A group law on smooth real quartics having at least $3$ real branches

J

OHAN

H

UISMAN

A group law on smooth real quartics having

at least 3 real branches

Journal de Théorie des Nombres de Bordeaux, tome

14, n

o

_{1 (2002),}

p. 249-256

<http://www.numdam.org/item?id=JTNB_2002__14_1_249_0>

© Université Bordeaux 1, 2002, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

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

A

_group

law

on

smooth real

quartics

having

at

least

3

real branches

par JOHAN HUISMAN

RÉSUMÉ. Soit C une courbe

quartique

réelle lisse dans

P2,

_qui

admet au moins trois branches réelles On pose B =

B1 x _B2 x _B3 et soit O ~ B. On note 03C4O

l’application

de B sur la

composante neutre

_{Jac(C) (R)0}

de l’ensemble des

_points

réels de

la jacobienne

de

_C,

définie en _posant

03C4 (P)

comme étant la classe du diviseur

_{03A3 Pi -}

_Oi.

_Alors,

03C4O est

bijective.

On montre que cela

permet une

_{description géométrique}

_explicite

de la loi de _groupe sur

_Jac(C)

(R)° .

Cela

_généralise

la

_description

_{géométrique}

clas-sique

de la loi de _groupe sur la _composante neutre de l’ensemble des

_points

réels de la

_jacobienne

d’une

_cubique.

Si la quartique

est définie sur un _corps de nombres

_réel,

alors on obtient une

de-scription

géométrique

d’un _sous-groupe du _groupe de Mordell-Weil d’indice un diviseur de 8.

ABSTRACT. Let C be a smooth real

_quartic

curve in P2.

Sup-pose that C has at least 3 real branches

B1,

B2, B3.

Let B = B1 x _B2 x _B3 and let O ~ B. Let _03C4o be the _map from B into the neutral _component

_{Jac(C) (R)°}

of the set of real _points of the Ja-cobian of

_C,

defined

_by

_letting

_{03C4o (P)}

be the divisor class of the

di-visor

_03A3

Pi -

Oi. Then,

03C4o is a

bijection.

We show that this allows an

_{explicit geometric description}

of the _group law on

It

_generalizes

the classical

_{geometric description}

of the _group law

on the neutral _component of the set of real

_points

of the Jacobian of a cubic curve. If the

_quartic

curve is defined over a real number field then one _gets a

_{geometric description}

of a

_subgroup

of its

Mordell-Weil _group of index a divisor of 8.

1. Introduction

The _group law on a cubic curve is a famous classical

_geometric

construc-tion

_[9].

It

_gives

a

_{geometric description}

of the Jacobian of the curve. For curves of

_higher-or

_Iower-degree

such a

geometric description

is nonex-istent. If one is

only

interested in real

points

of a curve then the

_present

paper may be of interest: We show that there is a _group law on the

Carte-sian

_product

of 3 real branches of a smooth real

_quartic

curve in

P~,

in

case, of course, it does have at least 3 real branches. It

_gives

rise to a geo-metric

_description

of the neutral

_component

of the set of real

_points

of the Jacobian of the

_quartic.

This

_generalizes

_{the group law} on a real branch of a smooth real cubic curve.

Other

_{generalizations}

_{of the group law} on a real branch of a cubic curve can be found in

[7,

2, 3,

4].

All

_present

generalizations

are based on an

explicit

class of

_nonspecial

divisors on real

_algebraic

curves

[6]

(see

also

Section

_4).

The _paper is

_organized

as follows. In Section

_2,

we will be more

_precise

about the

_object

of the paper. In Section

3,

we

give

the

geometric

descrip-tion of the neutral

_component

of the set of real

_points

of the Jacobian of a

_quartic.

It relies on a statement

_concerning

_points

in

_general

_position

whose

_proof

is

_given

in Section 5. In Section

_3,

we also

give

an

_application

to the Mordell-Weil _group if the

_quartic

is defined over a real number field. This

_application

is

_proved

in Section 6. Section 4 recalls some useful facts on

_nonspecial

divisors on real curves.

2. The

_object

Let C be smooth real

_quartic

curve in

1~2.

_By

the _genus

_formula,

the genus of C is

equal

to 3. A connected

component

of the set

C(R)

of real

points

of C is called a real branch of C.

_By

Harnack’s

_Inequality

_[5],

the number of real branches of C is less than or

_equal

to 4. Smooth real

_quartic

curves

having

at least 3 real branches abound.

We assume

_throughout

the _paper that C has at least 3 real branches.

Choose,

once and for

all,

3

mutually

distinct real branches

Bl, B2, B3

of C and

_put

The Jacobian

_Jac(C)

of C is a real Abelian

_variety

of dimension 3. Its set of real

_points

_{Jac(C) (R)}

is a

_compact

commutative real Lie group. We denote

_by

the connected

_component

of

_Jac(C)(R)

that con-tains 0.

Let 0 E B be a base

_point.

Define a _map

by

for all P E

_B,

where cl denotes the class of a divisor. Since = 0

and since B is

_connected,

_To(P)

_belongs

to

Jac(C)(R)O,

for all P E B.

Hence,

To maps B into

Applying

the

general

result of

[7]

to

Theorem 2.1. The _{map To :} B -+ a

bijection.

0

In

_particular,

one

_gets,

_by

_transport

of

_structure,

a _group law on B. The

object

of this paper is to

give

a

_{geometric description}

of this group law.

3. The

_geometric

_description

The

_following

statement is the main

_ingredient

in the

_geometric

descrip-tion of _group law on the neutral

_component

of the set of real

points

of the Jacobian of C. It

_states,

_among other

_things,

that _any 9 real

_points

of C

which are

uniformly

distributed over the real branches

_{B1, B2,}

B3,

are in

general position

with

_respect

to cubics.

Theorem 3.1. Let

_{P, Q, R}

E B.

_Then,

there is a

_unique

real cubic F

passing

through

the 9

_{points Pi,}

_Qi,

_R,-,

_for

i =

_{1, 2, 3, i. e.,}

Moreover,

there is a

_unique

S E B such that

We

_postpone

its

_proof

to Section 5 and

_explain

first how Theorem 3.1

gives

rise to the

_{geometric description}

of the _group law we are

looking

for. Choose 0 E B.

_According

to Theorem

_3.1,

there is a real cubic G such that

Moreover,

there is an element X E B such that

Now,

the

_{geometric description}

_{of the group law of} is

_given

in the

_following

statement. Recall

_{(Theorem 2.1)}

that To is a

bijective

_map from B onto and is defined

_by

_sending

P E B to

Theorem 3.2. Let

_{P, Q, R}

E B.

_Then,

Proof. Suppose

that

_{To(P) +To(Q) +To(R)}

= 0 in

Jac(C)(R).

By

Theo-rem

_3.1,

there is a real cubic F such that

Moreover,

there is an element S E B such that

Then,

Taking

divisor classes and

_using

the

_hypothesis,

one

_gets

that

in

_Jac(C)(R),

_i.e.,

_TX(S)

= 0.

By

Theorem

2.1,

S = X and it follows that

there is a real cubic F such that

In order to show the converse, suppose that there is a real cubic F

satisfying

the above

_equation.

_Then,

Remark 3.3. Theorem 3.2 allows to do calculations in the neutral

com-ponent

of the set of real

_points

of the Jacobian of

_{C, just}

as in the case of

elliptic

curves

[9]:

The inverse of an element P E B is obtained as follows.

_By

Theorem

_3.1,

there is a

_unique

real cubic F such that

_Moreover,

there is a

_unique

element

Q

E B such that F - C

_Oi

+ Pi

+

Qi

+

Xi.

Then, Q

is the inverse of P.

The sum of two elements

_P,

_Q

E B is obtained as follows.

_By

Theo-rem

_3.1,

there is a

_unique

real cubic F such that

_X"

-Moreover,

there is a

unique

element R E B such that

I would like to conclude this section with the

_{following application}

of Theorem 3.2 to the Mordell-Weil _group of a smooth

quartic

defined over a real number field.

Let K be a real number field and let C be a smooth

quartic

in

1~2 defined

over K. Assume that

CR

= C _{x K} R has at least 3 real branches

Bl,

B2,

B3.

Let

_{again B}

=

B1

x

_BZ

x

_B3.

An element P E B is a K-rational

point

of B if the divisor

_P,

+

P2

+

_P3

on

CR

comes from a divisor on C.

Equivalently,

P E B is K-rational if and

_only

if the divisor

Pi + P2 + P3

is invariant for the action of the Galois _group

_Gal(C/K).

The set of K-rational

_points

of B is denoted

_by

_B(K).

Denote

_by

_Jac(C)(K)O

the inverse

_image

of

_Jac(C)(R)o

_by

the natural

map from

Jac(C)(K)

into

Jac(C)(R).

Since the

subgroup

Jac(C)(R)o

is of

index 4 or 8 in

Jac(C) (R) [1,

Theorem

4.1.7],

the

subgroup

Jac(C)(K)°

has index

2i

in the Mordell-Weil _group

_Jac(C)(K)

of

_C,

for some i E

_{{0,1, 2, 3}.}

Theorem 3.4.

_Suppose

that 0 is a K-rational

_point

_of

B.

_Then,

the

map To maps the subset

B(K)

of

B

bijectively

onto

Jac(C)(K)O.

In

par-ticular,

B(K)

is a

subgroup of

B and the restriction

_of

_{Tp to}

B(K)

is an

isorraorphism

onto

Jac(C)(K)O.

A

_proof

is

_given

in Section 6.

4.

_Nonspecial

divisors

The

_{following proposition}

_[1,

_Corollary

_4.2.2]

_(cf.

_[6,

_Proposition

_2.1])

is

at the basis of the results. For the convenience of the

_reader,

a

proof

is

given.

Proposition

4.1. Let C be a

geometrically integral

_proper smooth real

al-gebraic

curve and let cv be a nonzero rational

differential form

on C.

Then,

for

all real branches B

_{of C,}

the

_{degree of}

the divisor

_of

w on B is even.

Proof.

The restriction of W to a real branch B of C is a nonzero real

meromorphic

differential on B. Since C is proper and

smooth,

B is real

analytically isomorphic

to the real

_projective

line

_Therefore,

we

may assume that B =

P (R) .

Then,

there is a nonzero real

meromorphic

function

_f

on

Pl (R)

such that

In

_particular,

the divisor

_div(wlb)

is

_equal

to the divisor

_{div( f )}

of

_{f .}

Note that

_{div( f )}

=

f *0 -

f *oo.

Since

deg( f *P)

is constant mod 2

[8],

for P E

P~(R),

the

_degree

of

_{div( f )}

is even. 0

Proposition

4.1 has the

_following

statement

_[6,

Theorem

_2.3]

as a

Theorem 4.2. Let C be a

geometrically integral

_proper smooth real

alge-braic curve and

let g

be its _genus. Let D be a divisor on C and let d be its

_degree.

Let k be the number

_of

real branches

_of

C on which D has odd

degree. If d + J~ &#x3E; 2g - 2

then D is

_nonspecial.

We need the

_following

lemma.

Lemma 4.3. Let C be a

geometricall y integral

_proper smooth real

algebraic

curve and

_{let g}

be its _genus. Let D be a divisor on C and let d be its

_degree.

Let k be the number

_of

real branches

of

C on which D has odd

degree.

then

_{h° (D)}

= 0.

Proo f. Suppose

0. Then there is an effective divisor E on C which is

_{linearly equivalent}

to D. Let

_f

be a nonzero rational function on C such that

_{div( f )}

= D - E. As we have seen

_before,

the

_degree

of

div( f )

is even on _any real branch of C.

_Hence,

the

degree

of E is odd on k real branches of C. Since E is

_effective,

there are real

points

Pl, ... ,

Pk

of

C,

each on a different real

_branch,

such that E &#x3E;

_Pl

_{+... +}

_Pk.

In

_particular,

deg(E) &#x3E;

k.

_But,

_deg(E)

=

deg(D)

= d k. Contradiction. 0

Proof of

Theorem

_4.2.

Let K be a canonical divisor on C. Consider the divisor K - D. It is a divisor of

degree

_{2g -}

2 - d.

_By

_Proposition

_4.1,

K-D is of odd

_degree

on k real branches of C.

_{By hypothesis, 2g-2-d}

k.

Hence,

D)

= 0

by

Lemma 4.3. 0

5. Points in

_{general position}

on a real

_quartic

In this

_section,

we _prove Theorem 3.1. Let us fix ourselves

_again

a smooth real

_quartic

C in

~2.

Let H be a

_hyperplane

section of

P2 and

let K = H ~ C.

Then, K

is _a canonical divisor _on C. The _statement that

will make

_things

work out is the

_following.

Proposition

5.1. The _map

_from

the linear

_system

_13HI

on

p2

into the linear

_system

_13KI

on C

defined by

sending

an element F E

_13HI

onto F . C E

_13KI

is an

_{isomorphism of projective}

_spaces.

Proo f .

Observe that the _map

₍

-~ ~

_13KI

is well defined since no divisor in can contain the curve C as a

_component.

Since this _map is a

linear map of

projective

spaces, it is

injective.

Now,

dim

13HI

= 9

and, by

Riemann-Roch,

dim

_13KI

= 9 _as well.

Therefore,

the

map -

_13KI

is

an

_isomorphism.

D

From now _on, we assume

again

that C has at least 3 real branches and we choose 3 of

_{them, Bl,}

_{B2, B3.}

As observed

_above,

each real branch

_Bi

is real

_{analytically isomorphic}

to the real

_projective

line.

_Moreover,

each

_Bi

is

_{homologically}

trivial in

_1~2(I~).

_Indeed,

this is a _consequence of

Proposi-tion

_4.1,

since C is a canonical curve. Put

and choose 0 E

_B,

as before.

Proof of

Theorem 3.1. Let D be the divisor +

Qi

on C.

By Proposition

5.1,

it suffices-for the first statement of Theorem 3.1-to show that the linear

_system

_IDI

on C is O-dimensional.

_{By Proposition 4.1,}

the divisor D has odd

_degree

on all real branches

Bl,

B2, B3.

Moreover,

its

degree

is

_equal

to 3. Since 3 + 3 &#x3E; 2 - 3 -

2,

the divisor D is

_nonspecial,

by

Theorem 4.2.

_By

_{Riemann-Roch,}

dim

_{I D}

0. This shows the first

statement of Theorem 3.1.

In order to show the last statement of Theorem

_3.1,

let F be the

_unique

real cubic

_{passing through}

the

_{points Pi, Qi,}

_R,-,

for i =

l, 2, 3.

Since each

real branch

Bi

of C is

_{homologically}

trivial in

_P2(R),

the

_degree

of the intersection

_{product F .}

C on

_Bi

is even.

Therefore,

there is S E

B,

such that

Now,

both divisors that intervene in this

_inequality

are of

_degree

12.

_Hence,

the

_inequality

is an

_equality

and the

points Si

are

_unique.

D

6. _{The Mordell-Weil group}

Proof of

Theorem

_3.4.

It is clear that To maps

B(K)

into

Jac(C)(K)°.

In

order to show that the

_image

of

_B(K)

is all of let J be an element of

_By

Theorem

_2.1,

there is an element P E B such that

_{To(P) =}

J. Let D be the divisor

_{E Pi}

on

CR.

By

Theorem

4.2,

D is

_{nonspecial. Hence,}

the linear

_system

_IDI

is 0-dimensional and consists of D

_only.

Since

_cl(D)

= J ₊

cl(~ Oi)

is a K-rational

_point

of the Picard scheme of

_C,

the divisor D on

CR

is

equal

to

_DR

for some divisor D’ on C.

Hence,

P E

_B(K)

and J is in the

_image

of

_B(K)

_by

To.

’

D

References

[1] C. _CILIBERTO, C. _PEDRINI, Real abelian varieties and real algebraic curves. Lectures in real

geometry (Madrid, 1994), 167-256, de _Gruyter Exp. Math. _23, de _{Gruyter, Berlin,} 1996.

[2] G. FICHOU, Loi de groupe sur la composante neutre de la _jacobienne d’une courbe réelle de

genre 2 ayant beaucoup de _composantes connexes réelles. Manuscripta Math. 104 _(2001), 459-466.

[3] G. FICHOU, Loi de groupe sur la composante neutre de la _jacobienne d’une courbe réelle

hyperelliptique ayant beaucoup de _composantes connexes réelles, (submitted).

[4] G. FICHOU, J. HUISMAN, A geometric description of the neutral _{component of} the Jacobian

of a real plane curve _{having many pseudo-lines,} (submitted).

[5] A. _HARNACK, Über die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann. 10 (1876), 189-198.

[6] J. _{HUISMAN, Nonspecial} divisors on real algebraic curves and _embeddings into real _projective

spaces. Ann. Math. Pura _Appl., to appear.

[7] J. HUISMAN, On the neutral _component of the Jacobian of a real _algebraic curve _having

many components. Indag. Math. 12 _(2001), 73-81.

[8] J. W. _{MILNOR, Topology from} the _{differentiable viewpoint.} Univ. _{Press, Virginia, 1965}

[9] J. H. _SILVERMAN, The arithmetic _{of elliptic} curves. Grad. Texts in Math. _106,

Springer-Verlag, 1986.

Johan HUISMAN

Institut de Recherche Math6matique de Rennes

Universite de Rennes 1

Campus de Beaulieu 35042 Rennes Cedex, France E-mail : hui smanQuniv-renne s 1. f r