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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�
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
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.
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.
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.
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
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
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 )
: