Rational points on $X_0^+ (N)$ and quadratic $\mathbb {Q}$-curves

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

S

TEVEN

D. G

ALBRAITH

Rational points on X

₀+

_{(N) and quadratic Q-curves}

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 205-219

<http://www.numdam.org/item?id=JTNB_2002__14_1_205_0>

© Université Bordeaux 1, 2002, tous droits réservés.

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

Rational

_points

on

and

quadratic Q-curves

par STEVEN D. GALBRAITH

RÉSUMÉ. Nous considérons les

_points

rationnels sur

dans le cas où N est un nombre

composé.

Nous faisons une étude

de certains cas

_qui

ne se déduisent _pas des resultats de Momose.

Des

_points

rationnels sont obtenus pour N = 91 _et N = 125.

Nous exhibons aussi les

_j-invariants

des

_Q-courbes

_quadratiques

correspondantes.

ABSTRACT. The rational _points on in the case where

N is a

_composite

number are considered. A

_{computational study}

of some of the cases not covered

_by

the results of Momose is

_given.

Exceptional

rational

_points

are found in the cases N = 91 and

N = ₁₂₅ and the

_j-invariants

of the

_{corresponding}

_quadratic

Q-curves are exhibited.

1. Introduction

Let N be an

_integer

_greater

than one and consider the modular curve

Xo (N)

whose _non-cusp

_points

_correspond

to

_isomorphism

classes of

isoge-nies between

_{elliptic curves § :}

E - E’ of

_degree

N with

_cyclic

kernel

_[5].

The rational

_points

of

_{Xo (N)}

have been studied

_by

_many authors. Results of Mazur

_[21],

Kenku

_[19]

and others have

_provided

a classification of them.

The conclusion is that rational

_{points usually}

arise from cusps or

elliptic

curves with

_complex

_{multiplication.}

There are a finite number of values of

N for which other rational

_{points arise,}

and we call such rational

points

’exceptional’.

For the

_largest

N for which there are

_exceptional

rational

_points

is the famous case N = 37.

The Fricke involution

_W~

on

Xo (N)

arises from

_taking

the dual

_isogeny

~ :

E’ -~ E. We define the modular curve

Xo (N)

to be the

_quotient

of

Xo (N)

by

the _group of two elements

_generated

_by

WN.

There is a model

for

_{Xo (N)}

_{over Q}

and one can

study

the

Q-rational

points

on this curve.

Rational

_points

on are an

_{interesting object}

of

_study.

Momose

[22],

[23]

has

_given

some results of a similar nature to those of

_Mazur,

but the results

_{only apply}

to certain

_composite

values of N.

_Therefore,

a

classification of rational

_points

on

Xj (N)

is not

_yet

complete.

In this _paper we use

_{computational}

methods to determine some

excep-tional raexcep-tional

_points

on in cases where N is

composite

and not

covered

_by

the results of Momose. This continues the work of

_[9]

which

gave a

_{computational study}

of the case when N is a

_prime

number.

The conclusions of this work and

_[9]

are the

following:

The modular curve

Xj (N)

has an

_exceptional

rational

_point

when N E

_{{73, 91,103,125,137,}

191,311}.

We

_conjecture

that these are the

only

values of N for which the

genus

of Xo (N)

is between 2 and 5 and for which has

exceptional

rational

_points.

The above

_conjecture

includes the statement that

_{Xsplit (p )}

(which

is

_isomorphic

to

_{xt (p2»}

has no

_exceptional

rational

points

when

p = 13.

2. Rational

_points

on

The case of _cusps can be

_easily

understood. Rational _cusps on

Xo(N)

give

rise to rational _cusps on

Xt(N).

_Using

the notation for _cusps

intro-duced

_by

_Ogg

_[25],

the

_following

result

_easily

follows.

Proposition

1. The

_only

_integers

N

_for

which non-rational _cusps

_of

Xo(N)

can

give

a rational _cusp on

Xo (N)

are N E

_{9,16,36}.

The

corre-sponding

cusps are

([1 :

~~,

[-1 :

The _non-cusp

_points

of

_{Xo (N)}

can be

_interpreted

as

_pairs

10 :

E

-E’, 0 :

E’ -~

_E~.

From

_[5]

it is known that if a _non-cusp

_point

of

Xo (N)

is

defined over a field L then the

corresponding pair

of

isogenies

and

elliptic

curves _may also be taken to be defined over L.

Therefore the

_{only possibilities}

for rational

_points

on

Xo (N)

are as

fol-lows : Either the rational

_point

is a _cusp, or else it

corresponds

to a

_pair

10 :

E -

E’, ~ :

E’ -~

_E~

such that one of the

following

holds.

1.

_E,

_{E’, 0}

and ~

are all defined over

_Q.

2. E

and E’

are defined over

Q,

the

_{isogeny 0}

is defined over a

quadratic

field

_L,

and the non-trivial element a E

_Gal(L/Q)

is such that

_0’

~

and so E ~ E’.

3.

_E,

_{E’, 0}

are defined over a

_quadratic

field

_L,

E’,

and the

non-trivial element a E

_Gal(L/Q)

is such that

_{~~ ^--’}

~

and E’ E£ E~.

Case 1 is the case of rational

points

on and these have been

classified. Rational

_points

on

Xo (N)

_{corresponding}

to

_elliptic

curves with

complex multiplication

can arise as

Heegner

points

(see

Section

3)

or not

(e.g.,

the case of

Xo(14)

where there is an

_{isogeny 0 :}

E --3 E’ of

_degree

14 such that

_End(E)

has discriminant -28 while

_End(E’)

has discriminant

In case 2 above we have E E£ E’ and so

End(E) ~

End(E’).

The existence

of a

_{cyclic isogeny}

of

_degree

N

_implies

that the

_elliptic

curves have

_complex

multiplication

and so the

_point

is a

_Heegner

_point

(and

the class number

of the

_{endomorphism ring}

is

_one).

In case 3 there are two

_{possibilities:}

either E has

_{complex multiplication}

(and

therefore

_End(E’)

and the

_point

is a

Heegner point

of class

number

_two)

or

_not,

in which case we call the

_point

an

_exceptional

rational

point.

In both cases we have an

elliptic

curve E over a

quadratic

field

(and

not defined over

Q)

which is

_isogenous

to its Galois

_conjugate.

Such an

elliptic

curve is called a

’quadratic Qcurve’

(see

[14], [27]).

Examples

of

_{quadratic Q-curves}

which do not have

_complex

multiplica-tion are

_interesting

and one of the main contributions of this _paper is to

provide

some new

examples

of these.

It can be shown that _every

_{quadratic Q-curve}

without

_complex

multipli-cation

_corresponds

to a rational

_point

on

Xj (N)

for some N &#x3E; 1. For a more

general

result

along

these lines see Elkies

[6].

We now recall the definition of the Atkin-Lehner involutions

[1]

for _any

nlN

such that

_gcd(n,

_N/n)

= 1. Over C

_they

may be defined as elements of

SL2(R)

as follows. Let _a,

_b,

_c, d E Z be such that

_adn-bcN/n

= 1 and define

Wn

=

na b

This construction is well-defined up to

_{multiplication by}

ro(N)

and therefore each

Wn

gives

an involution on the modular curve

Xo(N).

Note that if

_gcd(nl,

_n2)

= 1 then

_Wn¡n2

=

Wn¡ Wn2.

The

_W~

also

_give

rise to involutions on

Xo (N).

_{If n ¢}

~1, N~

then

Wn

acts

_{non-trivially}

and the action of

Wn

and

_{W N/n}

is identical. The

Atkin-Lehner involutions are defined

over Q

and so

_they

_map L-rational

points

of

_{Xo (N)}

to L-rational

_points

for _any field

_L/Q.

Furthermore the Atkin-Lehner involutions _{map cusps} to _cusps.

Proposition

2. Let

_w(N)

be the number

_of

distinct

_primes

_dividing

N. Then the

_exceptioraat

rational

_points

_of

_{Xo (N)}

_(if

there are

any)

fall

into

orbits under the Atkin-Lehner involutions

_of

size

2~~)"~.

The

_field

_of

def-inition

_of

the

_{corresponding j-invariants}

is the same

for

all the

exceptional

points

in a

given

orbit.

Proof. Suppose

we have a

_point

of

Xo (N)

which is fixed

by

some

Atkin-Lehner involution Such a

point

corresponds

to some T E such that

Wn(T) _

(or

WN/n(r) =

~(~-))

for some _q E

_ro(N).

It follows that r

satisfies a

_{quadratic equation}

over Z and so we either have a _cusp or a CM

point

and the

_point

is not

_exceptional.

The field of definition of the

_j-invariants

is the field of definition of the

points

on

Xo(N)

which

correspond

to the rational

point

on

Xt(N).

Since

the action of

_Wn

is rational on

Xo (N)

it follows that the field of definition

3.

_{Heegner points}

A

_Heegner

_point

of

_Xo(N)

_[2]

is a _non-cusp

_point

_{corresponding}

to an

isogeny

of

_elliptic

_{curves 0 :}

E --~

E’

such that both E

and E’

have

_complex

multiplication by

the same order ~? of discriminant D in the

quadratic

field

K

_{= Q(d5) .}

In this case we _say that the

Heegner

point

has discriminant

D.

It is well-known

_(see

_{[2], [15],}

_[9])

that

_{Heegner points}

on

Xo(N)(C)

are

in one-to-one

_{correspondence}

with

_{ro(N)-equivalence}

classes of

quadratic

forms

NAX 2

+ BX Y +

Cy2

where

_A,

_B,

C E Z are such that

A,

C &#x3E; 0 and

gcd(NA,

B,

C)

=

gcd(A,

B,

NC)

= 1. The

correspondence

is as follows.

Let T be the root of

N AT2

+ BT + C = 0 with

_{positive imaginary}

_part.

Then E =

C/(1,T)

is an

_elliptic

curve with

_{complex multiplication by}

the

order C~ of discriminant D

= B2 - 4NAC

and the

_{cyclic isogeny}

with kernel

( N,

Ty

maps to the

_elliptic

curve

E’ ^--’

cC/(1,

’ )

which also has CM

by

0

(see

Lang

[20]

Theorem

_8.1).

In

_particular,

a

_{Heegner point}

on

Xo(N)

of discriminant D can

only

arise when the

_primes

_satisfy

_{( D ) -1.}

However this condition is not

p

sufficient since there can be cases where all

piN

_split

or

_ramify

and

_yet

one

cannot find a suitable

triple

(A, B,

C)

as above.

The conductor of an order of discriminant D is the index of the order

in the maximal order of and it may be

computed

as the

_largest

positive integer

c such that

D/c2

=

_0,1

(mod

4).

We must recall a few well-known facts about

_isogenies

of

_{degree dividing}

the conductor c

(see

the

Appendix

of

[10]

for an

_elementary

_proof).

Let E be

an

_elliptic

curve over C such that

End(E) ~

C~ of discriminant D.

_Suppose

p is a

prime

dividing

the conductor of o.

Then,

_up to

_isomorphism,

there is

exactly

one

_p-isogeny

from E

_’up’

to an

elliptic

curve E’ such that

End(E’)

has discriminant

_{D / p2}

and there are

exactly

_p

_isogenies

of

degree

_p from E

’down’ to

_elliptic

curves whose

_{endomorphism ring}

has discriminant

p2D.

If p

does not divide the conductor of 0 then there are 1 +

(12)

isogenies

p

of

_degree

p to

elliptic

curves

E’

with

End(E’)

= d and there are _{p -}

(D)

p

isogenies

of

_degree

p down to

elliptic

curves whose

endomorphism ring

has

discriminant

_p2D.

Returning

to the context of

_Heegner

_points,

we have the

following

(where

we write

_{f o g}

for the

_composition

of functions

f (g(~))).

Proposition

3.

_{Suppose 0 :}

E -

E’ is

a

Heegner

point

on

Xo(N)

of

discriminant D and that

_{p is}

a

_prime

_dividing

gcd(N, c) .

_{Then ~ factors}

as

_~2

where

1. is an

_isogeny

_{of degree}

_{p up}

_from

E to an

_elliptic

curve

E1

whose

2. ~

is an

_{isogeny of degree}

N/p2

_frorra

E1

to some

_elliptic

curve

E2

such

that

_End(E2)

has discriminant

_D/p2.

3.

_{1b2 is}

an

isogeny

of degree

_p

from

E2

down to E’.

Proof.

We can write E as

~/(1, T)

where T satisfies

NAT2

+ BT + C = 0.

From the condition

_pID

= 4NAC we have

plB.

One can show that

p2 IN

(this

also _{follows from the fact that ’what goes up} must come

down’).

The

_{isogeny 0}

has kernel

_{(1/N, T)}

and we define

_1/11

to be the

_isogeny

_having

kernel

_{(1 /p,}

_T).

This

_isogeny

_maps E to

_Ei

=

cC/(1, pT),

where

pT is a root of the

_quadratic

+

_(B/P)X

+

_C,

and so the

_elliptic

curve E’ has

_{complex multiplication by}

the order of discriminant

D/p2.

The

_remaining

statements are now immediate. 0

Indeed,

when

_gcd(p,

= 1 then we can also

factor 0

as

1/1’

_{o 2 01/11}

or

_{02 0 01}

0

0’

(where 1b’

here is an

N/p2-isogeny

between

elliptic

curves

whose

_{endomorphism rings}

have discriminant

_D).

The

_following

result is now clear.

Proposition

4.

_Suppose

N is a

_{positive integer}

and that 0 is an order

of

discriminant D and conductor c in an

_{imaginary quadratic field}

K. Let d be

the

_largest

_positive

_integer

such that

_die

and

_d21N.

Then there are

Heegner

points

on

Xo(N)

_{corresponding}

to the order (7

_{only if}

_gcd(N/d2,

_cld)

=

1,

all

_primes

are such that

(Dpd2) ~

-1 and all

_primes

_p such that

p21(N/dl)

are such that

(Dp 2)

=+1.

Proof.

then there must be some

_p-isogeny

_up which

is not matched

_by

a

_p-isogeny

back down

_again

and it follows that the

corresponding

point

of

_Xo(N)

is not a

Heegner point.

The conditions on

_primes

_p

_dividing

N/d2

come from the fact that kernel

of the

_{corresponding}

_p-isogeny

can be viewed as an ideal. For the

composi-tion of these

_isogenies

to have

_cyclic

kernel it follows that the

_primes

must

split

or

_ramify

and that ramified

_primes

can

_only

occur with

multiplicity

one. D

The next result

_gives

further constraints on when a

Heegner

_point

can

exist.

Proposition

5. Let N be an

_{integer greater}

than one and 0 an order

of

discriminant D and conductor c.

Suppose

that

2allc,

that

( D 2 2a )

=

_+1,

and that Then a

Heegner

point

of

Xo(N)

of

discriminant D can

arise

_{only if}

_22a+1IN.

Proof. Suppose

instead that

_N/22a

is odd and that we have a

_Heegner

_point

on

Xo(N).

Without loss of

generality

we _may assume that c = 2.

The

_{isogeny 0}

factors as

₂

_{0 1/J 0 1/J1}

where

_1/J1

is an

_isogeny

_up of

degree

2 and

_~2

is an

_isogeny

down.

Indeed,

since

N / c2

is odd we _may instead

factor 0

as

1/J’

o

~2

0

_{lbi .}

= +1 the choice of the

isogeny’02

down is

unique.

It follows that

_{1/J2 ~}

and therefore the

_isogeny

does not have

_cyclic

kernel. 0 In Section 7 the case N = 64 and D = -28 _appears. This is an

_example

of how a rational

_point

of

Xo (N)

can arise when both c and

Nlc2

are even.

The Atkin-Lehner involutions

_Wn

_(where

is such that

_gcd(n,

_{N/n) _}

1)

map

Heegner points

to

Heegner points

with the same discriminant.

Therefore it makes sense to

_speak

of

_Heegner

_points

on

Xo (N).

4.

_Rationality

of

_Heegner

Points on

Xo (N)

In the context of this _{paper it} is

_important

to determine when

_Heegner

points

on

Xo(N)

can

_give

a rational

_point

of

Xo (N).

Following

Gross

_[15]

we write a for the

projective

(9-module

(1, T) (the

isomorphism

class of the

_elliptic

curve E

depends only

on the class of a

in

_Pic(O))

and write b for

_{(1 jN, T).}

The

_{isogeny 0}

then

_corresponds

to

the

_projective

0-module n = which in this case is the d-module

(N,

(-B +

-v/D-)/2).

The fact that the kernel is

_cyclic

_may be

_expressed

as

0/n

Gross uses the notation

(0, n,

~a~)

for the

_{Heegner point.}

From the results of Gross one can

easily

deduce the

following (see

[9]).

Theorem 1. Let x =

10 :

E -

E’, ~ :

E’ ~

E}

be _a

Heegner

point

of

with

_End(E)

= d. Let n be the

_projective

_{corresponding}

to the

_isogeny.

Then x is

defined over Q if

and

only if

either

1.

_{ho =1,}

or

2.

_ho

=

2,

n is not

principal.

and n = n.

We now discuss the

_meaning

of the condition n = n. In terms of the

representation

n =

(N, (B

₊

B/D)/2)

we see that n = n if and

only

if In the case when N is

_coprime

to the conductor of 0 it follows that every

prime

p

dividing

N must

_ramify

in

_0,

and therefore N must be

_square-free.

As seen in

[9],

rational

_{Heegner points coming}

from class number two

orders are rather rare.

Proposition

6.

_Suppose

N &#x3E; 89. Then there are no rational

Heegner

points

on

Xo (N)

_of

class number two.

Proo f .

Let D be the discriminant of a class number two discriminant. We consider first the case when N is

coprime

to the conductor of D.

In this case we

_require

that N be

_square-free

and that all

_primes

_piN

ramify

(i.e.,

The list of all class number two discriminants D is

_{-15, -20, -24, -32,}

---112, -115, -123, -147, -148, -187,

-232, -235, -267, -403,

-427. This

already severely

limits the number of

_possible

values of

_square-free

N for which

A further condition is that the

_projective

(9-module n which has norm

N must

_satisfy

n = n and be

non-principal.

For most of the

larger

discrim-inants in the list one sees that D is itself

square-free

and that

by unique

factorisation the

_only

ideal of norm N = -D is the ideal which is

principal.

One can check that 89 is the

_largest

N for which

gcd(N,

c)

= 1 and for which a suitable ideal n exists.

Indeed,

there is a rational

point

on the _genus one curve

Xt(89)

_{corresponding}

to the class number two

discriminant D = -267.

Now,

assume that

gcd(N, c) ~

1 and that we have some

_isogenies

_up

and down of

_degree

d

_(where

d divides

_c).

The

_remaining

_N/d2

_isogeny

is handled

_by

the

_previous

_case, and so it follows that It remains to

determine which

_possible

values for N can arise with d &#x3E; 1.

The

_only

values for D with non-trivial conductor are D

= -32, -36, -48,

-60

_{-64, -72, -75, -99, -100, -112}

and -147

_(for

which we have c =

2, 3, 4, 2, 4, 3, 5, 3,

5, 4

and 7

_{respectively).}

The

_{possibilities}

for N &#x3E; 89 are therefore

_99,100,112

and 147. These

cases do not have rational

_points

since the

_{corresponding}

ideal n would

necessarily

be

_principal.

0

As mentioned

_above,

_Xt(89)

has a class number two

_Heegner

_point.

Rational

_{Heegner points corresponding}

to class number two discriminants for which N is not

_coprime

to the conductor seem to be

_extremely

rare. In

fact,

the

_{only example}

I have noticed occurs with N = 8 and D = -32.

There are further

examples

of rational

Heegner points

of class number two.

For

_instance,

in Section 8 it is shown that the curve

Xo+ (74)

is an

example

of a

_composite

value of N for which there is a rational class number two

heegner

point.

We could end the

_analysis

_here,

since

_Proposition

4 and the fact that

Heegner

points

come from

quadratic

forms

NAT2

+ BT + C of discriminant D can be used as the basis of an

_algorithm

to list all

_{ro (N)-equivalence}

classes of suitable T and Theorem 1 tells when

they give

rational

points

of

Xt(N).

Thus,

from a

_{computational point}

of

_view,

we have all the tools we need.

However,

it is useful to have more information about the action

of the Atkin-Lehner involutions on

_{Heegner points.}

5. The Action of Atkin-Lehner Involutions on

Heegner

Points

We first consider the case where

gcd(N, c)

=1.

Proposition

7

_(Gross

_[15]).

Let

_(C~,

n,

[a])

be a

_Heegner

_point

ort

Xo(N)

where _p

_decomposes

as

~.p

in O. Then the Atkin-Lehner involution

Wpa

acts

by

_Wpa(O,

n,

[a])

=

(0,

pam,

The

_following

result is then immediate.

Proposition

8. Let N be a

_{positive integer.}

Let 0 be an order

of

conduc-tor

_coprime

to N

_for

which there exist rational

_{Heegner points}

on

Xo (N) .

Let

_w’(N)

be the number

_of

distinct

_{primes dividing}

N which

_split

in 0.

1.

_{If the}

class numbers

_{o f d}

is one then there are

maxfl,

2’ N-1

rational

Heegner

points

on

Xo (N)

_{corresponding}

to the order 0 and

_they

are

all

_mapped

to each other

_by

Atkin-Lehner involutions.

2.

_If

the class number

_of

0 is two then there is

_only

one such

Heegner

point.

Furthermore

_w’(N)

= 0 and the

point

is

f xed by

all the Atkin-Lehner involutions.

Proof.

From the notation

_(0,

n,

[a])

it follows that the set of all rational

Heegner

points

on

Xo (N)

corresponding

to an order (~ is obtained

by

tak-ing

the

_images

of one of them under the _group of Atkin-Lehner involutions.

The statements about the number of rational

_points

which arise then

follow from

_{Proposition 7}

and Theorem 1. 0

We now consider the case where N is not

coprime

to the conductor of

O.

~ ~ ~ ~

The

_{isogeny 0}

factors as

~2 0 "p

and

so 0

factors as

_{;¡¡;; 0 ;p}

o

_~2’

Since the

_isogeny

up is

always unique,

we have that

~1 and’02

are

uniquely

determined.

_However,

the

_{isogenies ~2}

and

₀₁

are

only

constrained

_by

the

condition that the full

_composition

has

_cyclic

kernel.

It follows that there may be several

non-isomorphic Heegner points

com-ing

from a

_given

discriminant D. It is useful to know when a

_Heegner

point

is fixed

_by

an Atkin-Lehner involution. We

give

one result in this

direction which can

_apply

when _p is 2 or 3

(which

are the most

_commonly

encountered

_cases).

Proposition

9. E -

E’ is

a

Heegner point

on

Xo (N)

of

dis-criminant D and

_having

_prime

conductor _p.

_Suppose

that the class number

of D is

one,

that p2 N

and

that p -

( D ~2 )

= 2. Then the

Heegner

point is

fixed by

the Atkzn-Lehner involutions

_{W2 .}

Proo f .

Write N =

p2m.

Since the class number of D is _one it follows that E ^--’ E’. We can

_{factor 0}

as

_~2

where

_~1

is a

_p-isogeny

_up and

_~2

is a

_p-isogeny

down

_{and 0}

has

_degree

m. Since _{p -}

( D r2 )

= 2 there _are

_only

two choices for the

_isogeny

down. It follows that

_~2

is

_{uniquely specified}

by

the condition

_{1b2 #}

It remains to show that the

_{m-isogeny 1b}

is fixed

_by

_Wp2

_(the

_argument

point,

so that where

B2 -

4NAC = D. We focus on

the

_{isogeny 0 given}

as

(cC/ ( 1, T), ( 1 ~m,

T) ) .

The class number one condition

implies

that = for _{some, E}

SL2(Z).

Using

the

quadratic

equation

for T one can deduce

_that,

E

_{ro (m)}

and that the

_{isogeny o}

is

preserved.

Therefore,

the involution

_Wp2

must fix the

_{Heegner point.}

0 An

_example

of the above situation occurs with N = 52 and D = -16. We have _p =

2,

(24)

= 0 and there is

only

one rational

_{Heegner point}

on

Xo (52)

corresponding

to the discriminant -16. In

_contrast,

with N = 52 and D = -12 we have

(23) = -1

and there are two rational

_{Heegner points}

on

Xo (52)

_arising

(each

mapped

to the other

_by

_W4).

6. Results of Momose

Momose

_{[22], [23]}

has studied the

_question

of whether there are

excep-tional raexcep-tional

_points

on

Xo

(N).

Theorem 2

_(Momose

_[23]).

Let N be a

_composite

number.

If

_{any one}

of

the

_following

conditions holds then

_Xt(N)(Q)

has no

_exceptional

ratio-nal

_points

_{(i. e.,}

all rational

_{points of}

_{Xo (N)}

are _cusps, rational

points

of

Xo (N),

or

Heegner

points. ).

1. N has a

_prime

divisor _p such that

_{p &#x3E;}

11, p ~

13, 37

and

#Jo (p) (~)

finite.

2. The _genus

_{of Xo (N) is at}

least 1 and N is

_{divisible by}

_{26, 27}

or 35.

3. The _genus

_{of Xo (N)}

is at least

_1,

N is divisible

_by

_{4 9,}

and m :=

N/49

is such that one

of

the

following

three conditions holds: 7 or 9 divides

m; a

_{prime q -}

-1

(mod

3)

divides _m; or m is not divisible

by

7 and

(~)=-1.

.

Regarding

the first condition

_above,

Momose states that the number of

points

of

_Ja

_{(p) (Q)}

is finite for p = 11 and all

primes

17

p

300

_except

151, 199,

227 and 277.

Of course, when the genus of

Xt(N)

is zero then there will be

_infinitely

many

exceptional

rational

points.

The N for which this occurs are N

21,

23 N

_27,

_{29, 31, 32, 35, 36, 39, 41, 47, 49, 50, 59}

and 71

_(see

_Ogg

[26]).

Information about the

_{quadratic Q-curves}

in the cases N =

_{2, 3, 5, 7}

and 13 was found

by Hasegawa

[18].

González and Lario

[12]

determined

the

_j-invariants

of

_Q-curves

when

_{Xo (N)}

_{has genus} zero or one and so

their results also contain all these cases of

quadratic Q-curves

(although

their results

_{give polyquadratic j-invariants}

and the

_quadratic

cases are

not

_{readily distinguishable}

from the

_others).

It is also

_possible

to have

_infinitely

many

exceptional points

in the case

7. Genus one cases

It can be shown

(see

Gonzalez and Lario

[12]

Section 3 for the

square-free

case)

that

_{Xo (N)}

_{has genus} one when N is

22,28,30,33,34,37, 38,40,43,

44, 45, 48, 51,

53, 54, 55, 56, 61, 63, 64, 65, ?5, ?9, 81, 83, 89, 95, I01,119

and 131. In the cases

_{37,43,53,61,65,79,83,89, 101}

and 131 the rank of the

elliptic

curve

Xo (N)

is one and so there are

infinitely

_many

exceptional

rational

_points.

For the

_remaining

cases the rank of the

elliptic

curve is zero and we can ask whether the

_{only points}

are _cusps and

Heegner

points.

Momose’s

result covers _many of these cases and so the

_only

N we must consider are

28, 30, 40, 45, 48, 56, 63, fi4

and 75. The

_following

table lists the results and

we see that there are no

exceptional

rational

_points

in these cases. Note

that in this table the number of rational

_points

on

Xo (N)

is known to be

correct.

(*) see

[4].

8.

_Higher

_genus cases

We now turn attention to the values of N for which the _genus of

is two or more. For these cases there are

only

finitely

_many rational

points.

The

_following

table lists all the

_composite

values of N _{for which the genus} of

_{Xo (N)}

is between 2 and 5 and for which Momose’s theorem does not

apply.

The

_prime

cases have

already

been studied in

[9].

We now embark on a

_{computational study}

of these cases

_using

the meth-ods of

_{[8], [9].}

The

_key

is to construct

_explicit

_equations

for

_using

the

_techniques

of

_~11~,

_[24]

and

_[28]

and _cusp form data from

_[3].

The results are

_given

in the

following

table. The value for

given

in the second column is _proven to be correct _{in many} cases

_{by using}

coverings

to rank zero

_elliptic

curves.

Nevertheless,

for the cases N E

{91,117,125,169,185}

it is

_simply

the number of

_points

of low

_height

found

by

a search as in

[9].

We

conjecture

that this is the correct number of

points

in each case.

9. The case N = 91 = 7 . 13

In this case there are

exceptional

rational

_points.

We

_give

the details of

the calculations in this case and we exhibit the

_j-invariants

of the

corre-sponding quadratic Q-curves.

A basis for the

_weight

two forms on

ro(91)

which have

_eigenvalue

₊₁

with

_respect

to

_W91

is

_given

_{(see [3])}

_by

the two forms

Following

the

_techniques

of

_{~11), [24], [17], [8]}

we set h =

( f -g)/2, x

=

f /h

for

_{Xo (91).}

The

_{hyperelliptic}

involution is not an Atkin-Lehner involution

in this case and so we find ourselves in an

analogous

situation to the case

Xo (37).

There are two rational _cusps on

Xo+(91)

and three candidate

discrimi-nants D =

-3, -12

and -27 for

Heegner

points.

The

primes

7 and 13 both

split

in each of the orders of these discriminants and so there are

_always

two rational

_Heegner

_points

for each of them.

The

_{Heegner point}

of discriminant -91 does not

_give

a rational

_point

since the

_{corresponding}

ramified ideal is

_principal.

It is _easy to find 10

_points

on the model

_above,

which confirms that there are two

_exceptional

rational

_points

on

Xo (91).

The Atkin-Lehner involution

_W7

_(which

is

_equivalent

to

_W13

on

Xt(91»

maps the

exceptional points

to each other. It can be shown

_{by considering}

the

_original

modular forms that

_W7

maps a

point

(x,

y)

to

_(x/(x-1),

y/(x-1)3).

.

The

_following

table lists all the data.

The

_{exceptional points correspond}

to

_{quadratic Q-curves.}

The

j-inva-riants can be

_{computed using}

the method of Elkies

[7].

The

_point

(3, 7)

corresponds

to the

_elliptic

curve

having

j-invariant equal

to

The

_point

_{(3/2, 7/8)}

_corresponds

to the

_elliptic

curve

having

_j-invariant

equal

to

As in

_[9]

and

_[13]

it is seen that these

_j-invariants

have some

_properties

similar to those

_{enjoyed by}

the

_{singular j-inva,riants}

_(see

Gross and

_Zagier

norm over the

quadratic

extension,

while ‘Coefficient’ means the coefficient

of

_{3 . 29}

in the

_{j-invariant).}

We see, as

usual,

that

N( j)

is

’neaxly

a cube’ and that

N(j - 1728)

is

square. Notice the similarities in the

primes arising above,

and that the

’coefficient’ is divisible

_by

7 in both cases but not 13.

Also note that the

_j-invariants

are of the form

ji

=

a/213

and

j2

=

c~/2~

where

_{a, a’}

are

_algebraic

_integers

such that the norms

satisfy

gcd(N(a),

2~3)

=

27-1

and

gcd(N(a’),

291)

=

291-1.

This

_suggests

an

analogue

of

Theorem 3.2 of Gonzilez

_[13].

10. The case N = 125

We are

_again

in the situation where

_{Xo (N)}

_{has genus} two and where the

_{hyperelliptic}

involution is not an Atkin-Lehner involution. From

[3]

we

find that the

_following

forms are a basis for the

weight

two cusp forms for

0

Taking

functions x =

f /g

and _y =

-q(dx/dq)lg

gives

the

following

equation

for

_{Xo (125)}

We find six rational

_points

on the curve. There is one rational _{cusp, and} we find

Heegner

points

for each of the discriminants D =

-4, -11,

-16 and -19. In each case there is

_only

one

_{Heegner point.}

It follows that there is an

exceptional point.

We first

_give

the table of

_points.

The

_exceptional

rational

_{point corresponds}

to a

quadratic Qcurve

with

The

_following

factorisations occur.

The

_j-inva,riant

is of the form

_{a/l l}

where a is an

_{algebraic integer}

and

the norm of a satisfies

gcd(N(a),115 )

= 115-1.

Acknowledgements

This research was

supported by

the

EPSRC,

the Centre for

Applied

Cryptographic

Research at the

_University

of

Waterloo,

the NRW-Initiative fair Wissenschaft und Wirtschaft "Innovationscluster fair Neue

_Medien",

and cv

_cryptovision

_gmbh

(Gelsenkirchen).

I thank the referee for their careful

_reading

of the _paper. References

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Royal Holloway University of London

Egham, Surrey TW20 OEX, UK E-mail : Steven. Galbraithtrhul. ac. uk