The Mumford-Tate group of 1-motives

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The Mumford-Tate group of 1-motives

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N

A

L

E

S

D

E

L ’IN

ST

IT

U

T

F

O

U

R

IE

ANNALES

DE

L’INSTITUT FOURIER

Cristiana BERTOLIN

The Mumford-Tate group of 1-motives Tome 52, no4 (2002), p. 1041-1059.

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THE

MUMFORD-TATE GROUP OF 1-MOTIVES

by

Cristiana BERTOLIN

52, 4

(2002), 1041-1059

Introduction.

The Mumford-Tate _group of a 1-motive M defined over

_C,

_MT(M),

is an

_{algebraic Q}

_{-group acting} on the

_Hodge

realization of M and endowed

with an

_increasing

filtration

W..

In this paper we

_study

the structure and

the

degeneracies

of this group.

After

_recalling

some

definitions,

in Section 1 we

_investigate

the

structure of

_MT(M):

the main result of this section is the structural

lemma

_(1.4),

where we _prove that the

unipotent

radical of

_MT(M),

which

is is an extension

_by

_W-2(MT(M))

of the

_unipotent

radical of the Mumford-Tate group of a 1-motive without toric

_part,

and that

_injects

into a

"generalized" Heisenberg

_group.

As a

_corollary,

we can

_compute

the dimension of

MT (M) (corollary

on

dimensions

_1.5).

We then

_explain

how to reduce to the

study

of the

Mumford-Tate _group of a direct sum of 1-motives whose torus’s character

group and whose lattice are both of rank 1

(Theorem 1.7).

In Section

_2,

we

classify

the

degeneracies

of

MT(M), i.e.,

those

phenomena

which

_imply

the decrease of the dimension of

_MT(M).

The

1-motive M is deficient if

_{W_2 (MT (M) )}

= 0. This

degeneracy

_was discovered

by

K. Ribet and O.

_Jacquinot

_(cf.

_[JR87]).

The 1-motive M is

quasi-deficient if

_W-,(MT(M))

is abelian and it is

depressive

if the dimension

Keywords: 1-motives - Mumford-Tate _{group - Degeneracies -} Poincaré biextension.

of

_{W_2 (MT (M) )}

is not maximal. The last two

_degeneracies

are new. Then

there are the "trivial

_{degeneracies" :}

the trivial deficient and the trivial

quasi-deficient.

We call them trivial because

_they

are induced

_by

trivial

phenomena:

for

_{example, they}

_appear when the 1-motive is without toric

part,

or without abelian

part,

or when its

_underlying

extension is

_split,

...

At the end of this section we

_give

several

_examples.

Using

the structural

_lemma,

in Section 3 we

_give

a

_geometrical

in-terpretation

of deficience and

_{quasi-deficience}

and we show that these two

degeneracies

are

_trivial;

more

_precisely

a deficient 1-motive is

_trivially

de-ficient and a

quasi-deficient

1-motive is

trivially quasi-deficient

(Theorems

3.5 and

_3.7).

Acknowledgements.

- The

study

of the Mumford-Tate group of

1-motives is

_part

of my doctoral

dissertation,

written under the

supervision

of Y. André. I want to _{express my}

_gratitude

to D. Bertrand for the various

discussions we have had on this

_subject.

I thank also B. Moonen for his

suggestions. Finally,

I am _{very grateful} to P.

_Deligne

and K. Ribet for their interest in this work.

1. The structure of the Mumford-Tate _group.

1.1. A 1-motive M over C consists of

(a)

a

_{finitely generated}

free Z-module

_X,

(b)

an extension

G,

defined over

C,

of an abelian

_variety

A

by

a torus

_T,

(c)

a

_{homomorphism u : X -}

G(C).

The 1-motive M =

(X,

A, T, G,

u)

can be view also as a

_complex

of

commutative _group schemes concentrated in

_degree

0 and 1: M =

[X

~G~ .

A

_morphism

of 1-motives is a

_morphism

of

_complexes

of commutative

group schemes. An

isogeny

between two 1-motives

_{Mi =}

_]

and

M2 =

[X2 #G2]

is a

_morphism

of 1-motives such that

_{fx :}

-

_X2

_is

injective

with finite

cokernel,

and

_{fG : GI}

-

G2

is

_surjective

with finite kernel.

We can define an

_increasing

filtration W. on M =

[X#G]

in the

following way:

If we denotes I

r - .."’I

The Cartier dual of M is the 1-motive M" -

_(XV,

_{Av, Tv,}

I

where X" =

_{Hom(T, Gm) ,}

AV is the dual abelian

_variety

of

_A,

TV is the

torus with character group

X,

Gv =

Ext’(M/W-2 (M),

Gm )

and uv comes

from the

_long

exact sequence

associated to the short exact sequence

The notion of biextension allows for a more

_{symmetric description}

of 1-motives: Consider the

7-uplet

where X and

Xv are two

_{finitely generated}

free

_Z-modules,

A and A" are two abelian

varieties dual to each

other, v : X -

A and vv : Xv --~ Av are two

homomorphisms, and 0

is a trivialization of the

_{pull-back by (v,}

of the

Poincaré biextension P of

_(A,

_A~).

From this

7-uplet

(A,

AV,

X,

X~,

v,

it is easy to reconstruct the 1-motive M =

(X,

A,

T,

G,

u)

and its Cartier

dual M" _

_(cf.

_{[D75] (10.2.12)).}

1.2. To each 1-motive M =

(X,

A,

T,

G,

u)

defined over

_C,

one can attach a

mixed

_Hodge

structure: Let =

Lie(G)

be the fibred

product

of

Lie(G)

and X over G.

_{The Q -vector}

_space

_{T~ (M) =}

is called

the

Hodge

realization of M. The filtration W, on M induces an

_increasing

filtration W. on

Tz(M) :

In

_particular

we have that

and

_{Hi (T, Z). According}

to

_{[D75] (10.1.3)}

on

we define a

decreasing

filtration F’ such that the

triplet

(Tz(M),

W.,

F")

we obtain is a Z -mixed

_Hodge

structure without

_torsion,

of level

_1,

of

type

~(0, 0), (0, -1), (-1, 0), (-1, -I)},

and with

_GrW

_{(Tz (M))}

_polarisable.

Let

_AHSQ

be the neutral tannakian

_{category of Q-mixed Hodge}

structures. Denote

_{T(M) =}

_{(TQ(M), We,Fe)}

_{the Q-mixed Hodge}

struc-ture attached to the 1-motive M and

_(T(M)~®

the neutral tannakian

sub-category

of

_MHSQ

_{generated by}

_T(M).

This

_subcategory

is endowed with

the fiber functor w which sends each

_object

of to its

algebraic Q-subgroup

P of

_GL(TQ(M))

and w defines an

_equivalence

of

tensorial

_categories

_(T(M))(8)

~ where is the

category

of finite dimensional

_{representations}

of P over

_Q.

We call P the

Mumford-Tate _group of M and often we denote it

by

MT(M).

By

defini-tion the notion of Mumford-Tate group is stable under

isogeny

and

duality

and so in this article we can work modulo

isogenies. By

[S72] Chapter

2

§2,

P is endowed with an

_{increasing filtration, W.,}

defined over

Q:

The inclusion that

implies

Since is a

_unipotent

_group, the derived _group of is

contained in

_{W- 2 (P).}

Let

_Grw

_(P)

=

PI W- I (P). According

to

[By831 32.2,

we know that acts

_trivially

on and

by

homotheties

on and that the

image

of in is

the Mumford-Tate group of

A,

MT(A).

In

particular,

is reductive

and

_W-i(P)

is the

unipotent

radical of P.

1.3. Let M = be a 1-motive defined over C and

P its Mumford-Tate group. Let = x x

Consider the

_following

group law on if

(9,:E, iv),

(t,

y, ~~)

are two elements of

~CM

we define

where

_{( , ~~ :}

x -

Q(l)

is

_the

Weil

_pairing.

We call

the

_{generalized Heisenberg}

group associaded to M.

Using

the canonical

isomorphisms Hom(

1.4. STRUCTURAL LEMMA.

(1)

The

_following

sequence:

is exact.

(2)

There exists an

_injective

_homomorphism

_{of groups}

I :

_{W- 1}

_(P)

--&#x3E;

such that the

_{following diagram}

commutes:

Proof.

(1)

Since is contained in

_(T(M))~,

by

2.21

_[DM82]

we

have the

surjective homomorphism

By

definition of

MSc,

we have

that g

E

_Ker(A)

if and

only

if both the

restrictions

_{of g}

to and to

_{W- ITQ (M)}

are trivial. But this

means that

W_2 (P) ^--J Ker(A),

and so we obtain the _{exact sequence}

which

implies

( 1.4.1 )

since P _preserves the filtration W..

(2)

The existence of I : - is

given by

(1.4.1)

and

by

the inclusions defined in

_(1.3.2).

We

_only

have to _prove that I is

an

homomorphism

of _groups. In order to

simplify notations,

we

identify

W-i (P)

with C and _{g2 -} are

any two elements of we have

where T is a _map from x

Hi (A, Q)

to

_Q(l).

In order

to determine

_T,

we have to understand how ₉₁ o _g2 acts on the

Hodge

Pi

=

MT(MI)

for i =

_{l, 2.}

According

_to 2.21

[DM82]

we have the

surjections

pri : W_ 1

(PSC)

-

_{w- i}

(Pi)

-Let 7r :

_W-,(P)

- be the

_surjection

_{constructed in}

_{( 1) .}

By

definition of we have

Hence modulo the canonical

_isomorphism

_Hom(X;

_HI

_{(A, Q)) ~}

_HI

_(A,

which allows us to

_identify

id with

_x,

we obtain that

Since

_{M2 - W_ 1 (M),}

_{pr2 ( 7r g2)}

acts on a contravariant _way on

and therefore we have

where the

_{symbol ’}

denote the contravariant action.

_{Consequently,}

modulo

the canonical

isomorphism

which

allows us to

_{identify pr2 (~r g2 )’ -}

id we have that

Again

modulo the canonical

_isomorphism

from (1.4.4) and (1.4.5)

we

_{finally get}

Remarks. -

(1)

If rkX = rkX’ = 1 then 1t is a

Heisenberg

group.

(2)

In

_[DO1]

P.

_{Deligne proposes}

to the author another

_approach

to

the

_study

of the structure of the Mumford-Tate group of M: he

suggests

one

_investigate

the of the Lie

_algebra

of the motivic Galois _group of M

_interpreted

as a 1-motive.

1.5. COROLLARY ON DIMENSIONS.

(1) If A fl 0, let

A be the connected

component

of the

_identity

in the

Zariski closure

_{of v x v"(X x X")}

and F = End A ®

Q.

Then

(2)

If A = 0,

let T be the connected

component

of the

_identity

in the

Zariski closure

_{of u(X)}

and F’ = End T 0

Q.

Then

where

remark that Hence

_applying

_[A92]

prop. 1 to MSc, we find that

1.6. Let M =

_(X,

_{X", A, AV,}

v, be a 1-motive defined over C. Denote

r = rank X and s = rankX~.

Moreover,

let

fxil (resp.

be a basis

of X

_(resp.

of

_X~).

We can also consider M as a

complex

M =

[X

---~

_G]

where G is an extension of A

_by

T =

_Hom(Xv,

_{Gm) .}

Denote

_by

parametrized

by

the

point

and

_by

the

_point

on the fibre of

_(xj)*(G)

above

_{v (xi )}

_{corresponding}

Consider the 1-motive

where and We can also write

~ in the form where and

are the

_{homomorphisms}

defined

_by

and

respectively,

and

_V)i3-

is the restriction of

1.7. THEOREM. - The 1-motives M and

_{®i 1}

_{EBj=1 Mij}

generate

the

same neutral tannakian

_subcategory

of

_M1tSQ’

In

particular, MT (M) _

Proof. Via the

_isomorphism

the extension G

_corresponds

to the

_product

of extensions

and so

which

_implies

that

To

_conclude,

it is

_enough

to show that

We remark that for each

i =

1,...,

r the

homomorphisms vrj

represent

the same extension

_Gj

and

so we can let

_v~

=

vYj.

In order to

simplify computations,

we

_identify

X

with zr and X~ with Z . If

_{dz :}

Z 2013~ 7~S is the

_{diagonal homomorphism,}

for each i =

1,...,

_{r we} have

Hence

taking

the sum for i =

1, ... , r,

we obtain

with the extension defined

_by

the

homomorphism

, The Cartier dual of

1-motive

If Z’ -

(Z’)’

is the

diagonal homomorphism

of Z’ in we

observe that

where is the 1-motive

with the extension defined

by

the

homomorphism

Now the

homomor-phism

defines a

_surjection

between the 1-motives and

which lifts to a

_surjection

from M to Hence M is a

quotient

of and so

_by

( 1. 7.1 )

and

( 1. 7. 2 ) ,

we can write M as a

_quotient

of

1.8. Let M =

[Z # G]

be a 1-motive over

C,

where G is the extension

of A

_{by Gm parametrized by}

=

Q

_E

AV(C),

and is the

point

S E

_G(C)

which lifts the

_{point v ( 1 ) -}

R E

_A(C).

Its Cartier dual is the

v

1-motive M" _ where Gv is defined

_by

the

_{point R}

E

_A(C)

and

E

_G~(C)

lifts the

_{point Q}

E

The

_{diagonal homomorphism}

d : Z ---~ Z x Z of Z and the

multipli-cation law _{p : Gm} x

_Gem

~

_Gm

of induce the

_morphisms

of 1-motives

and ~

J

In

_[Be98]

Thm

_1,

Bertrand constructs the 1-motive

where G is the extension of A = A x A v

_by

Cm parametrized by

the

_point

(Q, R)

E

_A~(C)

and

_U(l)

is the

point

S E

_G(C)

which lifts the

_point

1.9. LEMMA.

According

to

_(1.4.2),

we have that.

and . Since

we observe that and

we remark that

which

_implies

that Hence we have 1

and so we obtain that

We now have to

_verify

that

By

definition we have the inclusions

Since the notion of Mumford-Tate group is stable under

duality

we have

Therefore we can conclude that

which

_implies

that

2. The

_degeneracies

of

the Mumford-Tate _group.

2.1. Let M be a 1-motive defined over C and P its Mumford-Tate group.

Consider the

_{following subgroups}

of P:

Clearly

and

0,

then is a

_subcategory

of and so the natural

homomorphism

P - factorises via P -

which

_implies

that C

_{U~2&#x3E; (P).}

_Finally,

if P’ is the Mumford-Tate

group of

M",

then

U~3~ (P) =

U(4)

(Pv)

and

U(4)

(P) =

U(3)

(PV).

2.2. LEMMA

Proof.

_{By [A921}

Lemma 2

_(c),

P

_respects

the filtration

W. of

TQ(M).

Moreover each element of P acts

_trivially

on and

so

if g

E P we have

(g -

id) TQ (M)

C We remark also that

2.3. The Mumford-Tate _{group P} of a 1-motive M can

_present

some

dege-neracies which

correspond

to decreases of the dimension of P. We

_classify

these

_degeneracies

in the

_{following way:}

o M is deficient if

_{W-2 (P) =}

0.

_Among

the deficient

_1-motives,

there

are those which are

_trivially

deficient: M is

_trivially

deficient if

W- 1

(P) =

U~4~ (P)

and

_{W- 2 (P) -}

0

_{(or dually}

if

_{W_ 1 (P) -}

_U(3)(p)

n

_U(2)(p)

and

W-2 (P) = 0),

or if

_{W_ 1 (P) =}

0.

. M is

_{quasi-deficient}

if

W- 1

(P)

is abelian. Since the derived group of

W- i (P)

is contained in

_{W- 2 (P),}

deficience

_implies

_{quasi-deficience. Among}

the

quasi-deficient 1-motives,

there are those which are

_trivially

quasi-deficient : M is

_trivially

_{quasi-deficient}

if

_{W_1 (P)}

=

U(4)(p)

(or dually

if =

U (3) (P)

_n

U~~~ (P)),

or if =

W_2 (P).

o M is

_depressive

if 0

_dimQ

_W-2(P)

rankx - rankXv.

2.4.

_Examples

(1)

The 1-motives M = such that the

image

of v consists of torsion

_points

are

_examples

of

trivially quasi-deficient

1-motives.

(2)

The 1-motives without toric

_part

or the 1-motives M =

with the

_image

of u

_consisting

of torsion

points

are

_trivially

deficient

1-motives.

(3)

In

_{[JR87] Jacquinot}

and Ribet construct a deficient 1-motive

_using

an abelian

_variety

A with

_{complex multiplication,}

a

_point

in A" and a

homomorphism f :

Av --+ A.

(4)

In order to construct a

_{quasi-deficient}

1-motive it is

_enough

to take

a deficient 1-motive à la

Jacquinot-Ribet

and to

_"perturb"

its trivialization

~ by

an element of which is not a root of

_unity.

(5)

As

_example

of

_{quasi-deficient depressive}

1-motives we can take the

1-motive M =

[Z

-~

with u ( 1 ) _

(q, q2 )

and q

_{not a} root of

unity.

If

we want a

_depressive

but not

_{quasi-deficient 1-motive,}

it is

_enough

to take

the 1-motive M ©

M,

with _{llil any 1-motive} which is not

_{quasi-deficient.}

Unfortunately,

for the moment we have

_only

trivial

_examples

of

_depression,

i.e., examples

where the

depression

can be

_explain

in terms of deficience or

isogeny.

3.

Geometrical

_{interpretation}

of

deficience

and

of

_{quasi-deficience.}

3.1. We first introduce some notations due in

_{great part}

to D. Bertrand

(cf.

[Be94], [Be98]):

Let M = be a 1-motive over C and

P the Poincaré biextension of

_(A,

_A’).

An abelian

_{subvariety B}

of A x A"

is said to be

_isotropic

if the restriction of P to B is trivial or of order 2.

For each

_pair

_{(x, XV)}

in X x Xv consider the

_following

conditions:

(i)

there exists an

_isotropic

abelian

_subvariety

_B(x,xv)

of A x

_All,

which

contains a

_sufficiently

_{large multiple}

of

_(v(x),

We

_{denote ~}

the

canonical

_splitting

of the square of

x,x &#x3E;

(ii)

if

13

is the restriction of

_{(v, vv)}

to the _group Z

_{generated by}

a

sufficiently large multiple

of

_(x,

_xv)

in X x

_Xv,

the trivializations

_{01 z}

and

(3* ç

coincide.

We say that M is

quasi-isotropic

(resp. isotropic)

if

_{(i) (resp. (i)}

and

_(ii))

is

_{(resp. are)}

satisfied for each

_pair

_{(x, XV)}

in X x Xv. If we

consider the 1-motive M =

_(Z,

_{Z, E,}

_EV,

v,

vv, 0)

where E is an

elliptic

curve,

= P and

v" ( 1 ) =

Q

_E the conditions

(i)

and

(ii)

become

is

_trivial},

_{where £n}

_{(resp. S£)}

is the group of n-torsion

points

of E

_{(resp. ~" ) .}

(ii’)

corresponds

to one of the

following

E

is a root of

_{unity 1.}

Remark. - The condition

_(i)

is

_equivalent

to the existence of an

antisymmetric

homomorphism

A 2013~ ~ and of an

_integer

such that

_{g(x,xV)(P) ==}

for each

_{(P, Q)}

in The

antisymmetric homomorphisms

appear also in the construction of deficient

1-motives

_{by Jacquinot}

and Ribet

_(cf.

_[JR87]).

Also

_{using antisymmetric}

homomorphisms,

Breen

_interprets

the

_Jacquinot

and Ribet’s deficient

1-motives in terms of alternate biextensions

_(cf.

_[Br87]).

3.2. LEMMA. - Let M -

_Z,

A,

AV,

v,

vV,

w)

be a

_{quasi-isotropic}

1-motive over C. Then M has the same Mumford-Tate _group as the 1-motive

= x

_Gb]

where B is an

_isotropic

abelian

_subvariety

of A x AV

containing

a

_{sufficiently large multiple}

b of

(v (1),

B x

Gb

is the

and is the

_{point (b, a) of B x}

Cm

which lifts the

_{point (b, 0)}

of B. In

particular

if M is

_isotropic,

then

_{Ll (1) _ (b, a)}

with a a root of

_unity.

Proof.

_By

_hypothesis,

there exists an

_isotropic

abelian

_subvariety

B of A x A"

_containing

a

_{sufficiently large multiple}

b of

According

to

_[Be94]

Lemma

_1,

there exists an

_isogeny

S : B ----t A whose

restriction to B is the natural inclusion of B in A and such that is

a

multiple

of the Poincaré biextension of

(B, BV),

_P(B,BV),

Let M be the

1-motive associated to M in 1.8.

_Using

the

_isogeny

S : B -

_A,

we see

that M is

_isogeneous

to the

_following

1-motive

where B x

_Gb

is the extension of B

_{by Cm parametrized by}

the

point

(0, b)

E B~. Since M is

_{quasi-isotropic,}

admits a canonical

_splitting

~,

and therefore we obtain =

(b, a)

with a E

Gem.

If M is

isotropic,

we observe that

Ll ( 1 ) - (b, a)

with cx a root of

_{unity. Finally,}

since M is

isogeneous

to

_J1~1,

we have

_{MT (M) -}

and so

_by

1.9 we obtain

MT(M)

=

3.3. Since A is an abelian

_variety

defined over

C,

we can view it as

a

_quotient

V/A

of a C-vector _space V

_by

a lattice A.

_By

definition

A" =

V~/A~,

where Vv =

Hom, (V,

C)

is the C-vector _space ofC-antilinear

forms,

and A" -

_~l

1 E

_V"!

_I

_{Im l(A)}

C

_According

to

_[LB92]

_§4,

the

duality

between A and Av can be

expressed

in terms of the non

degenerate

R-bilinear form

By [LB92]

§5,

the Poincaré biextension P of

_{(A, A")}

is defined

by

the Hermitian form

The first Chern class

_{of P,}

_{cl (P),}

can be identified with the Hermitian form H or with the real valued alternate form

By

[D75]

(10.2.3),

in

_Hodge

realization the Poincaré biextension P

defines a

_pairing

(, ) z :

x 2013~ _{which coincides}

with the

_pairing

_{(, ) :}

V x Vv - R once we extend the scalars to R and

we

identify Z(l)

with Z

_{by sending}

2z7r to 1. From now _on, we

identify

these two

_pairings.

3.4. LEMMA. Let M = be a 1-motive over C.

The

_following

conditions are

_equivalent:

is

abelian,

HI

(B, Q),

where B is the connected

_component

of the

_identity

in the Zariski

closure

_{of v(X)}

x

_VV(XV).

Proof. Let P =

MT(M).

According

to the structural

lemma,

_we

have the

_following

inclusions:

and we know that the commutator of two elements g, =

(s,

~,

and

where

_(,

_)Q

is the

_pairing

obtained from

_{( , ) ~}

by

extension of scalars. The

result now follows from the fact that is abelian if and

_only

if

~gl, 92~ =

0 for each gl, g2 E W- I

(P).

3.5. THEOREM. - Let M =

_{(X, Xv, A, AV, v, vV, 1/J)}

be a 1-motive over C.

The

_following

conditions are

_equivalent:

(i)

M is

_{quasi-isotropic,}

(ii)

M is

_{quasi-deficient,}

Proof.

(i)

~

_(ii):

_By

_hypothesis

there exists an

_isotropic

abelian

_subvariety

B of A x AV. But this

_implies

that

_{ci (Pj B)}

= 0 and therefore

by (3.3.2)

and 3.4 we can conclude that is abelian.

(ii)

~

_(i):

From 3.4 and

_(3.3.2),

we have

_{ElvB XVB ==}

0.

Moreover,

if we consider a

special

case of 3.4

_(it

is

_enough

to take the

pairs

(i, 0)

and

_{(0, XV)}

of

_{H, (B, Q)),}

we find that

0 for

each in which

_{implies by (3.3.1)}

that 1. Since

and

1,

we have that the restriction of the Poincaré biextension to B is trivial.

(i)

-

_{(iii) :}

We have to _prove that =

U(4)

(MT(M)).

Since

_by

definition U(4)

_(MT(M))

C

_{W_ 1 (MT (M)) ,}

it is

_enough

to show

that

_{W_ 1 (MT (M))}

C

U(4)

_(MT(M)).

_Using

the same notations as in

_1.6,

by

Theorem 1.7 we have that where

There exists the

_injection

which

respects

the filtration

W, .

According

to 3.2 we know that for each

i, j

there is a 1-motive

_{A4 j}

such that =

MT(A4ij)

and

where is an

_isotropic

abelian

_subvariety

of A x A v

_containing

a

sufficiently large multiple

bij

of

_(Vij(Xi),

_Bij

x

_Gb,,

is the extension of

_Bij

x

_B2~

_by

parametrized by (0,

E

_Bfj,

and is the

point

_{(bij, aij)}

E

_Bij

x

_Gm

which lifts the

_point

0)

E

_B...

Consider the two 1-motives

where and is the

_diagonal

homomorphism

of

Z,

we have

therefore. But since.

we remark that .

_Using

Let g

be an element of Since

we have that But this

_implies

that

0,

and so we can conclude

_{that g}

E U (4)

_(MT(M)).

3.6. LEMMA. Let M = be a 1-motive defined over

C. The

_following

conditions are

_equivalent:

(i)

M is

_isotropic,

(ii)

M is deficient.

Proof.

(i)

~

_{(ii) :}

This is done in

_[Be98]

Thm

_1,

but

_unfortunatly

the

converse was

_{proved only}

for 1-motives defined over a number field.

(ii)

~

_{(i) :}

We will prove that if M is not

_isotropic

then it is

not deficient. If M is not

_{quasi-isotropic, by}

Theorem 3.5 it is not

quasi-deficient,

and so in

_particular

it is not deficient. If M is

_{quasi-isotropic,}

then

_according

to 3.2 M has the same Mumford-Tate _group as the 1-motive

= x

_Gb~,

where B is an

_isotropic

abelian

_subvariety

of A x Av

containing

a

_{sufficiently large}

_{multiple b}

of x

_Gb

is the

extension of B = B x Bv

_{by Cm}

_{parametrized by (0, b)}

E Bv and is

the

_{point (b, cx)}

c B x

_Gm

which lifts the

_{point (b, 0)}

E B. Consider the two 1-motives

where = a and

u2 ( 1 )

homomorphism

of

Z,

we have

is the

_diagonal

and so But with

and therefore

Since M is not

_isotropic,

a is not a root of

_unity

and so

_{by [B01]}

5.4 we

have

3.7. THEOREM. Let M = be a 1-motive defined over C. The

_following

conditions are

_equivalent:

(i)

M is

_isotropic,

(ii)

M is

_deficient,

(iii)

M is

_trivially

deficient.

Proof.

Using

the same notations as in

1.6, by

Theorem 1.7 we have

_ __

"I--, There

exists the

_injection

which

_respects

the filtration W..

(i)

~

_(ii):

_{By (3.7.1 )}

if all the

_Mij

are deficient then so is M.

Moreover, according

to 3.6 the 1-motive

_Mi3

is deficient if and

_only

if it is

isotropic.

Hence we can conclude that: M is deficient if and

_only

if

_Mij

is

deficient for each

_{i, j,}

if and

_only

if is

_isotropic

for each

_{i, j,}

if and

_only

if M is

_isotropic.

(i)

-

_(iii):

_According

to 3.2 for each

_{i, j}

there exists a 1-motive

_A4..

such that = and

where

_BZ~

is an

_isotropic

abelian

_subvariety

of A x A v

_containing

a

sufficiently large multiple

is the extension of

parametrized

by

we have

and so

using (3.7.1)

we obtain the

injection

By hypothesis

we

already

know that

W-2(MT(M)) =

0. In order to

conclude it is

enough

to prove that ’

we have and therefore

we can conclude that

Remark. - The direct

_proof

that

_(ii)

_{implies (i)}

answers a

_question

of Y. André

_(cf.

_[Be98]

Thm 1 Remark

_(iv)).

BIBLIOGRAPHY

[A92]

Y. ANDRÉ, Mumford-Tate groups of mixed Hodge structures and the theorem

of the fixed _{part, Compos. Math., 82, (1992).}

[B01]

C. _BERTOLIN, Périodes de 1-motifs et _{transcendance,} to _{appear in} the J. Number

Th.

[Be94]

D. _BERTRAND, 1-motifs et relations _{d’orthogonalité} dans les groupes de

Mordell-Weil, Publ. Math. Univ. Paris VI, 108, Problèmes Diophantiens 9293, exposé 2.

(Math. Zapiski, 2,

(1996)).

[Be98]

D. _BERTRAND, Relative splitting of one-motives, Number Theory

(Tiruchira-palli,

1996), Contemp. Math., 210,

Amer. Math. _{Soc., Providence,} RI

_(1998).

[Br87]

L. _BREEN, Biextensions _{alternées, Compos. Math., 63,}

_(1987).

[By83]

J.-L. _BRYLINSKI, 1-motifs et formes _automorphes

_(théorie

arithmétique des

domaines de

_Siegel),

Publ. Math. Univ. Paris VII, 15,

(1983).

[D75]

P. _DELIGNE, Théorie de Hodge III, Pub. Math. de _{l’I.H.E.S., 44, (1975).}

[D82]

P. _{DELIGNE, Hodge cycles} on abelian varieties, Hodge cycles, motives and

Shimura varieties, Springer L.N. 900,

(1982).

[D01]

P. _DELIGNE, letter to the author

_(2001).

[DM82]

P. DELIGNE, J.-S. MILNE, Tannakian categories, Hodge cycles, motives and

Shimura _{varieties, Springer} L.N. 900,

(1982).

[JR87]

O. JACQUINOT, K. _RIBET, Deficient _points on extensions of abelian varieties _by

Gm, J. Number _Th., 25

_(1987),

133-151.

[LB92]

H. _LANGE, C. BIRKENHAKE, Complex abelian varieties, Springer-Verlag 302

(1992).

[S72]

N. SAAVEDRA RIVANO, Catégories tannakiennes, Springer L.N. 265

_(1972).

Manuscrit reçu le 18 juin 2001,

revise le 19 novembre 2001,

accepté le 4 f6vrier 2002.

Cristiana BERTOLIN,

Universite Louis Pasteur

UFR de Mathématiques et _Informatique

7 rue Rene Descartes

67084 _{Strasbourg (France).}