Genus 2 curves with (3,3)-split Jacobian and large automorphism group

12 Pages·0.21 MB·English

by  T. Shaska

GENUS 2 CURVES WITH (3,3)-SPLIT JACOBIAN AND
LARGE AUTOMORPHISM GROUP
2
0
0 T. SHASKA
2

n Abstract. Let C be a genus 2 curve defined over k, char(k) = 0. If C has
a a (3,3)-split Jacobian then we show that the automorphism group Aut(C) is
J isomorphic to one of the following: Z2,V4,D8, or D12. Thereare exactly six

1 C-isomorphism classes of genus two curves C with Aut(C) isomorphic to D8
(resp., D12). We show that exactly four (resp., three) of these classes with

] group D8 (resp., D12) have representatives defined over Q. We discuss some
G
ofthesecurvesindetailandfindtheirrationalpoints.
A
.
h
t
a 1. Introduction
m
Let beagenus2curvedefinedoveranalgebraicallyclosedfieldk,ofcharacter-
[ C
istic zero. We denote by K := k( ) its function field and by Aut( ) := Aut(K/k)
C C
1 the automorphism group of . Let ψ : be a degree n maximal covering (i.e.
v does not factor through an iCsogeny) toCa→n eElliptic curve defined over k. We say
8 E
that has a degree n elliptic subcover. Degree n elliptic subcovers occur in pairs.
0 C
Let ( , ) be such a pair. It is well known that there is an isogeny of degree n2
0 ′
E E
1 betweentheJacobianJ of andtheproduct ′. Wesaythat has(n,n)-split
C C E×E C
0 Jacobian. The locus of such (denoted by ) is an algebraic subvariety of the
n
2 moduli space of genus twoC curves. For tLhe equation of in terms of Igusa
2 2
0 M L
invariants,see [19]. Computation of the equation of was the main focus of [18].
/ L3
h For n>3, equations of have not yet been computed.
n
t Equivalence classes oLf degree 2 coverings ψ : are in 1-1 correspondence
a
C → E
m with conjugacy classes of non-hyperelliptic involutions in Aut( ). In any charac-
C
teristic different from 2, the automorphism group Aut( ) is isomorphic to one of
:
v the following: Z , Z , V , D , D , Z ⋊D , GL (3), oCr 2+S ; see [19]. Here V
2 10 4 8 12 3 8 2 5 4
i
X is the Klein 4-group, D8 (resp., D12) denotes the dihedral group of order 8 (resp.,
12), and Z ,Z are cyclic groups of order 2 and 10. For a description of other
r 2 10
a groups, see [19]. If Aut( )=Z then is isomorphic to Y2 =X6 X. Thus, if
C ∼ 10 C − C
has extra automorphisms and it is not isomorphic to Y2 = X6 X then .
2
− C ∈ L
We say that a genus 2 curve has large automorphism group if the order of
C
Aut( ) is bigger then 4.
C
In section 2, we describe the loci for genus 2 curves with Aut( ) isomorphic
C
to D or D in terms of Igusa invariants. From these invariants we are able to
8 12
determine the fieldofdefinition ofa curve withAut( )=D orD . Further, we
C C ∼ 8 12
find the equation for this and j-invariants of degree 2 elliptic subcovers in terms
C
of i ,i ,i (cf. section2). This determines the fields of definition forthese elliptic
1 2 3
subcovers.
Let be a genus 2 curve with (3,3)-split Jacobian. In section 3 we give a short
C
description of the space . Results described in section 3 follow from [18], even
3
L
though sometimes nontrivially. We find equations of degree 3 elliptic subcovers
1
2 T. SHASKA
in terms of the coefficients of . In section 4, we show that Aut( ) is one of the
C C
following: Z , V , D , or D . Moreover, we show that there are exactly six C-
2 4 8 12
isomorphism classes of genus two curves with automorphism group D
3 8
C ∈ L
(resp., D ). We explicitly find the absolute invariants i ,i ,i which determine
12 1 2 3
these classes. For each such class we give the equation of a representative genus 2
curve . We notice that there are four cases (resp., three) such that the triple of
C
invariants (i ,i ,i ) Q3 when Aut(C)=D (resp., Aut(C)=D ). Using results
1 2 3 ∈ ∼ 8 ∼ 8
from section 2, we determine that there are exactly four (resp., three) genus 2
curves (up to Q¯-isomorphism) with group D (resp., D ) defined over Q
3 8 12
C ∈ L
and list their equations in Table 1. We discuss these curves and their degree 2 and
3ellipticsubcoversinmoredetailinsection5. Ourfocusisonthecaseswhichhave
elliptic subcovers defined over Q. In some of these cases we are able to use these
subcoversto find the rationalpoints ofthe genus 2 curve. This technique has been
used before by Flynn and Wetherell [5] for degree 2 elliptic subcovers.
Curves of genus 2 with degree 2 elliptic subcovers (or with elliptic involutions)
were first studied by Legendre and Jacobi. The genus 2 curve with the largest
knownnumberofrationalpointshas automorphismgroupisomorphictoD ;thus
12
ithasdegree2ellipticsubcovers. ItwasfoundbyKellerandKuleszanditisknown
tohaveatleast588rationalpoints;see[10]. Usingdegree2ellipticsubcoversHowe,
Leprevost,and Poonen [8] were able to construct a family of genus 2 curves whose
Jacobians each have large rational torsion subgroups. Similar techniques probably
could be applied using degree 3 elliptic subcovers. Curves of genus 2 with degree 3
elliptic subcovers have already occurredin the work of Clebsch, Hermite, Goursat,
Burkhardt, Brioschi,and Bolza in the context of elliptic integrals. For a history of
this topic see Krazer [11] (p. 479). For more recent work see Kuhn [12] and [18].
More generally, degree n elliptic subfields of genus 2 fields have been studied by
Frey [6], Frey and Kani [7], Kuhn [12], and this author [17].
Acknowledgements: The author wants to thank G. Cardona for pointing out
theparametrizationsinequations(5)and(6)whichmadesomeofthecomputations
easier.
2. Genus two curves and the moduli space .
2
M
Letk be analgebraicallyclosedfieldofcharacteristiczeroand agenus2curve
C
defined over k. Then can be described as a double cover of P1(k) ramified in 6
C
places w ,...,w . This sets up a bijection between isomorphism classes of genus 2
1 6
curves and unordered distinct 6-tuples w ,...,w P1(k) modulo automorphisms
1 6
∈
of P1(k). An unordered 6-tuple w 6 can be described by a binary sextic, (i.e.
{ i}i=1
a homogenous equation f(X,Z) of degree 6). Let denote the moduli space of
2
M
genus 2 curves; see [16]. To describe we need to find polynomial functions of
2
M
the coefficients of a binary sextic f(X,Z) invariant under linear substitutions in
X,Z of determinant one. These invariants were worked out by Clebsch and Bolza
in the case of zero characteristic and generalized by Igusa for any characteristic
different from 2; see [1], [9].
Consider a binary sextic i.e. homogeneous polynomial f(X,Z) in k[X,Z] of
degree 6:
f(X,Z)=a X6+a X5Z+ +a Z6
6 5 0
···
Classical invariants J off(X,Z)arehomogeneouspolynomials ofdegree2iin
2i
{ }
k[a ,...,a ],fori=1,2,3,5;see[9],[19]fortheirdefinitions. HereJ issimplythe
0 6 10
GENUS 2 CURVES WITH (3,3)-SPLIT JACOBIAN AND LARGE AUTOMORPHISM GROUP3
discriminant of f(X,Z). It vanishes if and only if the binary sextic has a multiple
linearfactor. TheseJ areinvariantunderthenaturalactionofSL (k)onsextics.
2i 2
Dividing such an invariant by another one of the same degree gives an invariant
under GL (k) action.
2
Two genus 2 fields K (resp., curves) in the standard form Y2 = f(X,1) are
isomorphic if and only if the corresponding sextics are GL (k) conjugate. Thus if
2
I isaGL (k)invariant(resp.,homogeneousSL (k)invariant),thentheexpression
2 2
I(K) (resp., the condition I(K) = 0) is well defined. Thus the GL (k) invariants
2
are functions on the moduli space of genus 2 curves. This is an affine
2 2
M M
variety with coordinate ring
k[M2]=k[a0,...,a6,J1−01]GL2(k) =subring of degree 0 elements in
k[J2,...,J10,J1−01]; see Igusa [9]. The absolute invariants
J J J 3J J
4 2 4 6 10
(1) i :=144 , i := 1728 − , i :=486
1 J2 2 − J3 3 J5
2 2 2
are even GL (k)-invariants. Two genus 2 curves with J =0 are isomorphic if and
2 2
6
only if they have the same absolute invariants. If J = 0 then we can define new
2
invariants as in [18]. For the rest of this paper if we say “there is a genus 2 curve
defined over k” we will mean the k-isomorphism class of .
C C
One can define GL (k) invariants with J in the denominator which will be
2 10
defined everywhere. However, this is not efficient in doing computations since the
degreesof these rationalfunctions in terms ofthe coefficients of will be multiples
C
of 10 and therefore higher then degrees of i ,i ,i . For the purposes of this paper
1 2 3
defining i ,i ,i as above is not a restriction as it will be seen in the proof of the
1 2 3
theorem 1. For the proofs of the following two lemmas, see [19].
Lemma 1. The automorphism group G of a genus 2 curve in characteristic =2
C 6
is isomorphic to Z , Z , V , D , D , Z ⋊D , GL (3), or 2+S . The case when
2 10 4 8 12 3 8 2 5
G=2+S occurs only in characteristic 5. If G=Z ⋊D (resp., GL (3)) then has
∼ 5 ∼ 3 8 2 C
equation Y2 = X6 1 (resp., Y2 = X(X4 1)). If G=Z then has equation
− − ∼ 10 C
Y2 =X6 X.
−
Remark 2. It is worth mentioning that the analogue of the above lemma has been
settled for all 2 g 48 in characteristic zero; see [14]. This was a major compu-
≤ ≤
tational effort using the computer algebra system GAP.
For the rest of this paper we assume that char(k)=0.
Lemma 3. i) The locus of genus 2 curves which have a degree 2 elliptic
2
L C
subcover is a closed subvariety of . The equation of is given by equation (17)
2 2
M L
in [19].
ii) The locus of genus 2 curves with Aut( )=D is given by the equation of
C C ∼ 8
and
2
L
(2) 1706J42J22+2560J43+27J4J24−81J23J6−14880J2J4J6+28800J62 =0
ii) The locus of genus 2 curves with Aut( )=D is
C C ∼ 12
−J4J24+12J23J6−52J42J22+80J43+960J2J4J6−3600J62 =0
(3) 864J10J25+3456000J10J42J2−43200J10J4J23−2332800000J120 −J42J26
−768J44J22+48J43J24+4096J45 =0
4 T. SHASKA
We we will refer to the locus of genus 2 curves with Aut( )=D (resp.,
C C ∼ 8
Aut( )=D ) as D -locus (resp., D -locus).
C ∼ 8 8 12
Eachgenus2curve hasanon-hyperellipticinvolutionv Aut( ). There
2 0
C ∈L ∈ C
is another non-hyperelliptic involution v := v w, where w is the hyperelliptic
0′ 0
involution. Thus, degree 2 elliptic subcovers come in pairs. We denote the pair of
degree 2 elliptic subcoversby (E ,E ). If Aut( )=D then E =E or E and E
0 0′ C ∼ 8 0∼ 0′ 0 0′
are 2-isogenous. If Aut( )=D , then E and E are isogenous of degree 3. See
C ∼ 12 0 0′
[19] for details.
Lemma 4. Let be a genus 2 curve defined over k. Then,
C
i) Aut( )=D if and only if is isomorphic to
C ∼ 8 C
(4) Y2 =X5+X3+tX
for some t k 0,1, 9 .
∈ \{ 4 100}
ii) Aut( )=D if and only if is isomorphic to
C ∼ 12 C
(5) Y2 =X6+X3+t
for some t k 0,1, 1 .
∈ \{ 4 −50}
Proof. i) Aut( )=D : Then is isomorphic to
C ∼ 8 C
Y2 =(X2 1)(X4 λX2+1)
− −
for λ= 2; see [19]. Denote τ := 2λ+6. The transformation
6 ± − l 2
−
qτx 1 4τ (λ+6)2
φ:(X,Y) ( − , )
→ τx+1 (τx+1)3 · λ 2
−
gives
Y2 =X5+X3+tX
where t=( λ 2 )2 and t=0,1. If t= 9 then Aut( ) has order 24.
2(λ−6) 6 4 100 C
Conversely,−the absolute invariants i ,i ,i of a genus 2 curve isomorphic to
1 2 3
C
Y2 =X5+X3+tX
satisfy the locus as described in lemma 2, part ii). Thus, Aut( )=D .
C ∼ 8
ii) Aut( )=D : In [19] it is shown that is isomorphic to
C ∼ 12 C
Y2 =(X3 1)(X3 λ)
− −
for λ=0,1 and λ2 38λ+1=0. Then,
6 − 6
1
φ:(X,Y) ((λ+1)3 X,(λ+1)Y)
→
transforms to the curve with equation
C
Y2 =X6+X3+t
where t= λ and t=0,1. If t= 1 then Aut( ) has order 48.
(λ+1)2 6 4 −50 C
The absolute invariants i ,i ,i of a genus 2 curve isomorphic to
1 2 3
C
Y2 =X6+X3+t
satisfy the locus as described in lemma 2, part iii). Thus, Aut( )=D . This
C ∼ 12
completes the proof.
Thefollowinglemmadeterminesagenus2curveforeachpointintheD orD
8 12
locus.
GENUS 2 CURVES WITH (3,3)-SPLIT JACOBIAN AND LARGE AUTOMORPHISM GROUP5
Lemma 5. Letp:=(J ,J ,J ,J )beapointin suchthatJ =0and(i ,i ,i )
2 4 6 10 2 2 1 2 3
L 6
the corresponding absolute invariants.
i) If p is in the D -locus, then the genus two curve corresponding to p is given
8
C
by:
3 345i2+50i i 90i 1296i
Y2 =X5+X3+ 1 1 2− 2− 1 X
−4 2925i2+250i i 9450i 54000i +139968
1 1 2− 2− 1
ii) If p is in the D -locus, then the genus two curve corresponding to p is given
8
C
by:
1 540i2+100i i 1728i +45i
Y2 =X6+X4+ 1 1 2− 1 2
4 2700i2+1000i i +204525i +40950i 708588
1 1 2 1 2−
Proof. i) By previous lemma every genus 2 curve with automorphism group D
8
C
is isomorphic to Y2 =X5+X3+tX. Since J =0 then t= 3 and the absolute
2 6 6 −20
invariants are:
(6) i1 = −144t((2200tt+−39))2, i2 = 3456t2 ((12400tt+−32)73), i3 = 243t3 ((240tt−+13))25
From the above system we have
3 345i2+50i i 90i 1296i
t= 1 1 2− 2− 1
−4 2925i2+250i i 9450i 54000i +139968
1 1 2− 2− 1
ii) By previous lemma every genus 2 curve with automorphism group D is
12
C
isomorphic to Y2 =X6+X3+t. The absolute invariants are:
t(5t+1) t(20t2+26t−1) 729 t2(4t−1)3
(7) i1 = 1296(40t−1)2, i2 = −11664 (40t−1)3 , i3 = 16 (40t−1)5
From the above system we have
1 540i2+100i i 1728i +45i
t= 1 1 2− 1 2
4 2700i2+1000i i +204525i +40950i 708588
1 1 2 1 2−
This completes the proof.
Note: If J = 0 then there is exactly one isomorphism class of genus 2 curves
2
with automorphism group D (resp., D ) given by Y2 = X5+X3 3 X (resp.,
8 12 − 20
Y2 =X6+X3 1 ).
− 40
Remark 6. If the invariants i ,i ,i Q then from above lemma there is a
1 2 3
∈ C
corresponding to these invariants defined over Q. If a genus 2 curve does not have
extra automorphisms (i.e. Aut( )=Z ), then an algorithm of Mestre determines
C ∼ 2
if the curve is defined over Q.
IftheorderoftheautomorphismgroupAut(C)isdivisibleby4,then hasdegree
C
2 elliptic subcovers. These elliptic subcovers are determined explicitly in [19]. Do
these elliptic subcovers of have the same field of definition as ? In general the
C C
answer is negative. The following lemma determines the field of definition of these
elliptic subcovers when Aut( ) isomorphic to D or D .
8 12
C
Lemma 7. Let be a genus 2 curve defined over k, char(k)=0.
C
i) If has equation
C
Y2 =X5+X3+tX
6 T. SHASKA
where t k =0,1, 9 , then its degree 2 elliptic subfields have j-invariants given
∈ \{6 4 100}
by
2000t2+1440t+27 (100t 9)3
j2 128 +4096 − =0
− (4t 1)2 (4t 1)3
− −
ii) If has equation
C
Y2 =X6+X3+t
where t k = 0,1, 1 , then its degree 2 elliptic subfields have j-invariants
∈ \{6 4 −50}
given by
500t2+965t+27 (25t 4)3
j2 13824t [phone]t − =0
− (4t 1)3 (4t 1)4
− −
Proof. The proof is elementary and follows from [19].
3. Curves of genus 2 with degree 3 elliptic subcovers
Inthissectionwewillgiveabriefdescriptionofspaces and . IncaseJ =0
2 3 2
L L 6
we take these spaces as equations in terms of i ,i ,i , otherwise as homogeneous
1 2 3
equations in terms of J ,J ,J ,J . By a point p we will mean a tuple
2 4 6 10 3
∈ L
(J ,J ,J ,J ) which satisfies equation of . When it is clear that J = 0 then
2 4 6 10 3 2
L 6
p would mean a triple (i ,i ,i ) . As before k is an algebraically closed
3 1 2 3 3
∈ L ∈ L
field of characteristic zero.
Definition 8. A non-degenerate pair (resp., degenerate pair) is a pair ( , )
C E
such that is a genus 2 field with a degree 3 elliptic subcover where ψ :
C E C → E
is ramified in two (resp., one) places. Two such pairs ( , ) and ( , ) are called
′ ′
C E C E
isomorphic if there is a k-isomorphism mapping .
′ ′
C →C E →E
If ( , ) is a non-degenerate pair, then can be parameterized as follows
C E C
(8) Y2 =(v2X3+uvX2+vX +1)(4v2X3+v2X2+2vX +1)
where u,v k and the discriminant
∈
∆= 16v17(v 27)(27v+4v2 u2v+4u3 18uv)3
− − − −
of the sextic is nonzero. We let R := (27v+4v2 u2v+4u3 18uv) = 0. For
− − 6
4u v 9=0 degree 3 coverings are given by
− − 6
vX2 (vX+3)2 ((4u−v−9)X+3u−v)
U1 = v2X3+uvX2+vX+1, U2 = v(4u−v−9)(4v2X3+v2X2+2vX+1)
and elliptic curves have equations:
E : V12 =RU13−(12u2−2uv−18v)U12+(12u−v)U1−4
(9)
E′: V22 =c3U23+c2U22+c1U2+c0
where
c0 =−(9u−2v−27)3
c1 =(4u−v−9)(729u2+54u2v−972uv−18uv2+189v2+729v+v3)
(10)
c2 =−v(4u−v−9)2(54u+uv−27v)
c3 =v2(4u−v−9)3
GENUS 2 CURVES WITH (3,3)-SPLIT JACOBIAN AND LARGE AUTOMORPHISM GROUP7
The above facts can be deducted from lemma 1 of [18]. The case 4u v 9=0 is
− −
treated separately in [18]. There is an automorphism β Gal
∈ k(u,v)/k(i1,i2,i3)
(v−3u)(324u2+15u2v−378uv−4uv2+243v+72v2)
β(u)=
(v−27)(4u3+27v−18uv−u2v+4v2)
(11)
4(v−3u)3
β(v)=−
4u3+27v−18uv−u2v+4v2
which permutes the j-invariants of and . The map
′
E E
θ :(u,v) (i ,i ,i )
1 2 3
→
defined when J = 0 and ∆ = 0 has degree 2. Denote by J the Jacobian matrix
2 θ
6 6
of θ. Then det(J )=0 consist of the (non-singular) curve X given by
θ
(12) X: 8v3+27v2 54uv2 u2v2+108u2v+4u3v 108u3 =0
− − −
and 6 isolated (u,v) solutions. These solutions correspond to the following values
for (i ,i ,i ):
1 2 3
8019 1240029 531441 729 1240029 531441 5103 729
(13) ( , , ), ( , , ), (81, , )
− 20 − 200 −100000 2116 97336 13181630464 − 25 −12500
We denote the image of X in the locus by Y. The map θ restricted to X is
3
L
unirational. ThecurveYcanbe computedasanaffinecurveinterms ofi ,i . For
1 2
eachpointp Ydegree3ellipticsubcoversareisomorphic. Ifpisanordinarypoint
∈
in Y and p = p (cf. Table 1) then the corresponding curve has automorphism
6 p
6 C
group V .
4
If ( , ) is a degenerate pair then can be parameterized as follows
C E C
Y2 =(3X2+4)(X3+X +c)
for some c such that c2 = 4 ; see [18]. We define w:=c2. The map
6 −27
w (i ,i ,i )
1 2 3
→
is injective as it was shown in [18].
Definition 9. Let p be a point in . We say p is a generic point in if the
3 3
L L
corresponding ( , ) is a non-degenerate pair. We define
p
C E
θ 1(p), if p is a generic point
−
e (p):= | |
3
( 1 otherwise
In [18] it is shownthat the pairs (u,v) with ∆(u,v)=0 bijectively parameterize
6
the isomorphism classes of non-degenerate pairs ( , ). Those w with w = 4
C E 6 −27
bijectively parameterize the isomorphism classes of degenerate pairs ( , ). Thus,
C E
the number e (p) is the number of isomorphismclasses of such pairs ( , ). In [18]
3
C E
it is shown that e (p) = 0,1,2, or 4. The following lemma describes the locus .
3 3
L
For details see [18].
Lemma 10. The locus of genus 2 curves with degree 3 elliptic subcovers is the
3
L
closed subvariety of defined by the equation
2
M
(14) C J8 + +C J +C =0
8 10 ··· 1 10 0
where coefficients C ,...,C k[J ,J ,J ] are displayed in [18].
0 8 2 6 10
∈
As noted above, with the assumption J = 0 equation (14) can be written in
2
6
terms of i ,i ,i .
1 2 3
8 T. SHASKA
4. Automorphism groups of genus 2 curves with degree 3 elliptic
subcovers
Let be a genus 2 curve defined over an algebraically closed field k,
3
C ∈ L
char(k)=0. The following theorem determines the automorphism group of .
C
Theorem 11. Let be a genus two curve which has a degree 3 elliptic subcover.
C
Then the automorphism group of is one of the following: Z ,V , D , or D .
2 4 8 12
C
Moreover, there are exactly six curves with automorphism group D and six
3 8
C ∈L
curves with automorphism group D .
3 12
C ∈L
Proof. WedenotebyG:=Aut( ). NoneofthecurvesY2 =X6 X,Y2 =X6 1,
C − −
Y2 = X5 X have degree 3 elliptic subcovers since their J ,J ,J ,J invariants
2 4 6 10
−
don’t satisfy equation (14). From lemma 1 we have the following cases:
i) If G=D , then is isomorphic to
∼ 8 C
Y2 =X5+X3+tX
as in Lemma 3. Classical invariants are:
J2 =40t+6, J4 =4t(9−20t), J6 =8t(22t+9−40t2), J10 =16t3(4t−1)2.
Substituting in the equation (14) we have the following equation:
(15) (196t−81)4(49t−12)(5t−1)4(700t+81)4(490000t2−136200t+2401)2 =0
For
81 12 1 81
t= , , ,
196 49 5 −700
the triple (i ,i ,i ) has the following values respectively:
1 2 3
729 1240029 531441 4288 243712 64
( , , ), ( , , ),
2116 97336 13181630464 1849 79507 1323075987
144 3456 243 8019 1240029 531441
( , , ), (− ,− ,− )
49 8575 52521875 20 200 10000
If
490000t2 136200t+2401=0
−
then we have two distinct triples (i ,i ,i ) which are in Q(√2). Thus, there are
1 2 3
exactly 6 genus 2 curves with automorphism group D and only four of
3 8
C ∈ L
them have rational invariants.
ii) If G=D then is isomorphic to a genus 2 curve in the form
∼ 12 C
Y2 =X6+X3+t
as in Lemma 3. Then, J = 6(40t 1) and
2
− −
J4 =324t(5t+1), J6 =−162t(740t2+62t−1), J10 =−729t2(4t−1)
Then the equation of becomes:
3
L
(16)
(25t−4)(11t+4)3(20t−1)6(111320000t3−60075600t2+13037748t+15625)3 =0
For
4 4 1
t= , ,
25 −11 20
the corresponding values for (i ,i ,i ) are respectively:
1 2 3
64 1088 1 576 60480 243 5103 729
( , , ), ( , , ), (81,− ,− )
5 25 84375 361 6859 2476099 25 12500
GENUS 2 CURVES WITH (3,3)-SPLIT JACOBIAN AND LARGE AUTOMORPHISM GROUP9
If
111320000t3 60075600t2=13037748t+15625=0
−
then there are three distinct triples (i ,i ,i ) none of which is rational. Hence,
1 2 3
there are exactly 6 classes of genus 2 curves with Aut( )=D of which
C ∈ L3 C ∼ 12
three have rational invariants.
iii) G=V . There is a 1-dimensional family of genus 2 curves with a degree 3
∼ 4
elliptic subcover and automorphism group V given by Y.
4
iv) Generically genus 2 curves have Aut( )=Z . For example, every point
C C ∼ 2
p correspondto a class of genus 2 curves with degree 3 elliptic subcovers
3 2
∈L \L
and automorphism group isomorphic to Z . This completes the proof.
2
The theorem determines that are exactly 12 genus 2 curves with auto-
3
C ∈ L
morphism group D or D . Only seven of them have rational invariants. From
8 12
Lemma 4, we have the following:
Corollary 12. There are exactly four (resp., three) genus 2 curves defined over
Q(uptoQ¯-isomorphism)withadegree3ellipticsubcoverwhichhaveaCutomorphism
group D (resp., D ). They are listed in Table 1.
8 12
p=(i1,i2,i3) e3(p) Aut( )
C C
p1 196X5+196X3+81X i1= 2712196,i2= 192743030629,i3= 13158311643401464 2 D8
p2 49X5+49X3+12X i1= 14824898,i2= 27493570172,i3= 132306745987 1 D8
p3 5X5+5X3+X i1= 14494,i2= 38455765,i3= 52522413875 2 D8
p4 700X5+700X3−81X i1=−802109,i2=−1224000029,i3=−51301040401 2 D8
p5 25X6+25X3+4 i1= 654,i2=−120588,i3=−841375 1 D12
p6 11X6+11X3−4 i1= 356716,i2= 660845890,i3= 2427463099 1 D12
p7 20X6+20X3+1 i1=81,i2=−521503,i3=−12752090 2 D12
Table 1. Rational points p with Aut(p) >4
3
∈L | |
Remark 13. All points p in Table 1 are in the locus det(J )=0. We have already
θ
seen cases p ,p , and p as the exceptional points of det(J )=0; see equation (13).
1 4 7 θ
The class p is a singular point of order 2 of Y, p is the only point which belong
3 2
to the degenerate case, and p is the only ordinary point in Y such that the order
6
of Aut(p) is greater then 4.
5. Computing elliptic subcovers
Next we will consider all points p in Table 1 and compute j-invariants of their
degree 2 and 3 elliptic subcovers. To compute j-invariants of degree 2 elliptic
subcoversweuselemma5andthevaluesoftfromtheproofoftheorem1. Werecall
that for p ,...,p there are four degree 2 elliptic subcovers which are two and two
1 4
isomorphic. We list the j-invariant of each isomorphic class. They are 2-isogenous
as mentioned before. For p ,p ,p there are two degree 2 elliptic subcovers which
5 6 7
are 3-isogenous to each other. To compute degree 3 elliptic subcovers for each p
we find the pairs (u,v) in the fiber θ 1(p) and then use equations (9). We focus
−
on cases which have elliptic subcovers defined over Q. There are techniques for
computing rational points of a genus two curves which have degree 2 subcovers
10 T. SHASKA
definedoverQasinFlynnandWetherell[5]. Sometimesdegree3ellipticsubcovers
are defined over Q even though degree 2 elliptic subcovers are not; see examples 2
and 6. These degree 3 subcovers help determine rational points of genus 2 curves
as illustrated in examples 2, 4, 5, and 6.
Example 1. p = p : The j-invariants of degree 3 elliptic subcovers are j = j =
1 ′
663. A genus 2 curve corresponding to p is
C
: Y2 =X6+3X4 6X2 8.
C − −
Claim: The equation above has no rational solutions.
Indeed, two of degree 2 elliptic subcovers (isomorphic to each other) have equa-
tions
: Y2 =x3+3x2 6x 8
1
E − −
: Y2 = 8x3 6x2+3x+1
2
E − −
where x = X2 (i.e. φ : of degree 2 such that φ(X,Y) = (X2,Y) ). The
1
C → E
elliptic curve has rank 0. Thus, the rational points of are the preimages of the
1
E C
torsion points of . The torsion group of has order 4 and is given by
1 1
E E
Tor( )= ,( 1,0),(2,0),( 4,0)
1
E {O − − }
None of the preimages is rational. Thus, has no rational points except the point
C
at infinity.
Example 2. p=p : The j-invariants of degree 2 elliptic subcovers are
2
76771008 44330496√3.
±
The point p belongs to the degenerate locus with w=0. Thus, the equation of the
2
genus 2 curve corresponding to p is
C
: Y2 =(3X2+4)(X3+X).
C
Indeed, this curve has both pairs ( , ) and ( , ) degenerate pairs. It is the only
′
C E C E
such genus 2 curve defined over Q. This fact was noted in [12] and [17]. Both
authors failed to identify the automorphism group. The degree 3 coverings are
X3
U =φ (X)=X3+X, U =φ (X)=
1 1 2 2 3X2+4
and equations of elliptic curves:
: V2 =27U3+4U , and : V2 =U3+U .
E 1 1 1 E′ 2 2 2
and are isomorphic with j-invariant 1728. They have rank 0 and rational
′
E E
torsion group of order 2, Tor( )= ,(0,0) . Thus, the only rational points of
E {O } C
are in fibers φ−11(0) and φ−21(∞). Hence, C(Q)={(0,0),∞}.
Example 3. p=p : All degree 2 and 3 elliptic subcovers are defined over Q(√5).
3
Example 4. p=p : Degree 2 elliptic subcovers have j-invariants
4
1728000 17496000
√I
2809 ± 2809
where I2 = 1. Thus, we can’t recover any information from degree 2 subcovers.
−
One corresponding value for (u,v) is (25,250). Then is
2 9 C
: 38 Y2 =(100X+9)(2500X2+400X+9)(25X+9)(2500X2+225X+9)
C ·