A geometric description of the class invariant homomorphism

A. A

GBOOLA

A geometric description of the class invariant

homomorphism

Journal de Théorie des Nombres de Bordeaux, tome

6, n

o

_{2 (1994),}

p. 273-280

<http://www.numdam.org/item?id=JTNB_1994__6_2_273_0>

© Université Bordeaux 1, 1994, tous droits réservés.

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

A

_geometric

_description

of the

class invariant

_homomorphism

par A. AGBOOLA

In recent _years, a certain amount of work has been done on the Galois

structure of

_{principal homogeneous}

_spaces of finite _group schemes that are

constructed via

_dividing

_points

on abelian varieties. This

study

was

_begun

by

M. J.

_Taylor

in

_[Tl]

and was

_originally

motivated

_by

the fact that

such

_{principal homogeneous}

_spaces are _very

_closely

connected with certain

rings

of

_integers.

The

_starting

_point

of the

_theory

is the so-called class

invariant

_homomorphism

which was first introduced

_by

Waterhouse in

_[W].

The _purpose of this note is to

_point

out a

_simple

_geometric

_description

of

this

_homomorphism

_(as

considered in

_[Tl])

in terms of the restriction of

certain line bundles to torsion

_subgroup

schemes of the abelian

_variety

in

question.

We shall also make a number of remarks

_concerning

both old

results and new

_questions

that arise in

_light

of this

_description.

In what

_follows,

we shall confine ourselves to the case of abelian varieties

defined over number fields. It should however be noted that there is no

difficulty

in

_carrying

out a similar

analysis

of the

analogous

situation over

global

function fields

_(cf.

_[A3]).

I would like to thank K. Ribet for

_drawing

_my attention to

_[R].

I am

also

_grateful

to T.

_Chinburg,

M. J.

_Taylor,

and W.

_Messing

for

_interesting

conversations.

1. In this section we shall recall the definition of the class invariant

homo-morphism

for abelian varieties. We refer the reader to

_{[W], [Tl]}

and

_[BT]

for further details.

Let F be a number field with

_ring

of

_{integers D F .}

_Suppose

that is

an abelian

variety

with

_{everywhere good}

_reduction,

and let

_AIIDF

denote the N6ron model of

_A/F.

Write

A (resp.

_Â)

for the dual abelian

_variety

of A

_{(resp. A). (We}

shall

_frequently

omit the

_dependence

_upon F from

our notation when there is no

danger

of

confusion.)

It follows from the

universal

_property

of the N6ron model that there are natural

_isomorphisms

A(F) (resp.

we shall feel free to make such identifications without further notice.

Let denote

_{the DF-group}

scheme of

p’-torsion

on ,4. Then

Apn

is

a,fhne with Cartier dual

_Apn,

and both

_Apn

and

_Âpn

are twisted constant

groups. Hence we _may

apply

Theorem 2 of

[W]

and deduce that there is an

_isomorphism

(Here

Gm)

is

_computed

in the

_category

of

_fpqc

_group

_schemes,

and we take

cohomology

with

_respect

to the

_fpqc

_site.)

This

_isomorphism

may be described as follows. Let V be an extension of

by

Gm ;

then there is an exact _sequence of commutative group schemes

Applying

the functor

_{Hom(.Apn, -)}

to

₍₁₎

_yields

the exact _sequence

Let c E denote the

_identity

map, and define Y to be the

inverse

_image

of c in

₍₃₎

above

_(i.e.

Y is the sheaf of

_{group-theoretic}

sections of

_(1)).

Then Y is a

principal homogeneous

_space

of Ap.

(see

_e.g. Theorem

2’ of

_[W])

and so determines an element

vl (V)

of

H 1

(D F,

We next observe that there is an obvious natural

homomorphism

and that Kummer

_theory

on .~4 affords us a natural _map

The class invariant

_homomorphism

is defined

_by

_qbn

_{= v2} v3.

This

_homomorphism

_may be described somewhat more

_concretely

as

follows. Let Fc be an

_algebraic

closure of

F,

and write

QF

for the absolute

let -,5

(resp. 93)

be an

D F-Hopf

algebra

that

represents

the group scheme

(resp.

It is shown in

_[Tl]

_{that §}

_{(resp. 23)}

is an

F-order in the

_algebrah

=

(resp. Map(G,

where here

OF

acts on both G and Fc.

Let

_Q

E

A(F),

and set

Define the Kummer

_algebra

_FQ

_by

_{FQ =}

and write

_DQ

for the

_integral

closure in

_{FQ .}

The

_algebra

H acts on

_F~

by

for

_f

E

_FQ

and a =

EgEG

agg E 7-l.

In

_general

_DQ

is not acted _upon

_by

Define the Kummer order

_CQ

to be the

_largest

S5-module

contained in It is shown in

_§3

of

_[Tl]

_(see

also

_p.186

of

_[BT])

that

_CQ

is a

principal

homogeneous

_space of the

algebra

23,

and that

_CQ

is a

locally

free

~-module.

Then

(ibid.)

0,,(Q) = (CQ)

E

2. Recall

that,

via standard

_theory,

there are canonical

_isomorphisms

Thus each

_point

_Q

E

A(F)

determines a

(rigidified)

line bundle

G~

on

A,

and an extension

of A

_by

G,

(cf.

[Mu,

§13

and

_§23]

or

_{[Mi, §11]).}

The inclusion _map

_{Apn ~}

A induces a natural

_homomorphism

Hence,

from

_(5),

we obtain an extension

g(Q)(n)

:=

of Ap.

by

G,.

We thus have a

_homomorphism

defined

_by

_{0’ n =}

v2 o _v4. Note that the

_{map ?P’ n}

_is

_simply

_{that induced}

_by

restricting

the line bundle

_,C~

to the

_subgroup

scheme of A. We are now in a

position

to state the main result of this note.

THEOREM 1.

_1/Jn

=

1/J~.

Proof..

To _prove the result it suffices to show that

We first of all note that the natural

_homomorphism

is an

_injection

(see

_e.g.

[BT,

Lemma

2.1]),

and so

(8)

will follow if we show

that

We next observe that

_{by reasoning exactly}

as above in the definition of

vi’s,

but this time

_working

over

Spec(F)

rather than

Spec(D F ) ,

we _may define

natural

_{homomorphisms}

_{vi :}

(It

is

_perhaps

worth

_remarking

that =

0,

and _so the

map v2

is

not

_particularly

_{interesting!)}

Let

_{G(Q) (resp.}

_G(Q)(n))

denote the

_generic

fibre of

_9(Q)IDF

_(resp.

g(Q)(n) IDF),

and set 0 =

v’

o

v4.

Then

G(Q)

is the extension of

AIF

_by

G,

afforded

_{by Q}

E

_A(F),

and so

by functoriality

we have

and

However, Proposition

3.1 of

_[R]

asserts that

(note

that

_{Ribet’s ç}

is our

v3,

and his P is our

?),

and this

implies

the

For the sake of

_{completeness,}

we shall now

_give

a brief sketch of the

proof

of

_(10).

The reader _may refer to

_§3

of

_[R]

for full details.

Let D be the divisor on A determined

_by

the

_point

Q

(cf. (4) above).

Let

_F°(A)

denote the function field of and for each a E write

Ta :

A - A for the

_{translation-by-a}

map. Ribet shows that

is

_isomorphic

to the group of

pairs

(a, f )

E x

_FC(A)X

_satisfying:

under the

_{multiplication}

where

_g(x)

=

f2(x+al)fl(x).

_

Now choose a

p~th

root

_Q’,

say, of

Q

on and let E be the

corre-sponding

divisor on Then

for some function h E and _{it may} be shown that the map s :

Ap- (Fc) -

defined

_by

is a

group-theoretic

section of the natural

epic

It

follows that E

HI (F, Âpn)

is

_represented

_by

the

_cocycle

₆₀

= a. s - s

(where a

E

_QF).

_{By explicitly computing 80"’}

_{it may} be deduced that

where e~, denotes the Weil

pairing.

This establishes

(10),

as

required.

It follows from Theorem 1 that there are certain constraints on the

_image

of 0,,.

Let W be any

flat,

commutative

_{D F-group}

_scheme,

and write m :

W x W - W for the

_{multiplication}

_map. Let _{pi :} W x W -&#x3E; W

_(i

=

1, 2)

denote

_projection

onto the ith factor. Recall

_(see

eg

[Se,

Chapter

VII,

§3])

that a class c E

_Pic(W)

is said to be

_primitive

if

_m*(c)

=

pi(c)

₊

p2(c).

Let

_PPic(W)

denote the

_subgroup

of

_Pic(W)

_consisting

of the

_primitive

elements of

_Pic(W).

Then

_{(oP. cit.) PPic(W)}

is an additive functor of

_W,

and if W is an abelian

variety,

then

PPic(W)

=

Pic°(W).

Hence _we deduce

that the

_image

of

_1/;n

is contained in i.e. that we in fact have

We remark that this fact also follows from the results contained in

_§l

of

[CM] .

3. Theorem 1

_implies

that the

_study

of the

_{homomorphism 0,,}

is

_really

the

_study

of the behaviour of line bundles on ,~4 under restriction to certain

torsion

_subgroup

schemes of ~4. It is

_interesting

to reformulate the results of

[A], [AT]

and

_[ST]

in

_light

of this fact. For

_{example, combining Proposition}

1 above with Theorem 1 of

_[ST]

_{immediately yields}

the

_following

result. THEOREM 2

_{(SRIVASTAV-TAYLOR).}

Let

_AIF

above be a CM

elliptic

curve.

Suppose

that _p &#x3E;

_3,

and let L E be a torsion line bundle. Then

£)Apn is

trivial

_{f or}

all n &#x3E; 1.

It is

_conjectured

that a similar result holds for an

_arbitrary

CM abelian

variety

with

_everywhere

_good

reduction.

_(Note

added in

_proof:

The author has

_recently

shown that Theorem 2 also holds in the non-CM case. See

his

_forthcoming

_paper ’Torsion

_points

on

_elliptic

curves and Galois module

structure’.)

We shall now introduce some further notation. Let denote the

normalisation of

_Composing

_’Øn

with the natural _map

-i

gives

us a

_homomorphism

The _{map ’Pn may} be described in terms of Kummer orders

_{(cf. § 1 )}

as

follows.

_{By definition,}

=

Spec(M),

where M is the

unique

max-imal

_{D F-order}

contained in 1i.

_Suppose

_{that Q}

E

A(F).

Then

_{pn (Q)}

=

(~:Q

g-i5

E

Next,

we recall that

Qn

(and

hence also

’Pn)

factors

through

the

projec-tion

_{~(F) 2013~}

_Also,

the inclusion map

/4pn

- induces

natural

_morphisms

--+ and

-

HI

(Apn,

Gm) .

_Taking

_inverse limits

_{of 0.,,}

_and

’Pn with

respect

to these maps, and

identifying I

with

Pic° (A)

as _per

₍₄₎

above

_yields

and

We remark that a line bundle G E lies in the kernel

_of’l/Joo

if and

_only

if

_£’IApn

is trivial for all n &#x3E; 1. We _may now _pose the

following

question.

QUESTION

1.

_(i)

Is the kernel

_of

_{finite ?}

(ii)

Is

_surjective

_(possibly

_only

up to

finite

It seems

possible

to

_regard

an affirmative answer to these

_questions

as a

loose

_analogue

of the Tate

_conjecture

for abelian varieties

_(now

a theorem

of

_Faltings

_(see

_[F]

or

[CS])).

This states that the natural _map

is an

isomorphism.

(Here

TP(A)

is the

p-adic

Tate module of

A,

and

EndF(A) (resp.

EndF(TP(A)))

is the

_ring

of

_{endomorphisms}

of A

_(resp.

Tp(A))

that are defined over

F.)

It is shown in

[AT]

that

Question

l(i)

has an affirmative answer

(subject

to certain technical

hypotheses)

in the case that

A/F

is a CM

_elliptic

curve and _{p is} a

_prime

of

_{good ordinary}

reduction.

We _may also ask similar

_questions

about the kernel of When

_A/F

is

a CM

elliptic

curve and _p is an

_ordinary

_prime,

it is shown in

_[Al]

and

_[AT]

that the kernel of

_is

_equal

to a canonical

subgroup

of

A(F)

_@z

_Zp

first

defined

_by

R.

_Greenberg

_using

Iwasawa

_theory

_(see

_[G]).

This

_subgroup

may be described in terms of the

p-adic height

pairing

on A

(see

[P])

and

is in

_general

of infinite order.

_(See

also

_[A2]

for a discussion of in the

context of

_p-adic

_{representations.)}

It would be of some interest to obtain a

better

_{understanding}

of the

_relationship

between class invariants and

_p-adic

heights.

(See

[A4]

for further remarks on this in the case of CM

elliptic

curves.)

We conclude with the

_following

_question.

QUESTION

2.

_Suppose

that the kernel

_of

is

_{of infinite}

order. Does this

_imply

that that the

_p-adic

_height

_Pairing

on A

(cf.

[PR]

or

(Sc~)

is

BIBLIOGRAPHIE

[A1]

A. _Agboola, Iwasawa _{theory of elliptic} curves and Galois module structure, Duke Math. J. 71 _{(1973), 441-462.}

[A2]

A. _Agboola, p-adic representations and Galois module structure, preprint.

[A3]

A. _Agboola, Abelian varieties and Galois module structure in _{global function}

fields, Math. Z. _{(to appear).}

[A4]

A. _Agboola, On _{p-adic height pairings} and _{locally free classgroups of Hopf}

or-ders, in _preparation.

[AT]

A. _Agboola, M. J. Taylor, Class invariants of Mordell-Weil groups, J. reine angew. Math. 447 (1994), 23-61.

[BT]

N .P. _Byott, M. J. Taylor, Hopf orders and Galois module structure, in: _Group

rings and _classgroups, K. W. _Roggenkamp, M. J. _{Taylor (eds.),} Birkhauser, 1992.

[CM]

L. Childs, A. _Magid, The Picard invariant of a _{principal homogeneous space,} J.

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[CS]

G. Cornell, J. Silverman (eds.), Arithmetic _{Geometry, Springer,} 1986.

[F]

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[G]

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[Mi]

J. Milne, Abelian varieties, in: Arithmetic Geometry, G. _Cornell, J. Silverman

(eds.), Springer, 1986.

[Mu]

D. Mumford, Abelian Varieties, OUP, 1970.

[PR]

B. _Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adique, Invent. Math. 109

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[P]

A. _{Plater, Height Pairings} on Elliptic Curves, Ph.D. Thesis, Cambridge

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[R]

K. Ribet, Kummer theory on extensions _of abelian varieties _by tori, Duke Math. J. 49 _(1979), 745-761.

[Sc]

P. Schneider, Iwasawa theory for abelian varieties2014a _{first approach,} Invent. Math. 71 _(1983), 251-293.

[Se]

J.-P. Serre, Algebraic Groups and Class _{Fields, Springer,} 1988.

[ST]

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AMS Transactions 153 _(1971), 181-189. A. _Agboola Department of Mathematics UC _{Berkeley Berkeley} CA 94720 USA