Arakelov computations in genus $3$ curves

J

ORDI

G

UÀRDIA

Arakelov computations in genus 3 curves

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 157-165

<http://www.numdam.org/item?id=JTNB_2001__13_1_157_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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

Arakelov

_computations

in

_genus

3

curves

par JORDI

GUÀRDIA

RÉSUMÉ. Les invariants d’Arakelov des surfaces

_{arithmétiques}

sont bien connus _pour le _{genre 1} et 2

_{([4], [2]).}

Dans cette _note,

nous étudions la hauteur modulaire et la self-intersection d’Arakelov _pour une famille de courbes de _genre 3

_possédant

beau-coup

d’automorphismes,

à savoir

Cn : Y4

=

X4 -

(4n - 2)

X2Z2

₊

Z4 .

La théorie d’Arakelov fait intervenir à la fois des calculs

arithméti-ques et des calculs

analytiques.

Nous

exprimons

les

périodes

de

Cn

en termes

_{d’intégrales elliptiques.}

Les substitutions utilisées dans les

_intégrales

fournissent une

_{décomposition}

de

_{la jacobienne}

de

Cn

en

_produit

de trois courbes

_elliptiques.

En utilisant

_{l’isogénie}

correspondante,

nous déterminons le modèle stable de la surface

arithmétique

définie _par

Cn .

Une fois calculés les

_périodes

et le modèle stable de

_Cn,

nous

sommes en mesure de déterminer la hauteur modulaire et la self

intersection du modèle

_canonique.

Nous donnons une bonne esti-mation de cette hauteur

_modulaire,

traduite _par son

comporte-ment

_{logarithmique.}

Nous donnons

_également

une minoration

de la self-intersection

_qui

montre

_qu’elle

_peut être arbitrairement

grande.

Nous

_présentons

ici nos calculs _presque sans démonstrations. Les détails _peuvent être lus dans

_{[5] .}

ABSTRACT. Arakelov invariants of arithmetic surfaces are well

known for _{genus 1} and 2

_{([4], [2]).}

In this _note, we

_study

the

modular

_height

and the Arakelov self-intersection for a

family

of curves of _genus 3 with many

automorphisms:

Cn: Y4

=

X4 -

(4n -

2)X2 Z2

₊

Z4.

Arakelov calculus involves both

analytic

and arithmetic

computa-tions. We _express the

_periods

of the curve

_Cn

in terms of

ellip-tic

_integrals.

The substitutions used in these

_integrals

_provide

a

splitting

of the

_jacobian

of

Cn

as a

product

of three

elliptic

curves.

Using

the

_{corresponding isogeny,}

we determine the stable model

of the arithmetic surface

_{given by}

Cn.

Once we have the

_periods

and the stable model of

_Cn,

we can

study

the modular

height

and Manuscrit reçu Ie 26 octobre 1999.

haviour. We

_provide

a lower bound for the

self-intersection,

which

shows that it can be

_{arbitrarily large.}

We _present here all our calculations on the curves

_Cn,

almost

without

_proofs.

Details can be found in

_[5].

1. A

_family

of curves of genus 3

We will

_study

the curves

_{given by}

the

_equation:

for a E

_{0, :i:l.}

These are

non-singular

curves of genus 3. We note

that each of the four values of the

_parameter

a,

-a,1/a,

-1/a

gives

rise

to the same curve.

_Equation

(1)

is

particularly

well-suited to

_study

the

geometry

of the curves

_{(7~) ,}

but to

_study

their arithmetic we should use

the

_equation

obtained from

₍₁₎

_through

the

_change

of

_parameter

a = ₊ In

either

_equation

one can observe that these curves have _many

_symmetries.

More

_{specifically,}

we _prove:

Proposition

1. Let

_{a ¥= 0, ::1:1, :1:1 ::I:}

_B/2.

The

_{automorphisms of}

_C(a)

are

the restrictions

_of

the

_following

16

_{projectivities}

_of

_p2(C):

The _group

_Aut(C(a»

is

_isomorphic

to a semidirect

product

Y4.

The

automorphismes

form

a

_system

of

_generators

for

Aut(C(a».

Proposition

2. The curve

isomorphic

to the Fermat curve

of

fourth degree,

F4

=

ly4

=

X4

₊

Z4}.

The

group

Aut(F4)

has order

96,

2.

_Geometry

of the curves

_C(a)

The determination of the

_period

lattice of the curves

_C(,,)

is

possible

thanks to their _many

_symmetries.

First of

_all,

we can find a basis for the

space of

holomorphic

differentials on

C(~) :

Proposition

3. Let x =

XIZ, y

=

Y/Z.

The

differential

formes

yield

an

orthogonal

basis

for

We can also determine a basis for the

_singular

_homology

of the curves.

We

_regard

the curve

_C~a~

as a 4-sheeted

_covering

of the

_{complex projective}

line, through

the map:

Figure 1 .

We can

explicitly

describe the

monodromy

of this map. For

instance,

a

lifts

_points

one sheet. With all these

data,

we can

proceed

to build closed

paths

on

_{G(a) ,}

_lifting

closed

_paths

in

_I~2(~),

as seen in

figure

1. We refer

to the

_path

sketched there as G. We

proceed analogously

to build the

_path

F indicated in

_figure

2

_(numbers

indicate the sheet on which each

_part

of

the

_path

_lies).

Figure 2

Proposition

4. The

_homology

classes

_of

the

_paths

form

a

symplectic

basis

for

H¡(C(a),

7~).

Once we have bases for the _spaces

HO (C(a),

and

_H¡(G(a)’

Z)

we

com-pute

the

_period

lattice of the curve

_{C(a) -}

The Abelian

_integrals

that we must

_compute

can be reduced to

_{elliptic integrals,}

and can thus be

_expressed

in terms of

_elliptic

functions. We obtain:

Proposition

5. A normalized

_period

matrix

_for

the curves

_C(a)

is

(IlZa),

where

and I is the

complete eltiptic integral

o f

the

_{first kind.}

For

_instance,

for the Fermat curve of fourth

_degree,

_isomorphic

to the

curve

_C1+V2’

we re-obtain its well-known

_period

matrix:

B 2 2 -’-/

The fact that all the Abelian

_integrals

on

_C(a)

can be reduced to

_elliptic

integrals

tells us that the

jacobian

of the curve should

_split

as a

product

of

_elliptic

curves. We can show the situation

_{exactly. Let p =}

Proposition

6. Let E be the

_elliptic

curve

and let

_En

be the

_elliptic

curve

which is a Weierstrass model

of

the curve

- - .

-We have

_isomorphisms

_defined

over

_Ko:

The

_quotient

_maps induce an

_isogeny

of degree

8

defined

over

_Ko.

’ ’

3. Arithmetic of the curves

Cn

From now _on, we will assume that n E N is a natural

_{number, and,}

for

technical reasons

only, that n 0

_0,1

(mod

25).

The

isogeny

described in

the

_previous

section is the

_key

to the

_study

of reduction of the curves

Cn.

It is well known that the the reduction of a curve is

_intimately

related with

that of its

_{jacobian. Moreover,}

at the level of Abelian

_varieties,

the

_type

of reduction is invariant under

_{isogenies. Thus,}

_studying

the reduction of the

Proposition

7. The curve

Cn

has

_good

reduction outside the

_primes

di-viding

n (n -1 ) .

Let

_Kn

= where a =

3 18 - 6 ~

and

24 ) 3 - j ( E~ )

(an -

27) =

0. Let

_On

be the

_ring

of

_integers

of

_K~.

The

_elliptic

curves

E,

En

acquire

good

reduction at 2 over the field since their

Deuring

models are defined over this field. The curve

En

has

multiplicative

reduc-tion at odd

_primes

_dividing

_{n (n -1 ) .}

From

_this,

we conclude:

Proposition

8.

_a)

The curve

Cn

has a stable model

Cnst

and a minimal

regular

model over

On.

b)

At the

_On

which

_{divide 2,}

_J(Cn)

has Abelian reduction over

C

c)

At the

_{odd primes p}

in

_On

which divides

_{n(n-1), J(Cn)}

has semi-Abelian reduction over

On,

and the tolsc

_part

of

its N6ron model has dimension 1.

We now describe the bad fibres of the stable model

Cn

of the curves

_Cn.

At the odd

_{primes, Prop.}

8

_{gives enough}

information to describe the bad fibres.

Theorem 9.

_a)

The

_{special fibre of}

at an odd

prime p

in

On dividing

n (n - 1 )

has two

_elliptic

_components,

_intersecting

at two

_different

_points.

b)

The intersection

_pairing

on this

_fibre

is

_{given by}

The situation at the

_{primes dividing}

2 is much more

complicated,

and

requires

a detailed

_study

of the reduction of the

automorphisms

of

Gn.

Once this is

_done,

we obtain that:

Theorem 10.

_a)

The

_{special fibre}

_of

_Gn

at a

_{prime p}

in

On dividing 2

consists

_of

three

_elliptic

_components,

which intersect a rational

_component

at three

_different

_points.

These intersection

_points

_map to

_{4-torsion points}

b)

The intersection

_pairing

on this

fibre

is

_given

_by

4. Arakelov invariants for arithmetic surfaces

Having

a

_great

deal of

_geometric

and arithmetic information about a

curve, one can

proceed

to

_study

it from the Arakelov

_viewpoint.

We will

now do so with the curves

Cn.

We will

_study

the two basic Arakelov invariants for curves: the modular

_height

and the self-intersection. We

briefly

recall their definitions

_(details

can be found in

[4]).

Given a curve X of _{genus g,} defined over a number field

K,

consider an

extension where X

_acquires

stable

_reduction,

and let

;rest ---+

T =

be its stable model. The relative

_dualizing

sheaf

_úJxest/T

can be

provided

with an admissible

_metric,

_giving

rise to an Arakelov line

sheaf

WxestlT-Definition 11. The modular

_{height of}

the curve X is

The Arakelov

_{self-intersection of}

X is

mhere

_{(.,.) Ar}

denotes the Arakelov intersection

_pairing

d4])-For _{genus g} =

1,

it is known that

e(X)

= 0 and

h(X)

_can be

expressed

as a function

h(X)

= in _terms of the

periods

of the

curve, its

discriminant and its locus of bad reduction. For _{genus g} =

2,

Bost - Mestre

-

Moret-Bailly

([3])give

explicit

formulas for both

_invariants,

where

_again

the

_periods

and the reduction of the curve

_play

an

important

role. But

there is not much more information about these two invariants in

_general.

Ullmo

_([8])

has

_proved

that

_e(X)

&#x3E; 0

_{for g}

&#x3E;

_1,

and Abbes and Ullmo

_([l])

have found upper and lower bounds for the self-intersection of modular

Theorem 12.

_{Let Tn}

The

_{height of}

the curve

Cn

is

_{given by}

Sketch of the

_proof

The modular

_height

of a curve coincides with that

of its

_jacobian

as an Abelian

_variety.

A result of

Raynaud

([7])

controls

the behaviour of the modular

_height

of Abelian varieties

_{through isogenies}

A

~ B

of

_degree

a

_prime

_power

_p’:

we have that

h(A)

=

h(B)

₊ k

logp,

with k E

Z,

Ikl

e/2.

Moreover,

the modular

_height

of a

product

of

Abelian varieties is the sum of the

_heights

of the factors. We thus obtain:

Now we use the

_explicit

formula for the

_height

of the

_elliptic

curves

_given

in

([4]),

to determine the terms

_{h(E), h(En).}

Note that Tn is the fundamental

period

of the

_elliptic

curve

_En.

0

6. A lower bound for

_e(Cn)

Lemma 13

_([6]).

Let X - S =

Spec(OK)

be a stable arithmetic

_surface

of

2. Let _{pl, ... , pt} be the

_primers

in

_OK

where the

_{fibres of}

X

are reducible. We have that

We can

_apply

this result to the curves

Cn,

since we have a

precise

de-scription

of the

_primes

where their stable model has reducible fibres:

_they

are the

_primes

which divide

n(n -

1).

We thus obtain:

Proposition

14.

As the

_degree

_{~.I~n :}

_K]

is

_always

less than or

equal

to

_1152,

this result

implies

that the Arakelov self-intersection of a curve can be

_arbitrarily

References

[1] A. ABBES, E. ULLMO, Auto-intersection du dualisant _relatif des courbes modulaires _X0(N). J. reine angew. Math. 484 (1997), 1-70.

[2] J.-B. BOST, Fonctions de _{Green-Anakelov, fonctions} théta et courbes de genre 2. C.R. Acad. Sci. Paris Série I 305 _(1987), 643-646.

[3] J.-B. BosT, J.-F. MESTRE, L. MORET-BAILLY, Sur le calcul _explicite des "classes de Chern" des surfaces arithmétiques de genre 2. Astérisque 183 _(1990), 69-105.

[4] G. FALTINGS, Calculus on arithmetic surfaces. Ann. of Math. 119 _(1984), 387-424.

[5] J. _GUÀRDIA, Geometria aritmética en una _famlia de corbes de _genere 3. Thesis, Universitat

de Barcelona, 1998.

[6] A. _MORIWAKI, Lower bound _{of self-intersection of dualizing} sheaves on arithmetic _surfaces

with reducible _{fibres. Compositio} Mathematica 105 _(1997), 125-140.

[7] M. RAYNAUD, Hauteurs et _isogénies. Séminaire sur les pinceaux arithmétiques: la _conjecture de Mordell, exp. VII. Astérisque 127 _(1985), 199-234.

[8] E. ULLMO, Positivité et discrétion des points algébriques des courbes. Annals of Math. 147

(1998), 167-179. Jordi GUARDIA

Dept. algebra i Geometria Universitat de Barcelona

Gran Via de les Corts Catalanes, 585 E-08007 Barcelona

Espana