Fields of definition of $\mathbb {Q}$-curves

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

ORDI

Q

UER

Fields of definition of Q-curves

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 275-285

<http://www.numdam.org/item?id=JTNB_2001__13_1_275_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

Fields of

definition of

_Q-curves

par JORDI

QUER

RÉSUMÉ. Soit C une

Q-courbe

sans

multiplication complexe.

Dans cet article, nous caractérisons les _corps de nombres K _pour

lesquels

il existe une courbe C’

isogène

à C dont toutes les

_isogénies

entre les

_conjuguées

par le groupe de Galois sont définies sur K. Nous caractérisons

_également

les courbes C’

_isogènes

à C définies

sur un _corps de nombres K telles _que la variété abélienne

_ResK/Q

déduite de C’ par restriction des scalaires est un

produit

de variétés

abéliennes de _type GL2.

ABSTRACT. Let C be a

Q-curve

with no

_{complex multiplication.}

In this note we characterize the number fields K such that there

is a curve C’ _isogenous to C

_having

all the _isogenies between its

Galois _conjugates defined over K, and also the curves C’ _isogenous

to C defined over a number field K such that the Abelian variety

ResK/Q (C’/K)

obtained

_by

restriction of scalars is a

_product

of

Abelian varieties of

_GL2-type.

1.

_Definitions,

notation and basic facts

We work in the

_category

of Abelian varieties up to

isogeny.

Endk (A)

will

denote the

_Q-algebra

of

_{endomorphisms}

defined over a field k of an Abelian

variety

A.

For a Galois

(profinite)

_group G all the G-modules are discrete and we

always

assume that the

_{corresponding cohomological objects}

are continu-ous.

A

Q-curve

is an

_elliptic

curve defined over a number field that is

_isogenous

to all of its Galois

_conjugates.

An Abelian

_variety

of

_GL2-type

is an Abelian

variety

A defined

_{over Q}

whose

_Q-algebra

of

_{endomorphisms}

is

a number field of

degree

equal

to its dimension

_(these

are the

_primitive

Abelian varieties of

_GL2-type

of Ribet’s definition in

_1.1).

Both families _{appear in}

_{generalizations}

of the

_{Shimura-Taniyama}

conjec-ture : the

_Q-curves

are

_{conjecturally}

the

_elliptic

curves

C /Q

for which there

is a nontrivial

morphism

_C;

the Abelian varieties of

_GL2-type

are

Manuscrit regu le 28 octobre 1999.

conjecturally

the varieties

_Q-isogenous

to a

_Q-simple

factor of some

Jl(N).

The two

_conjectures

are

equivalent

as a _consequence of the

following

Theorem 1.1

_{(Ribet [4]).}

An

_elliptic

curve

_{over Q is}

a

Q-curve

if,

and

only if, it is

a

_quotient

of

some Abelian

_variety

of GL2-type.

We will

_only

consider

_Q-curves

with no

complex multiplication,

the

study

of the

_{complex multiplication}

case

_requiring

different

techniques.

Let

_C/Q

be a

_Q-curve.

We will say that C is

completely defined

over a

number field K if all the Galois

_conjugates

of C and the

_isogenies

between them are defined over K.

Let

_K/Q

be a Galois extension such that C is

_completely

defined over

K. For _{every a} E choose an

_{isogeny ~~ :}

uC -+ C. The _map

is a

_two-cocycle

of the _group

Gal(K/Q)

with values in the _group

Q*,

iden-tified with the nonzero elements of

End(C),

and viewed as a Galois module

with trivial action. Another choice of

_isogenies

between Galois

_conjugates

or the

_change

of C

_by

a curve

_K-isogenous

to it modifies the

_two-cocycle

cK

by

a

coboundary.

Hence,

the

cohomology

class

[CK]

E

H2(K/Q,

_Q*)

depends

only

on the

_K-isogeny

class of the curve C.

Fix an invariant differential w defined over K for the curve

C/K,

and

take its

_conjugates

as invariant differentials for the

conjugate

curves. For

every

isogeny

0, :

u C -+

C,

defined over

K,

let

Àu

E K* be determined

_by

This shows that the

_cocycle

_CK, viewed with

_images

in the

Gal(K/Q)-module

_K*,

is a

coboundary.

In other

words,

the

image

of the

cocycle

class

[CK]

in the group

H 2(K/Q,

K*)

=

Br(K/Q)

is trivial.

Let c be the inflation of the

two-cocycle

cK to

GQ.

The

cohomology

class

[c]

E

H 2(GQ,

_Q*)

_{depends only}

on the

_isogeny

class of C and has trivial

image

in the Brauer _group

H2(GQ,Q*)

=

Br(Q).

In

_[5]

Ribet used the

two-cocycle

class

_[c]

for

_giving

a characterization of the smallest field of

definition for

_Q-curves

_{up to}

_isogeny:

Theorem 1.2

_{(Ribet [5]).}

A

_{Q-curve is isogenous}

to a curve

defined

over

a number

field

K

if,

and

only if,

the

_cocycle

class

[c]

is in the kernel

_of

the restriction map

This result can also be deduced from results

_by

Elkies

_using

a

_completely

different

_argument.

Our aim in the next section is to

_investigate

the anal-ogous situation for fields of

complete

definition.

2. Minimal fields of

_complete

definition

Theorem 2.1. A

_Q-curve

is

_isogenous

to a curve

completely defined

over a Galois number

field

K

if,

and

only if,

the

cocycle

class

[c]

is in the

image

of

the

_inflation

map

Inf :

_{H 2(K/Q,}

_{Q*) -+}

_{H 2(GQ,}

_Q*).

Moreover,

if

[c]

=

Inf ~ for

_an

element

_E

H 2(K/Q,

Q*)

then there is such an

_isogenous

curve

having

[CK] =

~-Proof.

The if

_part

_being

_obvious,

let C be a

_Q-curve

whose attached

two-cocycle

[c]

is the inflation of some

element ~

E

H 2(K/Q,Q*).

Then

_[c]

is also in the kernel of the restriction _{map to}

_{H2 (GK, ~* )}

_and,

_applying

Ribet’s theorem

_1.2,

we _{may assume, up to}

isogeny,

that the curve C is

defined over K.

Let c be the

two-cocycle

constructed from a

locally

constant set of

iso-genies

0, :

’C -+ C between the

_conjugates

of C and let

_Àu

E

U

be such that

_{§3(w) =}

for an invariant differential w defined over K of

_C/K.

Consider the commutative

_diagram

1

where the first row is the

_cohomology

exact _sequence

_{corresponding}

to the Galois module exact _sequence

Since

_{Inf(~) =}

_[c]

and

_~([c])

is

_trivial,

then Inf

_t*(ç)

is trivial and is trivial.

_{Hence ~}

is in the

_image

of the

_{boundary map 6}

and there is a

one-cocycle a r-+

pa :

Gal(K/Q) -

K*/Q*

with =

6( [p] ) .

Let us denote

also

_by

_{p any} lift with values in K* of the

_one-cocycle

_Gal(K/Q)

-K*/Q*

obtained

_by

inflation. Let

_bad

=

J-La)..;; 1.

The

map

(Q, T)

~ is a

two-cocycle

of

_G~

with

values in

_Q*

whose

_cohomology

class in

_Q*)

is

_(Inf

_~)(c~-1

=

1,

hence it is a

coboundary; changing

the choice of

_{isogenies §a}

_according

to

this

_coboundary

we _may assume that this

two-cocycle

is in fact the trivial map. Then the map a H

bad

is a

one-cocycle

of

G~

with values in

~*.

By

Hilbert’s Theorem 90 there is an element of that we _{may write} as

_Via

then both

_Acr

and _pa are rational

numbers,

ba

is also a rational

number,

and

~77 ~

=

b~

is a

_positive

rational

_number;

_taking

the norm from a number

field

_{containing y}

it follows that it is a root of

_unity,

hence it must be 1. Then

_u,

_{= ~y} for _{every a} E

_Gx

_and,

E K* .

Let

C’

be the K-twist of the curve C

_{corresponding}

to the field

and

_{let 0 :}

C’ -~

C be an

isomorphism.

The kernel of the

’00- 1 :

GK -~ ~ ~ 1 ~

has the field as its fixed field and we _may choose an

invariant differential W’ for

_C’/K

such that

_{0* (w) = .¡;ÿ}

cv’.

We claim that the curve

C’

is

completely

defined over and has attached

two-cocycle

class

_{[cK] =}

_.

_{Choose qba =}

_-1

as

isogenies

between the

conjugates

of

C’ .

Then

The

_isogenies

are defined over the field = K and the

two-cocycle

cK(a, T) =

is

alr

hence

_~.

0

3. Restriction of scalars and

_GL2-type

In his

_proof

of Theorem 1.1 Ribet starts with an

arbitrary

Q-curve

C and

he shows the existence of a Galois extension

K/Q

over which the curve is

completely defined,

and such that the Abelian

_variety

B =

obtained

_by

restriction of scalars has some

_Q-simple

factor of

_GL2-type.

In this section we find _necessary and sufficient conditions on the curve

C and the field K for the

_variety

B

_being

a

_product

of Abelian varieties of

_GL2-type.

We remark

_that,

since the zeta functions of the curve

C/K

and of the

_variety

_{B /Q}

are the _same, the curves

having

this

_property

_are,

assuming

the

_{generalized Shimura-Taniyama}

_conjecture,

the

_elliptic

curves over a number field whose zeta functions are

_products

of L-series of classical

modular forms for _congruence

_subgroups

_rl(N).

Let

_C/K

be a

Q-curve

defined over a number field K of

degree

[K : Q] =

n and let B = be the Abelian

variety

obtained

by

restriction

of scalars. Let E be a set of

representatives

of

right

cosets of

GQ

modulo

GK,

GQ

=

aG K.

The elements of E

correspond

to the

embeddings

K ~

_{Q. Over Q}

the

_variety

B is

_isomorphic

to the

_product

_IIuE’E

uC. The full

_endomorphism

_algebra

of B is

which is a

Q-algebra

of dimension

n2

since every

Hom(aC,

TC)

is a

one-di-mensional

_Q-vector

space.

and,

for every s E

_S’,

let

_E,

be the

_{representatives}

of a

decomposi-tion of the double coset of s as a

_disjoint

union of

_right

cosets

GxsGK

=

UtE’Es

tGK,

and let

_GK

be a

_{corresponding}

set of elements T E

_Gx

with

_TsGK

=

tGx.

For

every s E rS’ choose an

_isogeny

0, :

~C 2013~

C;

then

the set

is a basis of the

endomorphism algebra

End(B).

The orbit of an

_isogeny

§s by

the action of

_G~

contains

_isogenies

in

_Hom("’C,

_uG)

for T E

_{E’ 8}

and

~ E E.

_Then,

the

_{homomorphisms}

for all elements s E S‘ such that the

_{isogeny 4&#x3E;s :}

sC -+ C is defined over

the field SKK are a

Q-basis

of

EndQ(B),

since the elements of

_G~

fixing

are those in

GK

n

_GSK.

In

_particular,

n with

equality if,

and

only

if,

is

Galois and all the

_isogenies

between Galois

_conjugates

of the curve C are

defined over K. In this case the set :E = S

may be identified with

Gal(K/Q)

and one checks the formula

J , -.. --’--’---’

showing

that the

_{endomorphism algebra}

_{EndQ (B)}

is

_isomorphic

to the twisted _group

_algebra

_(G~.

Theorem 3.1. Let

_CIK

be a

Q-curve

defined

over a number

field

K.

Then,

is

_Q-isomorphic

to a

_{products of}

Abelian varieties

_of

GL2-type if,

and

_{only if,}

_K/Q

is Galois

_Abelian,

C is

_{completely defined}

over

_K,

and is in the kernel

_of

the _map

induced

_by

the

_{embedding t ~* --3}

_Q*,

with

_~*

viewed as a module with

trivial action.

Proof.

If an Abelian

_variety

B/Q

is a

product

of varieties of

GL2-type

over Q

then dim B with

_equality

_if,

and

_only

_if,

all the varieties in the

_{decomposition}

are

pairwise non-isogenous

_or,

equivalently,

the

_algebra

_EndQ(B)

is commutative. In case that B is obtained from a

Q-curve

by

restriction of scalars we have seen that

_dimQ

dim B with

_{equality if,}

and

_{only if,}

_K/Q

is Galois and the curve is

_completely

defined over K.

Then,

if B = is _a

product

of Abelian varieties of

GL2-type

we must have

_dimQ

_{EndQ (B)}

= dim

B,

K/Q

is

Galois,

C is

completely

defined over

K,

and

Endo (B )

is a commutative

algebra. Conversely,

if K/Q

algebra

(of

dimension

_equal

to

_n)

then the

_{decomposition}

of this

_algebra

as a

_product

of number fields induces a

Q-decomposition

of B as a

product

of

Abelian varieties of

_GL2-type.

Then the

_proof

of the theorem comes from a

general

fact about twisted

group

algebras. Namely,

that

given

a

cohomology class ~

E

H2 (G, ~* )

the

twisted group

algebra

Q~[G]

is commutative

if,

and

only if,

the group G is Abelian and the

_{element ~}

has trivial

_image

in the Schur

multiplier

group

H2(G,

Q*)

(see

proposition

5.2 of

_[3]).

D

4. Galois

_{cohomological computations}

Let C be a

Q-curve

and let

1011,EGQ

be a

(locally constant)

set of

isogenies

between its Galois

_conjugates.

For _every Q E

_G~

let

_{d( a) =}

deg (0,)

(mod

Q*2 )

be the

_degree

of the

_{isogeny 0,}

_{up to} squares of rational numbers. The _map d :

_{Q* /Q*2}

is a _group

_homomorphism

that

depends only

on the

isogeny

class of the curve C. We call it the

degree

_map

corresponding

to the curve C.

Let

_Kd

be the fixed field of ker d. It is a

compositum

of

quadratic

fields.

Lemma 4.1. Let C be a

Q-curve

with associated

[c]

E

H 2(GQ,

_Q*)

and let d be the

_{corresponding degree}

_map. The _map

is a

two-cocycle of

GQ

with values in

Q*

whose

cohomology

class is

[c].

Proof.

Let be a

locally

constant set of

_isogenies

between the

conju-gates

of C and let

_Àu

E

0*

be the elements as in

(1)

corresponding

to an

invariant differential of

_C/Q.

The map a -

A 2/

deg(o,)

is a

one-cocycle

of

_GQ

with values in

Q*.

_By

Hilbert’s Theorem 90 there is an element

y E

~

such that

_A2

= for

every Q E

_GQ.

Fix a _square root

_vff

E

Q*.

For _every Q E

_G~

one has

_Aa

= + deg(§a)

Hence,

the two maps Q H

Àu

and Q H viewed as

one-cocycles

with

values in

~ /~*,

are

cohomologous,

and determine the same element of

~*/~*).

The

_image

of this element in

_{H2 (GQ,}

_Q*)

_by

the cobound-ary map is

equal

to

_[c]

and is also

_equal

to the

_cohomology

class of the

two-cocycle

of the statement. D

Taking degrees

in the definition of the

_{two-cocycle cK}

one sees that

its square is a

_coboundary,

hence

[cK]

_belongs

to the 2-torsion

_subgroup

H 2(K/Q,

~*) ~2~.

Let G be the _group

_Gal(K/Q)

or

GQ (or

_any

_profinite

group).

There is a natural

decomposition

where P denotes the

_subgroup

of

_positive

numbers of

_Q*.

Under this

de-composition

a

cocycle

class

[c]

corresponds

to the

_pair

_{(~c~~, d),}

where

c-7~

is the

_two-cocycle

with values in

_1±11

_giving

the

_sign

of c and d is

deter-mined

_by

the

_identity

= We will call

[cl]

the

sign

component

and d the

_degree

_corrzponent

of the

_cohomology

class

_[c].

We

_identify

as usual the _group

H 2(GQ,

1±11)

with the 2-torsion

_subgroup

Br( Q)

~2~.

We say that two sets of elements al, ... , am,

dl, ... , dm

E

Q* /Q*2

such that K = and

the di

generate

the

_image

of the

_degree

_map are

dual bases with

_respect

to the

_degree

_map d if there are elements _{al, ... , am} E

G~

with

_d(Qi)

=

di

and

O"iý7Lj=

Proposition

4.2. Let C be a

_Q-curve

and

_{let al, ... ,}

an and

dl, ... ,

dn

be

dual bases with

_respect

to the

_{corresponding degree}

_map. Then the

coho-mology

class

_[c±l

E

_Br(Q)[2]

is the

_product

_of

_quaternion

_algebras

Proof. By

lemma 4.1 the

_sign

_component

_[cl]

of the

_cohomology

class

_[c]

is the class of the

_two-cocycle

and the formula follows

_{by straightforward}

_computation

_(see

_[3]

for

_details).

D

Using

the results of this section we can

_give

a lot of information about

the fields

_satisfying

theorems

_1.2,

2.1 and 3.1.

Fields of definition. The restriction _map Res :

_Q*)

_{H 2(K/Q,}

_Q*)

has a

_sign

and a

degree

_components

corresponding

to the

_{decomposition}

(2).

To trivialize the

_degree

_component

_{it is necessary} and sufficient that the field K contains the field

Kd.

The

_sign

component

is also trivialized

_by

the field

_Kd

as it is obvious from the formula of

_proposition

4.2.

Hence the field

_Kd

is the smallest

_possible

field of definition for the curve

C _up to

_isogeny.

Fields of

_complete

definition. The inflation _map Inf :

_{H2 (K/~, ~* ) --~}

Q*)

has a

_sign

and a

_degree

_components

_{corresponding}

to the

de-composition

(2).

The

_degree

_component

of

_[c]

_belongs

to the

_image

of the inflation _map

_if,

and

_{only if,}

the field K contains the field

_Kd.

The condi-tion for the

_sign

_component

is more

_complicated:

the _group

must contain an

_{element ~}

whose inflation to

H2 (G~, ~ ~ 1 ~ )

is the

prod-uct of

_quaternion

_algebras

_(3).

This condition can be

_interpreted

in terms

of

_embedding

_problems

in Galois

_theory:

the elements of

element to

_{H 2(GQ,}

_{f ±l 1)}

is the obstruction to the

_solvability

of the

corre-sponding

embedding problem.

Hence the Galois number fields K over which the curve C can be

com-pletely

defined _{up to}

_isogeny

are the fields

_containing

_Kd

as a subfield

and such that there is a double cover of

Gal(K/Q)

whose

corresponding

embedding problem

has obstruction

_given

_by

_(3).

An

_example

of a field

_having

this

_property

is the field

obtained

_adjoining

to

_Kd

the _square roots of the

_degrees

of the

_isogenies

between Galois

_conjugates

of the curve. It is a

_compositum

of at most 2m

quadratic

fields if

_Kd

is a

_compositum

of m

_quadratic

fields. In some cases

the curve C can be

completely

defined _up to

_isogeny

over a smaller

field;

for

example,

if the

_product

of

_quaternion

_algebras

₍₃₎

is trivial in the Brauer group then the curve can be

completely

defined _{up to}

_isogeny

over the field

Kd.

In

_general

there is no smallest field over which C can be

completely

de-fined _{up to}

_isogeny,

but _many different minimal fields

_having

that

_property.

Restriction of scalars a

product

of

GL2-type.

Let be Galois Abelian

and let C be

_completely

defined over K. Then K must contain

_Kd

as a subfield. The element in

H2 (K/~, ~ )

is the

_product

of the

images

of the

_sign

_component

and of the

_degree

_component.

The

_image

of the

_degree

_component

is

_{obviously always}

trivial. As for the

_image

of the

_sign

_component

_triviality

is

_equivalent

to the existence of a _map

m--&#x3E;

,Q(Q) :

Gal(K/Q) -

~*

with

The square of such a _{map must} be a character e =

(32 : Gal(K/Q) -

Q*

and the _map

_fl

can be recovered

(up

to a

coboundary)

from the character

just

defining

(3(a)

=

Q.

Let

_KE

be the field fixed

_by

the kernel of the character E.

What we have seen is that the

triviality

of

~* ( ~cK~ )

is

_equivalent

to the existence of a character e of

Gal(K/Q)

such that

]

is the

cohomology

class of the

_two-cocycle

Such a

two-cocycle

can be inflated from a

two-cocycle

of the _group

and the

_{corresponding}

element of is

as-sociated to the

_problem

of

_embedding

the

_cyclic

extension into a

cyclic

extension of double

_degree.

From the

_previous

remarks and

_using

theorems 2.1 and 3.1 one obtains

the

_following

Corollary

4.3. Let C be a

_Q-curve.

There is a

_Q-curve

C’

_isogenous

to

_C,

is a

_{product of}

Abelian varieties

_{of of GL2-type if,}

and

_{onLy if,}

the

_field

K contains the

_field

_Kd

and also contains a

_cyclic

extension

_Kg/Q

such that the obstruction to

_embedding

Kg

into a

_cyclic

extension

_of

double

_degree

is the

_{product of}

_quaternion

_algebras

_(3).

Every

element of

_{H 2(GQ,}

_{{ :f: 1 })}

can be realized as the obstruction to

embedding

cyclic

extensions

_{of Q}

into

_cyclic

extensions of double

_degree.

Hence it is

_{always possible}

to find fields

_satisfying

the

_properties

of the

corollary

and

_elliptic

curves

_isogenous

to a

_given

one with restriction of

scalars a

product

of varieties of

GL2-type.

For more details about the

fields

_satisfying

the

_properties

of the

_corollary

and for the

_splitting

of the varieties B as a

_product

of varieties of

_GL2-type

see

[3].

5.

_Examples:

_quadratic

_Q-curves

of

_degree

2

In this section we

_give

some

examples illustrating

the results of

previ-ous sections. For a _non-square rational number t the

elliptic

curve with

Weierstrass

_equation

is a

Q-curve

defined over the field

Kd

= with _an

isogeny

0, :

u Ct -+

Ct

of

_degree

2. Here Q denotes the nontrivial

automorphism

of

Kd. Moreover,

every

elliptic

curve defined over a

quadratic

field and

hav-ing

an

isogeny

of

degree

2 to its Galois

conjugate

is

isogenous

to a curve

_Ct

for some value of t. This is because the curves

Ct

have been obtained

by

parametrizing

the rational

_points

of the _genus zero curve

Xo(2)/W2

quo-tient of the modular curve

classifying

isogenies

of

degree

2 between

elliptic

curves

by

the Atkin-Lehner involution

W2;

see

[1]

for a

description

of the

parametrization

of

_Q-curves

_by

rational

_points

of modular curves. The curve

Ct

has no

complex multiplication

_except

for ten values of t

where it has

_{complex multiplications,}

_{respectively, by}

the order of discrim-inant

D =

-20, -24, -36, -40, -52, -72,

-88, -100, -148,

-232.

From now _on, we assume that

_Ct

has no

_complex

_{multiplication,}

_{i. e.,}

that t

is a _non-square rational number not among the ten values listed above. If K is a number field

_{containing Kd}

_{and 7}

E K* we denote

by

_Ct"

the K-twist

of the curve

Ct

over the field

K(., ,F-y);

it may be

_given

by

the Weierstrass

equation

Using

proposition

4.2 one

_computes

the

_sign

_component

of the

cohomol-ogy class

[c]

attached to the curve

Ct

to be

Let ca be an invariant differential defined over

_Kd

for

Ct/Kd.

_Using

V61u’s

explicit

formulas for

_isogenies

between

_elliptic

curves

_given

by

Weierstrass

equations

it is easy to

_compute

and deduce that the

_isogeny

_0,

is defined over the field

Q( Vi, B/~2).

The

_Kd-twist

_{Ct f}

has the

_isogeny

to its

_conjugate

defined over the field

Q

The curve

Ct

is

isomorphic

to a curve

completely

defined over the field

Kd

if,

and

_only

_if,

the

_cohomology

_group

_{H 2(Kd/Q,}

_~~1})

contains an element

whose inflation to

_{Br(Q) [2]}

is the class of the

_quaternion

_algebra

_{(t, 2).}

The group

H 2(Kd/Q,

~~1})

has two

elements,

the nontrivial one

_corresponds

to

the double cover of the _group

Z/2Z

by

a

_cyclic

_group of order

4,

and its inflation to

_{Br (Q) [2]}

is the class of the

_quaternion

_algebra

_(t,

_-1).

Hence,

the curve

Ct

can be

completely

defined over

Kd

_{up to}

isogeny if,

and

only

if,

one of the

_quaternion

_algebras

(t, 2)

or

(t,

-2)

is a matrix

_algebra.

Assume that

_{(t, 2)}

= 1 in

Br(~) ~2~.

Let _a, b rational numbers such that

a2 - tb2

= 2. Then the

Kd-twist

of

Ct

with 7

= tb ₊

av6

is

completely

defined over

Kd

and B = is _an Abelian surface of

GL2-type

with

_{Endo(B) cr}

_Analogously,

if

_{(t, -2)}

= 1 in

Br2(Q)[2]

and a, b are rational numbers

with a2 - tb2

=

-2,

then the

Kd-twist

Ct,y

with -y = _{a +}

bvfi

is

completely

defined _over

Kd

and B = is _an

Abelian surface of

_GL2-type

with

_{Endo (B) ci}

_{Q( 2) .}

Let t = 5. Since

(5,2)

and

(5, -2)

_are

(isomorphic)

division

alge-bras there is no curve

isogenous

to

C5

completely

defined over

Kd.

Let

K

_{= Ke}

=

~( 5 2 5 )

be the

cyclic

extension

of Q

of

degree

4 and conduc-tor

_20,

which is a

_quadratic

extension of

_Kd.

Then

Ke

can not be embedded into a

cyclic

extension of

degree

8 and the obstruction to such an

embedding

is

_precisely

_{(5, -2).}

In other

_words,

if E : character of order

_4,

then the

_two-cocycle

c, defined

by

(4)

has

(5, -2).

By corollary

4.3 we know that the curve

C5

is

_isogenous

to a curve

com-pletely

defined over the field K whose restriction of scalars is a

_product

of varieties of

_GL2-type.

In order to find such a curve we

proceed

as

fol-lows : let L =

K( 2),

the curve

_C5

is

_completely

defined over

L,

let

[ 1]

be the

_{corresponding two-cocycle class,}

and let

_[ce]

denote the infla-tion to

_Gal(L/Q)

of the element of

_{H 2 (K/Q,}

_{±1})

defined

_by

_(4).

_Then,

the element

_[ce][ct]-l

E

H 2(L/Q,

~~1})

has trivial

_image

in the Brauer group

Br(Q)[2]

and hence determines a solvable

embedding

problem

over

the extension

_{L /Q;}

one finds the

_following

solution

Then,

the K-twist

_C5,.y

is

_completely

defined over the field K.

Moreover,

the Abelian

_variety

of dimension 4

_splits

_{over Q}

as a

prod-uct x

_A2

of two Abelian surfaces of

_GL2-type

with

_Q(B/2013l).

There are

cyclic

extensions

K/Q

of

degree

8

containing Kd

such that there

is a K-twist

C5,’Y

of the curve

C5

completely

defined over K such that

B =

ReSK/Q(C5,,y/K)

is a

Q-simple

Abelian

variety

of

GL2-type

with

but I have not been able to

_compute

an element ~y

producing

such an

example.

Let t = -5. Then

(-5, 2)

and

(-5, -2)

_are both division

algebras

and

there is no curve

_isogenous

to

_C-5

and

_completely

defined over

_Kd.

Let

Ke

be the same

_cyclic

field of the

_previous

paragraph;

now

Kd

is not a

subfield of

Ke

and the

_{compositum K}

=

KdKE

is an Abelian number

field with Galois _group

_isomorphic

to

_Z/4Z

x

_Z/2Z.

The

_cocycle

class

E has inflation to

_Br(Q)[2]

_equal

to the

_quaternion

algebra

(-5, 2).

Solving

the

_appropriate

_{embedding problem}

over the field

L = one finds an element

such that the K-twist

_C-5{y

is

_completely

defined over the field K. More-over,

splits

over Q

as a

product

A1

x

_A2

of two Abelian varieties of

_GL2-type

of dimension 4 with

_{Q( V2, ~).}

In this case it can be shown that there does not exist a field K and a curve

C’ / K

isogenous

to

_C-5

with

_{ResK/Q(C’ / K)}

_being

a

(Q-simple)

Abelian

variety

of

_GL2-type.

References [1] N. ELKIES, Remarks on _elliptic k-curves. _Preprint, 1992.

[2] E. PYLE, Abelian varieties _{over Q} with _large endomorphism algebras and their simple

com-ponents over _Q. Ph.D. Thesis, Univ. of California at _Berkeley, 1995.

[3] J. QUER, Q-curves and Abelian varieties Proc. London Math. Soc. ₍₃₎ 81 _(2000),

285-317.

[4] K. RIBET, Abelian varieties _{over Q} and modular _{forms. Proceedings} of KAIST Mathematics

Workshop (1992), 53-79.

[5] K. RIBET, Fields of definition of Abelian varieties with real multiplication. Contemp. Math.

174 _(1994), 107-118.

Jordi _QUER

Dept. Matematica Aplicada II

Universitat Politecnica de Catalunya

Pau _Gargallo, 5

08028-Barcelona

Espana