations

_{diophantiennes}

ternaires aux

_techniques

mo-dernes issues de la théorie des formes modulaires.

ABSTRACT. We

_give

a _survey of methods used to connect the

study

of _ternary

_diophantine

equations to modern

_techniques

com-ing

from the

_theory

of modular forms.

INTRODUCTION AND BACKGROUND

In

_1952,

P.

_Denes,

from

_Budapest

_1,

_conjectured

that three non-zero

distinct n-th powers can not be in arithmetic

_progression

when n &#x3E; 2

_[15],

i. e. that the

_equation

has no solution in

_{integers x,}

_{y, z,} n

_~y,

and n &#x3E; 2. One cannot

fail to notice that it is a variant of the Fermat-Wiles theorem. We would

like to

_present

the ideas which led H. Darmon and the author to the solution of Dénes’

_problem

in

_[13].

_Many

of them are those

(due

to Y.

_{Hellegouarch,}

G.

_Frey,

J.-P.

_Serre,

B.

_Mazur,

K.

_Ribet,

A.

_Wiles,

R.

_Taylor,

_...)

which led

to the celebrated

_proof

of Fermat’s last theorem. Others

_originate

in earlier work of Darmon

_{(and Ribet).}

The

_proof

of Fermat’s last theorem can be understood as an advance in

the direction of the abc

_conjecture

of D. Masser and J. Oesterlé. We view

the solution to Denes’

_conjecture

as a modest further _step. We would like

to

_explain

the additional

_techniques

involved into this solution

_(and

_try to

avoid too much

_overlap

with the numerous other surveys on

closely

related

topics) .

1 Ce texte est la version 6crite de 1’expose donne lors du deuxième _colloque de la societe

mathématique europeenne a _Budapest. A la _{grande surprise} de _1’auteur, il ne _figurait pas dans les volumes _publics a cette occasion par Birkhafser Verlag, lorsque ceux-ci sont apparus sur les rayons des librairies. Suite a mon _expos6 sur le meme theme aux Journées _{Arithm6tiques,} les

6diteurs du pr6sent volume ont _accept6 de publier 1’article en _d6pit de de la nature _{plus sp6cialis6e}

A current

_approach

to

_{diophantine problems}

can be

_roughly

described as

follows. One is interested in

_problems

of the

_following

_type.

I. Determination of the solutions to a

_{diophantine equation}

of the form a+b+c=0.

Using

a

_machinery

_proposed

_by

_{Hellegouarch, Frey}

and

_Serre,

and estab-lished

_by

theorems of

_Mazur,

Ribet and Wiles

_(and

later refinements

_by

F.

Diamond and K.

_Kramer)

the

_problem

I

_might

be reduced to

_problems

in

the

_following

theme.

II. The action of

_Gal(Q/Q)

on a few torsion

_points

of an

elliptic

curve

over Q

characterizes the

_isogeny

class of the curve. or

_alternately

II’. same

problem

restricted to a certain class of

elliptic

curves

(see

section

1.1)

over Q

often called

_Frey

curves.

The

_problem

II and II’ can be reformulated in terms of inexistence of

rational

_points

on modular curves

(or

_generalized

modular

objects).

This

inexistence can sometimes be established

_using

some

algebraic

_geometry

and,

roughly speaking,

III. a

diophantine

_argument

_coming

from the

study

of the Galois

coho-mology

of some modular

_object.

We illustrate this _process

_{by examples}

_arranged

in

_increasing

order of

difficulty.

Fermat’s last Theorem.

According

to our

_picture,

the

proof

of Wiles’ theorem

decomposes

as

follows. Fermat’s Last Theorem is our

_problem

of

_type

I:

Theorem 0.1

_(Wiles).

The

_equation

xP +

_yP

= zP has _no solution in

gers x, y and z with _p

_prime

number &#x3E; 3 and

xyz =1=

0. The modular

_machinery

reduces it to a

problem

of

_type

II:

Theorem 0.2

_(Mazur).

An

_elliptic

curve

over Q

has no

Q-rational

sub-group

of prime

order p &#x3E; 163.

Weaker versions of

_type

II’ of the

_previous

theorem are sufficient for the

application

to Fermat’s Last Theorem:

Theorem 0.3

_(Mazur).

A

_Frey

curve has no

Q-rational

subgroup. of

prime

order _p &#x3E; 3

_(the

inexistence

_{of Q-rational points of prime}

order _p &#x3E; 3 on

The

_previous

two theorems of Mazur

_rely

in an essential _way on the

following

result of

_type

III:

Theorem 0.4

_(Mazur).

The Eisenstein

_quotient

_of

the

_{Jacobian of}

the mo-dular curve

Xo(p)

has

_,finitely

_many

_Q-rational

_points.

D6nes’

_equation.

D6nes’

_conjecture

has been

_proved

for n = 4

by

Euler,

for n = 3

by

Legendre,

for n = 5

by Dirichlet,

and for _n =

p

prime

with p 31

by

Denes

himself

_using

the methods of Kummer.

The

_problem

was reconsidered

by

Ribet

[63]

recently

in the

light

of the

proof

of Fermat’s last theorem: D6nes’

_equation

can not have _any non-trivial solution when n =

p is a

prime

_congruent

to 1 modulo 4

_{(see below).}

Darmon and the author

_proved

the

_following

theorem of

_type

I.

Theorem 0.5. The

_equation

xP +

_yP

= 2zP has _no solution in

_integers

x, y and z with p

prime

2,

and x 0

±y.

Using

the modular

_machinery

we reduced first the

problem

to a

_positice

answer to the

_{following problem}

of

_type

II.

Problem 0.6

_(Serre).

Does there exists a numbers B &#x3E; 0 such that

_for

every every

elliptic

curve E

over Q

without

complex multiplication

and every

prime

number _p &#x3E; B _any

_automorphism

_of

the _group

_of

_points

_of

_p-division

of

E is

_{given by}

the action

_of

an elements

of

Gal(Q/Q)?

We

_proved

a result of

_type

II’ in the direction of a

_positive

answer to that

question.

This result is sufficient for the

_application

to D6nes’

_conjecture.

Theorem 0.7. For _every

_Frey

curve E

_except

the one which has

_complex

rraultiplication

and every

prime

number p &#x3E; 3 _any

_{automorphism of}

the group

of

points

of

p-division

of

E is

given

by

the action

of

an element

of

Gal(Q/Q)

-We obtain this result as a _consequence of a theorem of

_type

III,

which is

a

special

case of the

_conjecture

of Birch and

Swinnerton-Dyer

(see

section

2.3.).

Theorem 0.8

_(Kolyvagin

and

_others).

_{Any quotients}

_defined

_of

the

jacobian

of

a modular curve whose

L-function

does not vanish at the

point

1 has

_finitely

_many

_{Q-rational points.}

Concerning

Serre’s

_problem

in full

_generality

we can offer

_only

wishful

thinking

for reasons

_explained

in section 2.4: One

might hope

to use as

argument

III more results in the direction of the

conjecture

of Birch and

Swinnerton-Dyer

such as the

Gross-Zagier

formula and the theorem of

The Generalized Fermat

_equation.

Fermat’s Last Theorem and Dénes’

_conjecture

are

essentially

_special

cases

of the

_following

_problem

of

_type

II.

Conjecture

0.9. Let a, b and c be three non-zero

_integers.

One has the

equality

axn +

_byn

+ = 0

for only finitely

many values

of

yn,

z’~)

where _{x, y, z,} and n &#x3E; 3 are

_integers

and _{x, y,} and z are

_coprime.

See

_[26]

for more

_background

on this

_problem.

Observe that in the case

where a + b + c =

0,

the

equation

has solution in

_{(x, y, z)}

for _any

_exponent

n. This

_difficulty

_appears

_already

in Dénes’

_equation.

Mazur

_proved

that

_conjecture

0.9 holds when a = b = 1 and _c is _a power

of an odd

_prime

number I which is not a Fermat

_prime

or a Mersenne

_prime

[66].

A. Kraus seems to have found

_{recently explicit}

bounds for the solutions

in terms of 1

_[40].

Frey proved

that the

_conjecture

0.9 is a _consequence of a

_conjecture

of

type

II.

_(Questions

of this

type

were raised

by

Mazur in

[50].)

Conjecture

0.10

_(Frey).

Let E be an

elliptic

curve over

Q.

There exists.

finitely

many

pairs

(E’, p)

where E’ is an

_elliptic

curve

over Q

and _p a

prime

number &#x3E; 5 such that the sets

_of

_{p-division points}

_of

E and E’ are

isomorphic as

Gal (Q/Q) -modules.

Frey

even shows

([24],

[25])

that the

_conjecture

0.9 is

_equivalent

to the

restriction to

_Frey

curves of the

_conjecture

0.10.

Darmon

_conjectures

that there exist

_{only finitely}

_many

_triples

_{(E, E’, p)}

as in

_conjecture

0.10 with E and E’

_{non-isogenous}

and _p &#x3E; 5.

No

_argument

of

_type

III seems to be available to solve the

_conjecture

0.10. One can

only hope

that the Birch and

Swinnerton-Dyer

conjecture

is

still relevant to the

_study

of the arithmetic of twisted modular curves and

their

_jacobian.

The Generalized Fermat-Catalan

_equation.

Here is another well-known

_{generalization}

of Fermat’s

_equation.

Conjecture

0.11.

_{Let a,}

b and c be three non-zero

_integers.

One has the

equality

axr +

_bys

+

czt

= 0

for

only

finitely

_many values

of

(xr,

ys, zt)

with

x, y, z, r, s, and t

integers,

x, y, z

coprime,

xyz =,4

0,

and 1 + s + t

1. The abc

_conjecture

_{easily implies}

_conjecture

0.11. For a = b = _-c =

1,

the

_conjecture

0.11 is sometimes called the Fermat-Catalan

_conjecture

since it combines Fermat’s theorem with the Catalan

_conjecture;

The ten known

triple

(xr,

ys,

zt)

which

_satisfy

the

_equality

xr +

ys

=

zt

_are listed in

[3].

In the direction of the Generalized Fermat-Catalan

conjecture,

we have the

following

two results. The first is obtained as an

application

of

Faltings’

Theorem 0.12

_{(Darmon, Granville).}

Let

_{r, s and t}

be three

_positive

inte-gers such

that -1 + ~ + 1

1. Let _a,

_b,

c be three non-zero

_integers.

Then

the

_equation

axr +

_bys

=

czt

has

finitely

many solutions in

coprime integers

x, y and z.

Moreover the

_techniques

_developped

_by

Darmon and the author to

_study

Denes’

_conjecture

combined with earlier work

_by

Darmon

_[8]

led us to the

following

[13]:

Theorem

_{0.13. 1)}

Let n be an

_{integer &#x3E;}

3. The

equation xn + yn

= z2

has no solution in

_coprime

and non-zero

_integers

_{x, y,} z.

2)

Suppose

that _every

_elliptic

curve

over Q

is modular. Let n be an

integer &#x3E;

2. Then the

_equation

xn

_{+ yn}

=

Z3

has _no solution in

coprime

and non-zero

_integers

_{x, y,} z.

The

_study

of these

_equations

has some

_history.

For instance the case

n = 4 in the first

_equation

_was solved

by

Fermat and the _{case n} = 3 in

the second

_equation

was solved

by

Euler. We relied on work of B. Poonen

to solve these

_{equations by elementary}

methods for a few small values of n

[62].

There is a considerable

_bibliography

on the

subject

of

diophantine

equa-tions of this

_type.

We are _very far from

being

complete

here. The interested

reader

_might

consult

_{[10], [12], [18],}

and

_[53].

The abc

_conjecture.

We recall its statement for the convenience of the reader

_[59].

Conjecture

0.14

_{(Masser-Oesterle).}

For _every real number E &#x3E; 0 there

exists a real

_KE

&#x3E; 0 such that

_for

every

triple

of

coprime

non-zero

_integers

(a,

b,

c)

satisfying

a + b + c = 0 _we have the

following

inequaLity:

ibl, icl)

where

_Rad(n)

_for

_any

_{integer n}

is the

_product

_of

the

_prime

numbers

_dividing

n.

The abc

_conjecture

illustrates

_(and

_maybe

_{concentrates)}

the

_difficulty

of

understanding

how the additive and

_{multiplicative}

structures of the

_integers

relate.

Its resolution would affect our

understanding

of _many

problems

in number

theory

([51],

[58], [76]).

As

_explained

in

_[25]

_(see

also

_{[44], [73]}

and

_[74]),

it is now a _consequence of the

_following

_{degree conjectures,}

which is a

_problem

of

_type

II:

Conjecture

0.15. Let E be a modielar

elliptic

curve

_{over Q}

_of

conductor

N. Let

6E

be the

_degree

_{of the}

_{corresponding}

minirraal modular

independent of

E such that one has

Remark that this is no

longer

a

uniformity

statement since the

_degree

6E

is

_expected

to be bounded

_by

a number

depending

on the conductor of

E. This

_problem

is

_apparently

related to the existence of rational

_points

on

certain Hilbert modular varieties.

This _paper is

_organized

to be accessible to a wide audience in its first

part

which

_explains

the _process of

_going

from

_problem

I to

_problem

II. The second

_part

_might

contain results of interest to the

_specialist

_(see

sections

2.4 and

_2.5)

and deals

_mainly

with Serre’s

_problem.

In the third _part we

explain

how the

_degree

_conjecture

can be studied from an

elementary

_point

of view.

Acknowledgement.

I would like to thank H.

_Darmon,

B.

_Edixhoven,

I.

_Chen,

and K. Ribet for comments useful in the course of

_writing

this _paper

_during

my

stay

at the

_University

of

_California,

_Berkeley.

1. THE MODULAR MACHINERY

1.1. The curves of

_Frey

and

_{Hellegouarch.}

In order to

_study

_equations

of the

_type

a + b + c =

0,

with

a, b

and _c

coprime

_non-zero

integers,

Frey,

following

earlier work

_by

_Hellegouarch

about Fermat’s

_equation

_{(~30~, [31]),}

proposed

to consider the

_elliptic

curve

_Ea,b,c

over Q given

_by

the cubic

equation

We can assume without loss of

generality

that b is even and that a is

congruent

to -1 modulo 4. We call

_elliptic

curves

_{given by}

cubic

_equations

of this form

_Frey

curves.

Elementary

facts

_concerning

them can be found in

various

_places

of the literature

_([22],

_{~23~, [67], [59]).}

The modular invariant

of

_Ea,b,c

is

_given

_by

the formula

In is not

_integral

_{(and Ea,b,c}

has no

complex

multiplica-tion)

except in the case

(a, b, c) = (-1, 2, -1).

All the 2-division

_points

of

_Ea,b,c

are

Q-rational.

The exact formula for the conductor

_Na,b,c

of

_Ea,b,c

has been calculated

_by

Diamond and Kramer

[17].

One has

Na,b,c = Rad(abc)EZ(b),

where

_{E2 (b)}

= 1 if

32 ( b, E2 (b)

= 2

if

_{16 ~ b}

and 32

_{E2 ( b)}

= 4 if

41b

and 16

_{¡fb, E2 (b) -}

16 if 4

_lb.

In

_particular

_Ea,b,c

is a semi-stable

_elliptic

curve if and

_only

if

_{16 ( b.}

1.2. Wiles’ theorem. The work of Wiles and the notion of modular

el-liptic

curve has been

_explained

in _many

_expository

articles

_recently.

For a different

_perspective

we

explain

the

meaning

of

_modularity

for an

_elliptic

curve in a non-standard _way. The

following

formulation has the

_advantage

of

_{bringing directly}

to mind the

_{quadratic reciprocity}

law of Gauss

_(the

function ~ below is

_analogous

to the

_Legendre

_symbol).

This

_point

of view

was introduced

_by

Y. Manin

_(and

_maybe

B.

_Birch)

more than

_twenty

_years

ago

~45~, 1461&#x3E; [52].

Let E be an

elliptic

curve over

Q.

Let 1 be a

prime

number. Let

IE(IFl)1 I

be the number of

_points

of the reduction modulo 1 of a minimal Weierstrass model of E. Let

_al(E)

= 1 +

_{1 -IE(IFl)l.}

Let

AN

be the set of functions

satisfying

for _any

_{(u, v)}

E the

_equations

_(the

Manin

_relations)

Let

_AE

be the set of elements ~ of

AN

satisfying

for _every

_prime

number 1 an

_equation

which can take several

_forms,

the most

_simple

that I know

being

the

_{ai(E)-modular}

relations

_(we

abuse the notations here since the relations refers to two numbers: al and

l):

where

_(a,

_b,

_c,

_d)

E

Z4.

The curve E is said to be modular

_of

level N if there is a non-zero element

in

_{AE .}

_Any

element

of AE

must be

_homogeneous,

i. e. it satisfies the

_equality

Remark. The existence of a non-zero element of

AE

is

equivalent

to the

existence of a normalized

eigenform

of

weight

2 for

rl(N’)

whose

1-th Fourier coefficient is hence the connection with the standard

notion of

_modularity

_(see

_[55]).

This can be

_{proved by}

_remarking

that

there is a

_bijection

between the set of elements of

_AN

which vanish on

non-primitive

elements of and The functions

of

_N

_satisfying

the

_{al(E)-modular}

relations

_correspond

_bijectively

to the

elements of

_HI

_(Yl

_{(N) (C), Z)}

which are

eigenvalue

of the Hecke _operator

7l

with the

_eigenvalue

_{ai (E)}

see

~55~.

The

_concept

of

_modularity

is

_captured

in the last

_family

of

_equations.

It is the

_expression

of a law

ruling

all the reductions modulo 1 of the

elliptic

curve E. Remark that the

_{integers a, b,}

c, d involved in the sum

_depend

on

The curve E is said to be modular if it is modular of some level. If E

is a modular

_elliptic

_curve, then a theorem of H.

Carayol

([5])

implies

that

it is modular of level the conductor of E. The set of functions 4D which establish the

_modularity

at that level is a free Z-module of rank 2. The

recent work of Wiles

_[77]

_(completed

_by

a

_joint

work with

_Taylor

_[75]

and

subsequent

work of Diamond and Kramer

_[17])

leads

_amongst

other

_things

to the

_following

_result,

as

_explained

in

[17]

(K.

Rubin and A.

_Silverberg

also remarked in

_[65]

that the next theorem can be deduced from another work

of Diamond

_[16]).

Theorem 1.1

_(Wiles,

_{Taylor-Wiles,}

_{Diamond-Kramer).}

The

_elliptic

curve

Ea,b,c

is modular.

1.3. Serre’s

_conjectures.

Let p be a

_prime

number. Consider an

irre-ducible Galois

_{representation}

which is

_odd,

i.e. the determinant of the

_image

of a

complex

_conjugation

is

-1.

There is a notion of

modularity

for such a

_{representation.}

For _every

prime

number 1

_~

_{p at} which the

_{representation}

_{p is}

_unramified,

denote

by

al(p) (resp. bl (p))

the trace

_(resp.

the

_determinant)

of the

_image

in

GL2(JFp)

of a Frobenius

_endomorphism

at the

_prime

l. Once

_again

we

_give

a non-standard formulation of

_modularity

as above.

Let

_AN,p

be the set of functions

satisfying

for _any

_{(u, v)}

E

(Z /NZ)

the Manin relations

Let

_Ap

be the set of elements of

_AN,p

_satisfying

for _every

_prime

number 1 the

_{al(p)-modular}

relations:

where a, b,

c and d are

_integers,

and

We _say that the

_{representation}

_p is modular

_of

level

_N,

if there exists a

non-zero function in

_Ap.

The

_{representation}

_p is said to be modular if it is modular of some level.

.Remark. This definition of

_modularity

can be shown to be

_equivalent

to the

one used

_{originally by}

Serre. One can first reformulate Serre’s

_conjecture

using

modular forms of

_weight

2

_only

_{by using}

results of Ash and Stevens

[2].

Making

use of the

_{interpretation}

of functions

satisfying

the Manin

relations in terms of the

_homology

of modular curves as we did in the

previous

section

_{( i. e.}

one shows that the existence of an element of

Ap

is

equivalent

to the existence of a non-zero modular

_{form f}

of

_weight

2 for

r1(N)

in characteristic

_p

which satisfies =

(l

prime

~’Np,

Tl

is

the 1-th Hecke

_operator).

Take note that we allow the

prime

_p to divide N here.

Serre

_conjectures

that any

odd,

irreducible

representation

is modular.

Moreover he

_predicts

the minimal level of

_modularity

of such a

representa-tion

_[67].

Let E be an

_elliptic

curve over

Q.

The set

_E[p]

of

_Q-rational

_p-division

points

of E is a

_Fp-vector

_space of dimension 2. Let us

choose,

once and

for

_all,

an identification between this set and

_~.

This defines a Galois

representation

pE,p of the

type

considered above. The coefficients

_al(PE,p)

and are the reductions modulo _p of and 1

respectively.

If the

_elliptic

curve E is modular of level N then the

_{representation}

_pE,p is modular of level

_N,

as one can see

_by

_reducing

a function (D attached to

E modulo _p.

_By

_choosing

an

appropriate multiple

one can insure that

the reduction is non-zero. Therefore is

_homogeneous.

But it

_might

occur

that pE,p is modular of level

_strictly

smaller than N. This is for instance

the case if the reduction modulo _p of 4D is non-zero and factorizes

through

where N’ is

_strictly

smaller than N.

1.4. The theorems of Ribet and Mazur.

_Suppose

that _p &#x3E; 2. For our

purpose we need

only

the

following

_recipe

predicted

by

Serre and

proved by

Ribet for the level of

_modularity

of the

_{representation}

_{pa,b,c,[p] =}

For any

integer

n &#x3E;

_0,

denote

_by

_np the

_largest

_p-th

_power

dividing n

and let =

Rad(4/np).

Let =

Radp(abc)E2(b).

Theorem 1.2

_(Ribet).

_Suppose

that _p &#x3E; 2 and that the

_{representation}

modular and

_absolutely

irreducible. Then zs modular

_of

level

At this

_point

one needs the

_following

theorem of B. Mazur. We do not

insist on it for the moment since results of this

_type

will be the

_subject

of the second

_part

of the paper.

Theorem 1.3

_(Mazur).

Let _p be a

_prime

number,. Let E be an

elliptic

curve

over Q

having

all its 2-division

points

defined

over

Q.

Then _PE,p is

This theorem

_implies

that we can not have al

(Pa,b,c,(p])

=1 + l for almost all

_{prime 1}

_{when p} &#x3E; 3.

_(If

= 1 + 1 for almost all

_{primes l,}

then

1 is an

_eigenvalue

of almost _every

_image

of a Frobenius

_{endomorphism.}

By

Chebotarev’s

_{density theorem,}

the number 1 is an

_eigenvalue

of _every

element in the

_image

of

_Pa,b,c,[p].

This

_implies

that the

_{representation}

_Pa,b,c,[p]

is

_reducible.)

1.5. Denes’

_conjecture.

We

_apply

now the

machinery

described above to D6nes’

_equation.

We

_merely

_repeat

in essence here what is contained in

[13], [64],

and

_[67].

In view of the results mentioned in the

_background,

we need

_only

_{to prove}

D6nes’

_conjecture

for

_prime

_{exponents &#x3E;}

31. Let us consider a solution

(x, y, z)

to the

_{equation xP}

= 2zP

(p

prime

number &#x3E;

31,

xyz =1=

0).

We _{may suppose} that _{x, y,} and z are

_coprime

and that x is

_congruent

to

-1 modulo 4. We want to show that we have x =

y = z = 1.

Proposition

1.4.

_Suppose

that _{p is} a

_{prime &#x3E;}

3. The Galois

representa-lion

_{PxP,-2zP,yP,[p] ZS}

not

_surjective.

Let us consider the

_Frey

curve

If z is even, then

3212zP

and the Galois

representation

associated to

p-division

_points

of the

_Frey

curve is modular of level 2. The _space of ho-mogeneous functions

(Z/32Z)2

--~

_Fp

satisfying

the Manin relations is

of small dimension. All its elements

_satisfy

modular

_equations

with

= l ₊ 1

(l

prime

~’2p) .

This is

impossible by

Mazur’s theorem.

If z is

_odd,

the Galois

_{representation}

associated to

_points

_{of p-division}

is

modular of level 32

_by

Ribet’s theorem.

One can check

_{by elementary}

but tedious

_computations

of linear

_algebra

that the

_IFp-vector

_space of

_homogeneous

functions of level 32

_satisfying

the Manin relations and the

_{al (p)}

modular relation with

_{al (p) =1=}

t+1

_(for

at least

a

prime 1

)’2p)

is of dimension 2.

Using

the fact that

_P-1,2,-l,[P]

is modular of level

_32,

one obtains that the function

_4)p

attached to

_{PxP,-2zP,yP,(P]}

is

pro-portional

to one attached to

_P-1,2,-,,[p]

and therefore that

_{al(P-1,2,-I,Ipl) =}

al (PxP ,-2zP ,yP ,(P]).

This

_implies

_by

Chebotarev’s

_density

theorem and Schur’s lemma that the

_{representations}

and

_{p_1 ~,_i,pj}

are

_isomorphic.

Note that the

_Frey

curve

_E-1,2,-l

corresponds

to the trivial solution D6nes’

_equation.

It is an

elliptic

curve with

_complex

multiplication

by

the

_{( i. e.}

its

_ring

of

_endomorphism

is

_isomorphic

to

The

_theory

of

_{complex multiplication}

informs us that the

_image

of

P-1,2,-l,[p]

(and

therefore of is

strictly

smaller than

GL2(IFp).

Specifically

it is contained in the normalizer of a

split

(resp. non-split)

Cartan

_subgroup

of

_{if p}

is

_congruent

to 1

_{(resp. -1)}

modulo 4

We now turn to the second

_part

of our _paper to

_explain

_why

the

_image

of a Galois

_{representation}

of the

_type

is

_surjective

when

(a, b, c) ~

(-l, 2, -1)

and p &#x3E; 3.

2. RATIONAL POINTS ON MODULAR CURVES

2.1. Serre’s

_problem.

In

_[66],

Serre

_proved

the

_following

theorem. Theorem 2.1

_(Serre).

Let E be an

_elliptic

curve

over Q

which does not

have

_complexe

_{multiplication.}

There exists a number

BE

such that

for

_every

prime

numbers _p &#x3E;

_BE

the _map

is

_surjective.

The interested reader

_might

consult

_[68]

to see this theorem _put in a

larger

theoretical context. Serre

_proposed

the

_following

_problem:

Can the number

_BE

be chosen

_uniformly

_(i.e.

_{independently of}

_E)?

This

_problem

is still unsolved. The smallest

_possible

candidate for a

uniform

_BE,

_{up to} the current

_knowledge,

is 37

_~50~.

There are formulas for

_BE

in terms of various invariants of E

_[47].

One

might

try

to find an

_expression

for

BE

in terms of the conductor of E.

Building

on

[70],

A. Kraus

_proved

the

_following

result

[40]:

Theorem 2.2

_(Kraus).

Let E be a modular

elliptic

curve without

complex

multiplication.

Then the

_{representation}

_PE,p is

_surjective

when one has

where

NE

is the

_{product of}

the

_prime

numbers

_dividing

the conductor

_of

E.

Note that one can

_{get stronger}

bounds

_assuming

the Generalized Rie-mann

Hypothesis

[70].

In the case of a

Frey

curve is

_equal

to

_{~2(&#x26;)Rad(a&#x26;c).}

Kraus’ theorem

_{gives directly}

an estimate

(left

to the

reader)

for the size of

_hypothetic

solutions to D6nes’

_equation.

To determine the

_image

_{of pE,p,} we have to take into account that its

determinant is

_Pp.

To show that is

_surjective

it is

_enough

to show that its

_image

is not contained in any of the

_following

proper maximal

subgroups

of

_[66]:

- A Borel

subgroup,

i. e. up to

_conjugacy

the _{group of upper}

_triangular

matrices.

- The normalizer of

a

_split

Cartan

_subgroup

of i. e. _{up to}

conjugacy

the set of

_diagonal

or

antidiagonal

matrices.

- The normalizer of

a

_non-split

Cartan

_subgroup

of

GL2(Fp),

i. e. _up to

conjugacy

the normalizer of the

_image

of an

embedding

of

_~2

into

GL2(Fp)

- An

exceptional

subgroup,

i. e. a

subgroup

of

GL2(Fp)

whose

_image

in

_PGL2(Fp)

is

_isomorphic

to the

_symmetric

_group

_S4

or to one of the

alternate _groups

_A4

or

A5.

The two latter cases can not occur because of

the

_surjectivity

of the determinant. The case of

_S4

can occur

only if p

is

congruent

to ~3 modulo 8.

Serre showed that the

_image

_{of PE,p} can not be contained in an

excep-tional

_subgroup

when _p &#x3E; 13 or _p = 7

[66].

2.2. Mazur’s theorems. Serre’s

_problem

can be translated in terms of rational

_points

on modular _curves, i. e. in

diophantine

terms.

Let N be an

_{integer &#x3E;}

0. Let

Y(N)

be the curve which is the moduli

space of

pairs

0),

where S is a

Q-scheme,

E/S

is an

_elliptic

curve

and 0

is an

_isomorphism

of _group scheme between and the _group

E[N]

of N-division

_points

of E. It is a curve defined

over Q

endowed

with an action of

GL2(Z/NZ)

defined over

Q.

Let K be a

_subgroup

of

GL2(Z/NZ).

Let

_YK

=

Y(N)/K.

This curve classifies

_elliptic

curves over

Q

up to

Q-isomorphism

such that the

image

of the _map

is contained in K.

A

_positive

answer to Serre’s

_problem

is

_equivalent

to show that none of

these curves has a

Q-rational

_point

which does not

_correspond

to an

_elliptic

curve with

_{complex multiplication}

when N = _p is _a

large enough

_prime

number and K is a _proper

_subgroup

of

GL2(Z/pZ).

We describe

_briefly

Mazur’s

_approach

to Serre’s

_problem.

Let

XK

be the

complete

modular curve attached to the

_{subgroup K}

of Let

JK

be the

_{jacobian variety}

of

XK.

For any cusp

Q

of

XK,

denote

by

kQ

C

_Q(J-tN)

its field of definition and

by

jQ

the

_morphism

XK

-

_JK

which sends P to the class of the divisor

(P) - (Q).

Let 0 = Let I be _a

prime

number which does not

divides N. Let A be a

_prime

ideal of 0 C above 1. Let

_Ox

be the

completion

of C~ at A.

There is a canonical smooth model

_XK

of

_XK

over C~

[14].

Let A be an

optimal quotient

of

_JK,

i. e. an abelian

_variety

defined

over Q

such that

there exists a

surjective homomorphism

of abelian varieties

over Q

JK

--7 A

with connected kernel. Let

_jQ,A

be the

_morphism

XK -&#x3E;

A obtained

_by

composing

the

_morphism

_JK

- A with

_jQ.

_By

universal

_property

of

Neron model

_jQ,A

extends to a

_morphism

XK ooth

- A defined over L~.

Proposition

2.3. Let 1 be a

prime

number

¡f2N.

Let A be a

_prime

_of

C~

dividing

l.

_Suppose

that there exists an abelian

variety

A

defined over Q

such that

1)

A is a non-trivial

_{optimal quotients}

_{of JK}

with

_finitely

_many

_Q-rational

2)

The

_morphism

_jQ,A

is a

formal

immersion at

Q

in characteristic l

for

each cusp

Q of

XK.

Then there is no

_elliptic

curve E

over Q

such that 1 divides the numerator

of

j(E)

and such that the

_image

_of

is contained in K.

Proof. Suppose

that the conclusion of the theorem is false. Then the modu-lar curve

XK

has a

Q-rational

_point

P which is not a _cusp. Let us _prove that

jQ,A(P)

is a torsion

_point

of A for _{any cusp}

_Q.

Let D be a

_Q-rational

divi-sor of

_{degree n}

&#x3E; 0 with

_support

on the cusps of

XK

(There

exists

some).

By

the Drinfeld-Manin theorem

_[19]

the class x of the divisor D -

_n(Q)

of

degree

0 is torsion in

_JK.

The

_point

_{njQ(P) - x}

of

_JK

is

_Q-rational.

Its

image

in A is of finite order since

_A(Q)

is finite. Therefore

_njQ,A(P)

is of finite order. Therefore

_{jQ A(P)}

is a torsion

_point

of A for _{any cusp}

_Q

of

XK.

Since 1 divides the denominator

_{of j(E),}

the sections

_Spec(9

_XK

defined

_by

P and some _cusp

Q

coincide at

_A,

for some

_prime

divisor A of 1.

This

_implies

that

_{jQ,A (P)}

crosses

_jQ,A

= 0 in the

special

fiber at A of A.

Since 1 &#x3E; 2 and since 1 is unramified in

_{kQ (because 1 /N)}

and since

_JK

(and

therefore

_A)

has

_good

reduction

_{at l,}

a standard

_{specialization}

lemma can be

applied

to show that

_jQ,A(P)

= 0

(see

~50~).

Therefore we have

jQ,A(P)

=

jQ,A(Q)

= 0 and the sections defined

by

P

and

_Q

coincide in characteristic I. This contradicts the fact that

_jQ,A

is a formal immersion in characteristic 1 at the _cusp

_Q.

Remarks.

₁₎

Mazur’s method _{can, to} a certain

_extent,

be

_generalized

to

algebraic

number fields of

_{higher degree}

_{([1], [34], [54]).}

2)

Mazur showed how the formal immersion

_property

can often be checked

using

the

_theory

of Hecke

_operators

and

_q-expansions

of modular forms

_[50].

3)

Central to the method is the existence of a non-trivial

_quotient

abelian

variety

of the

_jacobian

with

_finitely

_many rational

_points.

In

_[49],

Mazur introduced the Eisenstein

quotient

of

_Jo(p)

and

_proved

the finiteness of the group of rational

points

of this

quotient

by

a method which made use of

the

_{semi-stability}

of

_Jo(p).

Mazur’s construction and

_proof

have not been

extended to non semi-stable

jacobians

of modular curves

(see

[27]

for an

attempt).

The Eisenstein

_quotient

was used in

[50]

as the

_auxiliary

abelian

variety

A _{to prove} the

_following

result in the direction of Serre’s

_problem

(from

which one deduces the theorem

1.3).

Theorem 2.4

_(Mazur).

Let _p be a

prime

number. Let E be an

_elliptic

curve

over Q

such that the

_image

of

contained in a Bored

subgroup

of

GL2(Fp).

Then one has

_p

163.

If

one _supposes that all the

_{points of}

2-division

_of

E are

rational,

then one has _p 3.

By

the same

_method,

and still

_using

the Eisenstein

_quotient,

F. Momose

Theorem 2.5

_(Momose).

Let _p be an odd

_prime

number. Let E be an

elliptic

curve such

that j(E) ~

Z[ 1

]

and such that the

image of

is contained in the normalizer

_of

a

split

Cartan

subgroup

of GL2(Fp).

Then

one has

2.3.

_{Winding quotients.}

The

_conjecture

of Birch and

_{Swinnerton-Dyer}

provides

a criterion to decide whether an abelian

_{variety over Q}

has

_finitely

many rational

points:

one has to check whether the L-function of the abelian

variety

vanishes at the

_point

1.

_By

a

_{winding quotient,}

we mean a maximal

quotient

abelian

_variety

of the

_jacobian

of a modular curve

(or

alternately

of the new

_part

of the

jacobian)

whose L-function does not vanish at the

point

1. This

_terminology

has its

_origin

in

_[48]

and

_~49~.

Let N be an

integer &#x3E;

0. Let

J1

(N)

be the

jacobian

of the modular curve

Xl(N).

Let T be the

_subring

of

_End(Ji(N))

_generated

_by

the Hecke

opera-tor

_Ti

It

_operates

on the

new-quotient

JîeW(N)

of

J¡(N).

Let I be

an ideal of T. Let

_JI

be the

_quotient

abelian

_variety

A classical theorem of

_{Eichler, Shimura, Igusa}

and

_Carayol

_asserts, after an

easy

reformulation,

that

where

_f

runs

_through

the newforms

(i.e.

the normalized

_eigenforms

of

weight

2 which are new for

rl(N)),

and where

In the direction of the

_conjecture

of Birch and

_{Swinnerton-Dyer,}

K. Kato

has announced

_that,

_amongst

other

_things,

he has obtained the

_following

theorem

_by

an

original

method

[36],

[37], [38].

Theorem 2.6

_(Kato).

_Suppose

that

_L(JI,

_{1) ~}

_0,

then the _group

_of

ratio-nal

_points

_of

_JI

is

_finite.

This theorem should be sufficient to decide whether the

_jacobian

of _any modular curve has a non-trivial

quotient

with

finitely

_many rational

points.

Let us mention the earlier work of V.

Kolyvagin

and D.

Logachev

[39]

which treated the case of

_quotients

of jacobians

of the form

_They

built on earlier work

by

_Kolyvagin,

on a formula of B. Gross and D.

Zagier

(28~,

and on a result established

_{independently}

by

K.

_Murty

and R.

_Murty

[57]

and

_by

D.

_Bump,

S.

_Friedberg,

and J. Hoffstein

_[4].

This last case is

sufficient for the

_{practical applications}

which have been considered _up to

now. Indeed all three families of modular curves considered to solve Serre’s

problem

are related to

_jacobian

varieties of the

_type

_{Jo(N) (in}

the Borel and normalizer of a

split

Cartan cases this is

evident,

in the normalizer of

a

_non-split

Cartan case it is a theorem of Chen

explained

below).

We turn to that case in more detail.

Let N be an

_{integer &#x3E;}

0. Let d be a

square-free integer

dividing

N

(in

particular

one

_might

have d =

1).

Denote

by

Wd the Fricke involution

Xo(N).

Denote

_by

_Jo(N)d

the

_jacobian

of the

_quotient

curve

Xo(N)/wd

Let us describe a

winding

quotient

of the

jacobian

Jo(N)d

of the curve

Xo(N). (A

finer version is described in P. Parent’s

_forthcoming

_thesis.)

Let T be the

_subring

of

_End(Jo(N)d)

_generated

_by

the Hecke _operators

Tn

(n

&#x3E;

_1,

_{(N, n) = 1).}

It

_operates

on the

singular

homology

_group

H1

Let e be the

_unique

element such that the

_integral

of _any

_holomorphic

differential form ca on

_along

a

_cycle

of

class e coincides with the

integral along

the

path

from 0 to ioo in the _upper

half-plane

of the

_pullback

of Mazur has called e the

winding

element

(see

[48]

for the

_origin

of this

_{terminology).}

Let

_Ie

be the annihilator of e in

T. Let be the old abelian

_subvariety

of

_Jo(N)d.

Let be

the

quotient

abelian

_variety

_{Jo(N)d/(IeJo(N)d}

+ The

_following

proposition

can be

_proved

_{by mimicking}

what is in

[55]

(the

case N

_prime

and d =

1)

and in

[13]

(the

_case N =

2p2

and d =

p).

Proposition

2.7. The

_{L-function of}

does not vanish at the

_point

1.

By

application

of the theorem of

_Kolyvagin

and

_Logachev

one obtains.

Theorem 2.8

_{(Kolyvagin-Logachev).}

The _group

_ofQ-rational

_{points of}

the

abelian

_variety

is

_finite.

Therefore one obtains the

_following

criterion which is a

_prior

curious

since it involves

_only

the

_complex

structure of the modular curve.

Corollary

2.9.

_{If e}

does not

_belong

to the old

_part

_of

_{H1(Xo(N)d(C),Q),}

then has a non trivial

_quotient

with

finitely

_many rational

points.

To check such a criterion one

might

have to make calculations on

mod-ular

_{symbols using}

Manin’s

_presentation

of the

_homology

of

_Xo(N)

_([45],

[54], [61]).

The verification amounts then to a

_study

of a

graph

with

ver-tices indexed

_by

_(see

_{[54], [61])}

which is

_essentially

the "dessin d’enfant" associated to the curve

Xo(N)

in the sense of

[29].

However in

[13],

this was done

by

a

_simpler

_argument

that we do not describe here. 2.4. Chen’s

_isogeny.

In

_[13]

a result of I. Chen was used to obtain our

main result

_(6~.

Theorem 2.10

_(Chen).

Let

Kns

(resp. Ks) be the

normalizer

_of

a

corresponding

modular curves. Let

_Jns

and

Js

be the

jacobians

of

Xns

and

Xs.

There is an

_isogeny

_of

abelian varieties

_{defined over Q}

between

_J,

and

Jns

x

_{Jo (p) .}

Chen uses trace formulas to _{prove that} the two abelian varieties have the

same L-function.

_Making

use of a theorem of

Faltings

he concludes that

the abelian varieties are

_isogenous.

The

problem

was reconsidered

_by

B.

Edixhoven in

_[20],

who _gave a more

elementary

proof

of Chen’s theorem

based on the

_theory

of

_{representation}

of

GL2(Fp).

Neither of these two

proofs

gave an

explicit description

of the

_isogeny,

nor is the short

proof

that we

_give

below.

Since

_JS

is

_isomorphic

to the

_jacobian

of

_XO(p2)/wp

_(where

_wp is the Fricke

_involution),

an

elementary

_argument

of

sign

of functional

_equation

of L-function leads to the

_following

_consequence.

Corollary

2.11. The

_{L-function of}

_any non-trivial

_quotient

abelian

_variety

of

Jns

vanishes at 1.

If one believes in the

_conjecture

of Birch and

Swinnerton-Dyer,

no

non-trivial

_quotient

abelian

_variety

of

_Jns

has

_finitely

_many rational

_points.

And

we are embarrassed to

_apply

Mazur’s method to

_Xns.

One is

_tempted

now to consider other results in the direction of the

conjecture

of Birch and

_{Swinnerton-Dyer,}

such as the formula of Gross and

Zagier

[28]

and the result of

_Kolyvagin

on

elliptic

curves of

analytic

rank one to understand the arithmetic of

_Jns.

We _expect that Chen’s

_isogeny

can be deduced from from an

explicit

correspondence

between

_XS

and

_Xns.

Let

_Hp

=

("The

upper

half-plane

over

It is a finite set of

_cardinality

_{P(p - 1)/2}

endowed with a transitive

action of

_GL2(Fp)

deduced from the

_homographies

on Let A =

f t, -tl

E

_Hp

such

that t2

E

_{1FP .}

Let be the stabilizer of A in

_{GL2(Fp ).}

It is the normalizer of a

non-split

Cartan

subgroup

of

GL2(Fp).

Let ca C

_Hp.

_{To any g} E

_{GL2 (Fp )}

we associate a subset _gca of

_Hp

(the

support

of a

_"geodesic

path"

of

HP)

and a

_pair

of elements of the set

_{f g0, goo)}

_{(two "cusps").}

For _every

_pair

_{P2 ~}

of distinct elements of there is an element

g E

_GL2(Fp)

such that

_{P1,P2~.

This element is well defined

up to

multiplication

by

a

_diagonal

or

antidiagonal

matrix. This

implies

that

the set _gca

_{depends only}

on

f Pi, P2 1.

Let us denote it

_by

_{Pl, P2}

&#x3E;. Let

Cp

be the set of

_pairs

of distinct elements of

_{P (Fp ) .}

The _group of

_diagonal

or

antidiagonal

matrices is the normalizer

_KS

of

the

_split

Cartan

_subgroup

of

_{GL2(Fp )}

formed

_{by diagonal}

matrices. The

map

GL2(Fp )

-

_Cp

which

_{to g}

associates

f g0, goo}

defines a

_bijection

Let

be the _group

_homomorphism

which associates to

_{[IP,, P2}]}

the sum of

el-ements of

_{Pi, P2}

&#x3E;

_(the

_map which to the

_support

of a

"geodesic

_path"

associates the sum of its

_points).

Let

be the group

homomorphism

which to

_~P~

associates the sum of elements

of

_Cp

_containing

P.

The _group

_homomorphism

- . , .. ,. - - , r ,.

-which to

_gKs

associates

_{¿hEGL2(JFp)/(KsnKns)}

coincides with

_f

if one

identifies

_Hp

with and

_Cp

with as above. Therefore one deduces from

f~°

a

correspondence

_fp:

X,

-~

_{Xns given}

by

the canonical

_morphism

_XKasnxs

_{- Xns x Xs}

and of

_degree

Let

_Bo

be the Borel

_subgroup

of

_GL2(Fp)

_consisting

of _upper

_triangular

matrices. The _map

_GL2(Fp)

-

P1(IFp)

which to

_{to g}

associates _goo

defines a

_bijection

between

G L2 (JFp ) / Bo

and

P (Fp ) .

The _group

_homomorphism

which to

_gBo

associates

_{¿hEGL2(JFp)/(KsnBo)}

_ghKs

coincides

_{with 0}

if one identifies

_{P (Fp )}

with

_GL2(Fp)/Bo

and

_Cp

with as above.

Denote,

as

_{usual, by}

Xp(p)

the modular curve

_XBo.

One deduces from

gp

a

correspondence

_gp:

Xo(P)

--7

_Xs

of

_degree

_p.

, A

straightforward

calcula-Problem 2.12. What is the

_image

_of

_f2?

To our embarassment we could not

give

an answer to this

_question.

It is

likely

that this

_image

_generates

_Q[Hp]

as a vector _space. This was verified

by

Chen in a few non-trivial cases.

If this is _true, the inverse

_{image by}

₁₂

of 20 is

_{generated by}

the

_image

of

_92.

In

_particular

the kernel of

_fP

is

_{generated by}

the

_image

_{by 0}

of elements of

_degree

0.

Let us _say that a divisor of

_XS

is

p-old

if it is the

_image

by

the

correspon-dence

_{Xo (p)}

--&#x3E;

_XS

and that a divisor of

Xns

is

p-old

if it is the inverse

image

of a divisor of

X(1)

_by

the

_morphism

Xns

--&#x3E;

_X(1).

Since

_X(1)

is a

curve of _genus

_0,

the class _any

p-old

divisor of

degree

0 is 0 in

Jns(p).

The

properties

of

_fp

translate

_immediately

into

_properties

of the

_{correspondence}

The

_{correspondence}

_fp

defines

_by

Albanese and Picard

_{functiorality}

two

homomorphisms

of abelian varieties

_fp*:

_JS

)

_Jns

and

_fP:

_{Jns 2013~ ~}

The

correspondence

gp defines

similarly

an

homomorphism

of abelian

_variety

_gp,:

~ JS.

Since one has

_fp*

o

gP*(Jo(p))

=

0,

_{one can}

expect

that

_fP*

is

the

_explicit

_description

of Chen’s

_isogeny.

To _prove this it would suffice to

show that the cokernel

_{of fp}

is finite.

Remark.

_Using

the

_techniques

of

_[43],

the determination of the

_image

of

_{i f p 0}

should lead to the

_explicit

_description

of the kernel of It would not be

surprising

if the

_image

of

_f~

contained

Let us

_give

a

quick proof,

essentially

suggested by

Edixhoven,

of Chen’s

theorem. Let E be a set with one element considered as a trivial

GL2(IFP)-set.

Lemma 2.13. The sets

PI

_{(Fp )}

U

_Hp

and

_Cp

U E are

weakly isomorphic

as

i.e. _any element

_of

_GL2(Fp)

has the same number

_{of fixed}

points

in each set.

Proof. Let g

E

_GL2(Fp).

The number of fixed

_points

_{of 9}

_{depends only}

on the

_conjugacy

class of _g. There are four

_types

of

_conjugacy

classes. In each case we indicate the number of fixed

_points

of g

in

P1(IFp),

_{Hp, Cp}

and E

respectively.

First case: g is a scalar

matrix;

The numbers of fixed

points

are _{p +}

1,

P~ 2 1~ ,

1 and 1. Second case: g has two distinct

eigenvalues

in

_{Fp -}

We have then

_{2, 0, 1,}

and 1. Third _{case: g is} not a scalar matrix and

has one double root in its characteristic

_polynomial;

The numbers of fixed

points

are

_{1, 0, 0,}

and 1. Fourth and final case: g has no

eigenvalue

in

_Fp;

One obtains

_{2, 1, 0,}

and 1.

In each case the sum of the first two numbers

_equals

the sum of the

remaining

two.

As a _consequence the

Q[GL 2 (Fp )]-modules

Q[P

and

Q[Cp

U E~

are

_isomorphic

(see

[72]

exercise

_5,

_chapter

13).

Using

the identifications of

Hp, Cp,

and E as cosets of

_{GL2 (Fp )}

_{given above,}

such an

isomor-phism

defines a

correspondence

from

Xo ( 1 ) U XS (p)

to

Xo (p) U Xns (p) ,

which

defines in _turn,

_by

Albanese

_{fonctoriality}

an

homomorphism

of abelian

va-rieties

Since

_{Jo (I)}

is

_trivial,

Chen’s theorem follows.

One can deduce mutates mutandis similar results

_by

_adding

an extra and

prime

to p level structure. For the purpose of the

_study

of

_Frey

curves and