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On the invariants of the quotients of the Jacobian of a curve of genus 2

Pierrick Gaudry, Éric Schost

To cite this version:

Pierrick Gaudry, Éric Schost. On the invariants of the quotients of the Jacobian of a curve of genus 2. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC 14, Nov 2001, Mel- bourne, Australia. pp.373-386, �10.1007/3-540-45624-4_39�. �inria-00514434�

(2)

urve of genus 2

P. Gaudry

LIX,

Eole polytehnique

91128 Palaiseau Cedex, Frane

gaudrylix.polytehnique.fr

E. Shost

Laboratoire GAGE, UMS MEDICIS,

Eole polytehnique

91128 Palaiseau Cedex, Frane

shostgage.polytehnique.fr

(orresponding author)

January 31, 2001

Abstrat

Let C be a urve of genus 2 that admits a non-hyperellipti involution. We show

that there are at most 2 isomorphism lasses of ellipti urves that are quotients of

degree 2 oftheJaobian ofC.

Our proofis onstrutive,and wepresent expliitformulae,lassied aording to

theinvolutionsofC,thatgivetheminimalpolynomialofthej-invariantoftheseurves

in terms of the moduliof C. The oeÆients of these minimalpolynomials are given

asrationalfuntionsof themoduli.

keywords: urve ofgenus2,group of involutions,Igusa invariants,reduible

Jaobian

(3)

Amongtheurvesofgenus2,thosewithreduibleJaobianhaveapartiularinterest.

For instane, the present reords for rank or torsion are obtained on suh urves [3 ℄.

Also, it is in thispartiular setting that Dem'janenko-Manin's method yields all the

rationalpointsof aurve [7℄.

The aimof thispaperisto give a onstrutive proofof thefollowingtheorem.

Theorem 1 Let Cbeaurveof genus2with(2,2)-reduibleJaobian. Thenthereare

at most 2 ellipti urves that are quotients of degree 2 of its Jaobian, up to isomor-

phism.

Ifthisisthease,wepresentrationalformulaethatgivethej-invariantoftheseellipti

urvesinterms ofthe moduliof C.

Themodulioftheurvesofgenus2forma3-dimensionalvarietythatwasrstdesribed

byIgusain[4℄. Hisonstrutionrelieson4ovariantsoftheassoiatedsexti,denoted

by(A;B;C ;D); theformulae forthese ovariants are given again in [11 ℄. We usethe

moduli (j

1

;j

2

;j

3

) proposed in[5 ℄, whih are ratios of these ovariants. If we suppose

thatA is notzero, they aregiven by

j

1

= 144 B

A 2

;

j

2

= 1728

AB 3C

A 3

;

j

3

= 486 D

A 5

:

ThespeialaseA=0 isdealt withinappendix5.3. Allalong thepaper, thehara-

teristi of the baseeld will be supposed dierent from 2, 3 and 5. We will regularly

feelfreeto work overan algebrailosure oftheinitialeldofdenitionoftheurves.

Aknowledgements

Theomputations neessary to obtaintheformulae given herewere doneonthe om-

putersofUMSMEDICIS658(CNRS{

Eolepolytehnique,http://mediis.polyte-

hnique.fr). We thank Philippe Satge for his areful reading of this paper, and

Franois Morain forhisnumerous ommentsand suggestions.

1 Preliminaries

Denition 2 TheJaobian of a urve C of genus2 is(2,2)-reduibleif there exists a

(2,2)-isogeny between Ja(C) and a produt E

1 E

2

of ellipti urves. The urve E

1 is

then alled a quotient of Ja(C) of degree 2.

Asusual,theprex(2;2) meansthatthekerneloftheisogenyisisomorphitoZ=2Z

Z=2Z. Aurve ofgenus2alwaysadmits thehyperelliptiinvolution,denoted, whih

ommutes with all other automorphisms. The following lemma, in substane in [4 ℄,

relates thereduibilityto the existeneof other involutions.

(4)

of C is mapped onto the isomorphisms lasses of ellipti urves whih are quotient of

degree 2 of the Jaobian of C, via 7! C=. As a onsequene the Jaobian of C is

(2,2)-reduible if and only ifC admits a non-hyperellipti involution.

Proof. Let be a non-hyperelliptiinvolution of C. The quotient of C by is a urve

E of genus 1 [4℄; this urve is a also quotient of the Jaobian of C. The Jaobian

projetsonto E,and thekernel ofthismap isanotherelliptiurve E 0

. Consequently,

theJaobianof C splitsasEE 0

.

On theother hand, let E be an elliptiquotient of degree 2 of Ja(C). Thereexists a

morphism'ofdegree2fromC onto E. Fora generipointponC,theber' 1

('(p))

an be written fp;q(p)g, where q is a rational funtion of p. We dene as themap

p7!q(p). Sine theurve E has genus one, isnotthehyperelliptiinvolution. 2

Bolza[1 ℄, Igusa [4 ℄ and Lange [8 ℄ have lassied the urveswith automorphisms, and

in partiular the urves with involutions. The moduli of suh urves desribe a 2-

dimensional subvariety of the moduli spae; we will denote this set by H

2

. In our

loal oordinates, this hypersurfae is desribed by the following equation R , whose

onstrution isdonein[11℄.

R: 839390038939659468275712 j 2

3

+921141332169722324582400 000j 3

3

+32983576347223130112000 j 2

1 j

2

3

+182200942574622720j3j1j 2

2

374813367582081024j

3 j

2

1 j

2

+9995023135522160640000j 2

3 j

1 j

2

+94143178827j 4

2

562220051373121536j

3 j

2

2

562220051373121536j

3 j

3

1

+43381176803481600j3j 3

2

7196416657575955660800 0j 2

3 j2

38860649950910160568320 0j 2

3 j

1

1156831381426176j 5

1 j

3

31381059609j 7

1

+62762119218j 4

1 j

2

2

+13947137604j 3

1 j

3

2

31381059609j

1 j

4

2

188286357654j 3

1 j

2

2

6973568802j 6

1 j

2

+192612425007458304j 4

1

j3+94143178827j 6

1

6973568802j 5

2

+28920784535654400j 2

1 j

3 j

2

2

+164848471853230080j 3

1 j

3 j

2

=0:

We will all redued group of automorphisms of a urve the quotient of its group of

automorphismsbyf1;g. ThepointsonH

2

anbelassiedaordingtotheirredued

groupof automorphismsG.

G is thedihedralgroup D

6

;thisisthease forthepointon H

2

assoiatedtothe

urve y 2

=x 6

+1.

G isthesymmetrigroupS

4

;thisistheaseforthepointassoiatedto theurve

y 2

=x 5

x.

G is thedihedralgroup D

3

; theorresponding pointsdesribe a urve D on H

2 ,

exluding thetwo previouspoints.

G is Klein's group V

4

. The orresponding points desribe a urve V on H

2 ,

exluding thetwo previouspoints;these 2 pointsform theintersetionof D and

V.

G isthegroupZ=2Z.Thisorrespondsto theopen subsetU =H

2

D V;this

situation willbe alledthegeneri ase.

In thesequel, we haraterize all these ases, exept thetwo isolatedpoints, interms

of the moduli of C, desribe the involutions of C and ompute the orresponding j-

invariants.

(5)

lasses. Ourexpliitformuaethengiveaneasy proofofthefatthattheurveswhose

modulilieonDadmitarealmultipliationby p

3. Finally,theinvolutionsarenaturally

pairedas(;),and theseinvolutionsorrespondingeneraltodistintelliptiurves;

we show thaton theurve V,eah pair(;) yieldsasingle elliptiurve.

TheproofofTheorem1ouldbeahieved throughtheexhaustivestudyofallpossible

automorphism groups, whih would requireto onsidergroupsof order up to 48. We

followanotherapproah,whihreliesontheomputeralgebraofpolynomialssystems.

This method brings to treat many polynomial systems. While most of them an be

easily treated by the Grobner bases pakage of the Magma Computer Algebra Sys-

tem [10 ℄, the more diÆult one in setion 2 requires anotherapproah, whih we will

briey desribe. The systems we solved annot given here, forlakof spae; they are

availableuponrequest. Thestudyofthegroupationinsetion2waspartlyonduted

usingthefailitiesof Magmaforomputinginnitegroups.

2 The generi ase

In the open set U, the redued group of automorphisms is Z=2Z. Consequently, the

whole group of automorphisms has the form f1;;;g, and lemma 3 implies that

there areat mosttwo elliptiquotients. Our goal isthen to omputea polynomialof

degree 2 givingtheirj-invariants intermsof themoduli(j

1

;j

2

;j

3 ).

2.1 The minimal polynomial from a Rosenhain form

Asarststep,weobtainthej-invariantsfromaRosenhainform. The followingresult

isbased on [4℄,whih givestheRosenhainform of a(2;2)-reduibleurve.

Theorem 4 Let C bea urve of genus2 whose moduli belongto H

2

. On an algebrai

losure of its denition eld, C isisomorphi to a urve of equation

y 2

=x(x 1)(x )(x )(x ); where=

1

1

;

and , , are pairwise distint, dierent from 0 and 1. The Jaobian of C is (2;2)-

isogeneousto theprodut of the ellipti urvesof equationy 2

=x(x 1)(x ), where

is a solution of

2

2

2

+2( 2+)+ 2

=0: (1)

Proof. The urve C has 6 Weierstra points, and an isomorphism from C to another

urve is determined by the images of 3 of these points. Let be a non-hyperellipti

involution of C, and P

1 , P

2 ,P

3

beWeierstrapointson C that represent theorbits of

. Theurve C 0

dened bysending fP

1

;P

2

;P

3

g to f0;1;1g admits theequation

y 2

=x(x 1)(x )(x )(x ):

Thisurve isnotsingular, so,,are pairwisedistint,and dierentfrom 0 and1.

(6)

The imageofthe involution ofC on C is stilldenoted by. This involution permutes

theWeierstra pointsof C 0

;up to ahangeof names, we have (0)=,(1)= and

(1)=. Onanotherhand, an bewritten

(x;y)=

ax+b

x+d

; wy

(x+d) 3

;

and sineit has order 2,we have a= d and w=(ad b) 3=2

. The involution is

determinedby(0)=and (1)=,whih gives

(x;y)=

x

x

; u

3

y

(x ) 3

;

where

u= p

( ):

Changingthesignofuisequivalenttoomposing with. Therelation(1)=then

yieldstherst assertion

=

1

1

:

WenowlookforaurveisomorphitoC 0

,wheretheinvolutionanbewritten(x;y)7!

( x;y). This meansthatweare interested ina transformation

' : x7!

ax+b

x+d

suhthat'(0) = '(),'(1)= '(),'(1)= '(). Itisstraightforwardtohek

that

'(x)=

x u

x +u

;

is suh a transformation. As a result, the urve C is isomorphi to the urve C 00

of

equation y 2

=(x 2

x 2

1 )(x

2

x 2

2 )(x

2

x 2

3

),where

x

1

='(1)=1; x

2

='(0)=

u

+u

; x

3

='(1)=

1 ( u)

1 (+u) :

The morphism(x;y)7!(x 2

;y) maps C 00

onto the elliptiurve E of equation

y 2

=(x 1)(x x 2

2

)(x x 2

3 ):

The urveE hasLegendreform y 2

=x(x 1)(x ),where

= x

2

2 x

2

3

1 x 2

3

=

p

( )

2 :

Computingtheminimalpolynomialof proves thetheorem. Theonditions on, ,

showthatnone ofthedenominatorsvanishes,and thatE isnotsingular. 2

(7)

The j-invariants of the quotients of degree 2 of the Jaobian of C are the solutions of

the equation

j 2

+

1

(;)j+

0

(;)=0; (2)

where(

0

;

1

) arerational funtions.

Proof. Thej-invariantofanelliptiurveunderLegendreformisgivenbytherelation

2

( 1) 2

j 2 8

( 2

+1) 3

=0: (3)

The previous theorem yields2 elliptiurves that are quotients of the Jaobian of C,

and on the open set U, they are the only ones. The polynomial equation giving j is

obtainedastheresultantof equations3 and 1,usingtherelation= 1

1

. 2

We do notprint thevaluesof

0

(;) and

1

(;) for lakof spae. Sine the moduli

(j

1

;j

2

;j

3

) an bewrittenintermsof and ,anelimination proedure ouldgivethe

oeÆients

0 and

1

intermsof themoduli. Ourapproah islessdiret,butyieldsto

lighteromputations.

2.2 The group ating on Rosenhain forms

In thissetion,we introdue two invariantsthat haraterize the isomorphismlasses

of(2,2)-reduible urves.

Theorem 6 Let C be a urve of genus 2 whose moduli belong to H

2

. There are 24

triples(;= 1

1

;)forwhihtheurveofequationy 2

=x(x 1)(x )(x )(x )

isisomorphi to C. Theunique subgroup of order 24of PGL(2;5) atstransitively on

the set of these triples.

Proof. Theorem 4 yields a triple(

1

;

1

;

1

) that satises the ondition, so from now

on,weonsiderthatCistheorrespondingurve. EveryurveisomorphitoCisgiven

byabirational transformation

x7!

ax+b

x+d :

Sine this urve must be under Rosenhain form, the transformation must map 3 of

the6 Weierstra points(0;1;1;

1

;

1

;

1

) on thepoints(0;1;1). Theorresponding

homographi transformations form a group of order 6:5:4 = 120, and an exhaustive

searh shows thatonly24 ofthem satisfytherelation onthenew values(;;),

=

1

1

:

Let us denote by (

i

;

i

;

i )

i=1;:::;24

the orresponding triples. The exhaustive study

shows that the urve of Rosenhain form f0;1;1;

i

;

i

;

i

g is sent to the urve of

Rosenhainformf0;1;1;

j

;

j

;

j

g bysuessive appliationsonthese 6pointsofthe

maps

1

(x)=1=x,

2

(x)=1 x,

3 (x)=

x

1 ,

4

(x)=x=. Thesemaps generate a

(8)

thisgroupon thetriples(;;) isgiven bythe followingtable.

map

1

2

3

4

1

1

1

1

1

1 1

1

1

1

2

The24triples(

i

;

i

;

i

)areexpliitelygiveninappendix5.3. Thesymmetrifuntions

in these triples are invariants of the isomorphismlass of C. We will now dene two

spei invariantsthatharaterize these lasses.

Denition 7 LetCbeaurveofgenus2whosemodulibelongtoH

2

,andletf(

i

;

i

;

i )g

bethe set of triples dened above. We denote by and the following funtions:

=

P

24

i=1

2

i

;

=

P

24

i=1

i

i :

Thefollowingpropositionshowsthatand haraterize theisomorphismlassesof

suh urves. It isstraightforward to hek all thefollowingformulae,sine (j

1

;j

2

;j

3 ),

(

0

;

1

) and (;)an be writteninterms of(;).

Proposition 8 Let C be a urve of genus 2 whose moduli belong to H

2

, and (;)

dened as above. If all termsaredened, then the following holds:

j

1

=

36( 2) 2

( 8)(2 3) 2

;

j

2

=

216 2

(+ 27)

( 8)(2 3) 3

;

j

3

=

243 4

64( 8) 2

(2 3) 5

:

The previous system an be solved for (j

1

;j

2

;j

3

) only if the point (j

1

;j

2

;j

3

) belongs

to H

2

. Inthisase, and are givenbythefollowingproposition.

Proposition 9 Let C be a urve of genus 2 whose moduli belong to H

2

, and (;)

dened as above. If all termsaredened, then the following holds:

= (349360128j1j3 29859840j3j2+1911029760000j 2

3 +972j

2

1

j2 110730240j 2

1 j3

45j1j 2

2

12441600j1j3j2+6j 3

2 +45j

4

1 330j

3

1 j2 56j

2

1 j

2

2 16j

5

1 )=

( 26873856j

1 j

3

14929920j

3 j

2

+955514880000j 2

3

+3732480j 2

1 j

3 9j

1 j

2

2

+4147200j

1 j

3 j

2 +3j

3

2 +9j

4

1 3j

3

1 j

2 +2j

2

1 j

2

2 2j

5

1 );

= 3=4(162j 4

1

483729408j1j3+17199267840000j 2

3

+67184640j 2

1 j3 36j

5

1

134369280j

3 j

2 +162j

1 j

2

2 +45j

3

2

+35251200j

1 j

3 j

2 45j

3

1 j

2 72j

2

1 j

2

2

6912000j

3 j

2

2 20j

1 j

3

2 4j

4

1 j

2

)(349360128j

1 j

3

29859840j

3 j

2

+1911029760000j 2

3 +972j

2

1

j2 110730240j 2

1

j3 45j1j 2

2

12441600j1j3j2

+6j 3

2 +45j

4

1 330j

3

1 j2 56j

2

1 j

2

2 16j

5

1 )=

(27j 4

1

+161243136j

1 j

3

+1433272320000j 2

3

53498880j 2

1 j

3 9j

5

1

+44789760j

3 j

2 +486j

2

1 j

2 +135j

1 j

2

2

23846400j

1 j

3 j

2 162j

3

1 j

2 81j

2

1 j

2

2

3456000j3j 2

2 10j1j

3

2 2j

4

1

j2)( 26873856j1j3 14929920j3j2

+955514880000j 2

3

+3732480j 2

1 j

3 9j

1 j

2

2

+4147200j

1 j

3 j

2 +3j

3

2 +9j

4

1

3j 3

1 j

2 +2j

2

1 j

2

2 2j

5

1 )

:

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