Plane quartics over Q with complex multiplication

0.47 MB·

by  Pinar Kilicer

PLANE QUARTICS OVER Q WITH COMPLEX
MULTIPLICATION
7
1
PINARKILICER,HUGOLABRANDE,REYNALDLERCIER,
0
CHRISTOPHERITZENTHALER,JEROENSIJSLING, ANDMARCOSTRENG
2

n
Abstract. Wegiveexamplesofsmoothplanequartics overQwithcomplex
a
J multiplication over Q by a maximal order with primitive CM type. We de-
scribethe required algorithms aswego; theseinvolvethereduction ofperiod
3
matrices, the fast computation of Dixmier–Ohno invariants, and reconstruc-
2
tionfromtheseinvariants. Finally,wediscusssomeofthereductionproperties

ofthecurves thatweobtain.
]
T
N
h. Introduction
t
a Abelian varieties with complex multiplication (CM) are a fascinating common
m
groundbetweenalgebraicgeometryandnumber theory,andaccordinglyhavebeen
[ studied since a long time ago. One of the highlights of their theoretical study

was the proof of Kronecker’s Jugendtraum, which describes the ray class groups of
1
v imaginaryquadraticfieldsintermsofthedivisionpointsofellipticcurves. Hilbert’s
9 twelfth problem asked for the generalization of this theorem to arbitrary number
8 fields, and while the general version of this question is still open, Shimura and
4
Taniyama [54] gave an extensive partial answer for CM fields by using abelian
6
varieties whose endomorphism algebras are isomorphic to these fields. A current
0
. concrete applicationof the theory of CM abelian varieties is in public key cryptog-
1
raphy,where onetypically usesthis theory to constructelliptic curveswith agiven
0
number of points [8].
7
1 Beyond the theoretically well-understood case of elliptic curves, there are con-
: structionsofcurveswithCMJacobiansinbothgenus2[56,64,7]and3[31,67,36].
v
i Note that in genus 2 every curve is hyperelliptic, which leads to a relatively sim-
X
ple moduli space; moreover, the examples in genus 3 that we know up to now are
r either hyperelliptic or Picard curves, which again simplifies considerations. This
a
paper gives the first 19 conjectural examples of “generic” CM curves of genus 3,
in the sense that the curves obtained are smooth plane quartics with trivial auto-
morphism group. More precisely, it completes the list of curves of genus 3 over Q
whose endomorphism rings over Q are maximal orders of sextic fields (see Theo-
rem 1.1). The other curves of genus 3 with such endomorphism rings are either
hyperellipticorPicard. ThehyperellipticoneswereknowntoWeng[67],exceptfor
threecurvesthatwerecomputedbyBalakrishnan,Ionica,Kılıçer,Lauter,Vincent,
SomozaandStrengbyusingthemethodsandSageMathimplementationof[4,3].
Date:January24,2017.
2010 Mathematics Subject Classification. 13A50,14H25,14H45, 14K22,14Q05.
Key words and phrases. complexmultiplication,genus3,planequartics, explicitaspects.
ThethirdandfourthauthoracknowledgesupportfromtheCysMoLog“défiscientifiqueémer-
gent” oftheUniversitédeRennes 1.
1
2 KILICER,LABRANDE,LERCIER,RITZENTHALER,SIJSLING,ANDSTRENG
The Picard curves had all previously appeared in work by Koike-Weng [31] and
Lario-Somoza[36].
To construct our curves, we essentially follow the classical path; first we deter-
mine the period matrices, then the corresponding invariants, then we reconstruct
the curves from rational approximations of these invariants, and finally we heuris-
tically check that the curves obtained indeed have CM by the correct order. In
genus 3, however,all of these steps are somewhatmore complicated than was clas-
sically the case.
The provenverificationthat the curvesobtainedindeed haveCM by the correct
order is left for another occasion; we restrict ourselves to a few remarks. First
of all, there are no known equivalents in genus 3 of the results that bound the
denominators of Igusa class polynomials [38]. In fact very little is known on the
arithmetic nature of the Shioda and Dixmier–Ohno invariants that are used in
genus 3, and a theoretical motivation for finding our list was to have concrete
examples to aid with the generalization of the results in loc. cit.
Using the methods in [10] one could still verify the endomorphism rings of our
curves directly; this has already been done for the simplest of our curves, namely
X :x4 x3y+2x3z+2x2yz+2x2z2 2xy2z+4xyz2 y3z+3y2z2+2yz3+z4 =0.
15
− − −
The main restriction for applying these methods to the other examples is the time
required for this verification. At any rate, the results in the final section of this
paper are coherent with the existence of a CM structure with the given order.
TheCMfieldsthatgiverisetoourcurvesweredeterminedbyarithmeticmethods
in[26,29]. ThisalsogivesusRiemannmatricesthatwecanusetodetermineperiods
and hence the invariants of our quartic curves. However, we do need to take care
to reduce our matrices in order to get good convergence properties for their theta
values. The theory and techniques involved are discussed in Section 1.
With our reduced Riemann matrices in hand, we want to calculate the corre-
sponding theta values. We will need these values to high precision so as to later
recognize the corresponding invariants. The fast algorithms needed to make this
feasible were first developed in [34] for genus 2; further improvement is discussed
in Section 2.1. In the subsequent Section 2.2 we indicate how these values allow us
to obtainthe Dixmier–Ohnoinvariantsofoursmoothplane quartic curves. This is
based on formulas obtained by Weber [66, 17].
The theory of reconstructing smooth plane quartics from their invariants was
developed in [46] and is a main theme of Section 3. Equally important is the per-
formanceofthese algorithms,whichwas substantiallyimprovedduringthe writing
of this paper; starting from a reasonable tuple of Dixmier–Ohno invariants over
Q, we now actually obtain corresponding plane quartics over Q with acceptable
coefficients, which was not always the case before. In particular, we developed a
“conictrick” whichenablesustofindconicswithsmalldiscriminantinthecourseof
Mestre’sreconstructionalgorithmsfor generalhyperelliptic curves(byloc.cit., the
reconstruction methods for non-hyperelliptic curves of genus 3 reduce to Mestre’s
algorithmsforthehyperellipticcase). Section3discussestheseandotherspeed-ups
andthemathematicalbackgroundfromwhichtheysprang,sincewithoutthemour
final results would have been too large to even write down.
We finally take a step back in Section 4 to examine the reduction properties of
these curves, as well as directions for future work, before giving our explicit list of
curves in Section 5.
PLANE QUARTICS OVER Q WITH COMPLEX MULTIPLICATION 3
1. Riemann matrices
LetAbeaprincipallypolarizedabelianvarietyofdimensiongoverC,suchasthe
JacobianA=J(C)ofoneofthecurvesthatwearelookingfor. Thenbyintegrating
overa symplectic basis of the homologyand normalizing,the manifold A givesrise
to a point τ in the Siegel upper half space , well-defined up to the action of
g
H
the symplectic group Sp (Z). The elements of are also known as Riemann
2g Hg
matrices. In Section 1.1, we give the list, due to Kılıçer and Streng, of all fields K
that can occur as endomorphism algebra of a simple abelian threefold over Q with
complex multiplication over Q. In Section 1.2, we recall Van Wamelen’s methods
for listing all Riemannmatrices with complex multiplicationby the maximalorder
of a given field. In Section 1.3, we show how to reduce Riemann matrices to get
Riemann matrices with better convergence properties.
1.1. The CM fields. Let A be an abelian variety of dimension g over a field k of
characteristic 0, let K be a number field of degree 2g and let be an order in K.
O
WesaythatAhas CM by (overk)ifthereexistsanembedding End(A ).
OK O → k
If A is simple over k and has CM by the full ring of integers of K, then we
K
O
haveinfact =End(A )andK isaCM field,i.e.,atotallyimaginaryquadratic
OK ∼ k
extension K of a totally real number field F [35].
The field of moduli of a principally polarized abelian variety A/k is the residue
fieldofthe correspondingpointinthemodulispaceofprincipallypolarizedabelian
varieties. It is alsothe intersectionof the fields ofdefinition ofA in k [32, p.37]. In
particular,if A is defined over Q, then its field of moduli is Q. The field of moduli
ofa curve or anabelian varietyis not alwaysa field of definition [52]. However,we
have the following theorem.
Theorem 1.1. There are exactly 37isomorphism classes of CM fields K for which
there exist principally polarized abelian threefolds A/Q with field of moduli Q and
End(A)= . The set of such fields is exactly the list of fields given in Table 1.
∼OK
For each such field K, there is exactly one such principally polarized abelian
variety A up to Q-isomorphism, and this variety is the Jacobian of a curve X of
genus3 defined over Q. In particular, the abelian variety A itself is defined over Q.
Conjectural models over Q, correct to some precision over C, of all curves have
been computed; the cases 1 to 20 except case 4 are in Section 5 and the other cases
have non-trivial automorphisms and have been computed before (see Table 1).
Proof. Everythingexcept the models of the curvesandthe factthat the curvesare
definedoverQisfromKılıçer[26,Thm.4.1.1],whichwillbepublishedas[29]. Inthe
restofthis paperwe workout the equationsofthe ones with trivialautomorphism
group (cases 1 to 20 except case 4) which we give in Section 5.
Therefore we need only provethe statement onthe field of definition, which can
be done here directly from the knowledge of CM type. By the theorem of Torelli
[37, Appendix] and by Galois descent of the Jacobian, k is a field of definition for
the principally polarized abelian threefold A if and only if it is a field of definition
for X. In particular the field of moduli of X is Q. In genus 3 all curves descend to
theirfieldofmoduli,exceptforplanequarticswithautomorphismgroupZ/2Zand
hyperelliptic curveswith automorphismgroupZ/2Z Z/2Z (see [42, 44]). Neither
×
ofthese occursinTable1. Indeed, the planequarticsinitallhavetrivialautomor-
phismgroupsincethenegationoftheJacobianisnotinducedbyanautomorphism
of the corresponding curve. And while the table includes hyperelliptic curves with
4 KILICER,LABRANDE,LERCIER,RITZENTHALER,SIJSLING,ANDSTRENG
an automorphism group of order 4, the group is always isomorphic with Z/4Z in
these cases because Q(i) is then a subfield of K. This concludes the proof that A
is defined over Q. (cid:3)
Table1givesalistofcyclicsexticCMfieldsK asfollows. LetK be suchafield.
Thenit has animaginaryquadratic subfield k andtotally realcubic subfield F. In
Table 1, the number d is the discriminant of k; the polynomial p is a defining
k F
polynomial for F. These two entries of the table define the field K. The number
f is the conductor of F, and d is the discriminant of K. The entry # is the
F K
orderoftheautomorphismgroupoftheJacobianofthecorrespondingcurve,which
is nothing but the number of roots of unity in K. The “Type” column indicates
whether the conjectured model of the curve is hyperelliptic (H), Picard (P), or a
plane quartic with trivial automorphism group (G). The “Curve” column gives a
reference to the conjectured model over Q of the curve. The cases 1, 2, 3, 5, ...,
20 correspond to the smooth plane quartics X in Section 5.
i
Ca1234se −7777dk XXXX3333+−++X386XXX2p22−−F−4159X1XX++1+127 732f13·F32371 773335−88··d·13K77313544 h1111∗K #2222 TyGGGHpe [3]XXX+Cs123ue(((er§§§vb555ee)))low
11111111112256789012345678901 11111114677778884111999937 XXXXXXXXXXXXXXXXX33333333333333333+++++++++++−+++++XXX42XXXX3X229722XXXXXXXXX2222222222222222−−−−−−−−−−−−−−−−−3242121113555362504X044XX9X061XXXXXXXXXXXX+−−+−+−+++++1+11+−−2281881112228777176 13773273441111613··77··31339399792·11113999 1117777227332132185394865399343416·········397·····11111111735557411999319933143995444334444454 11441141441411111 22222222222222224 GGGGGGGGGGGGGGGGH [67XXXXXXXXXXX,XXXXX§1111111111246789601234567890((((((((((((((((3§§§§§§§§§§§§§§§§r55555d55555555555))))))))))))))))ex.]
22 4 X3 3X −1 32 26· 38 1 4 H [67,§62ndex.]
23 4 X3−+X2−2X 1 7 26·74 1 4 H [67,§61stex.]and[61]
24 4 X3+X2−10X− 8 31 26 ·314 4 4 H [67,§64thex.]
25 4 X3+X2−14X−+8 43 26·434 4 4 H [3]+seebelow
26 4 X3+3X2− 18X+8 32 7 26 ·3874 4 4 H [3]+seebelow
27 3 X3+X2 −4X+1 13· 33·134 1 6 P [31,6.1(3)](also[36,4.1.3])
28 3 X3+X2−2X 1 7 33· 74 1 6 P [31,6.1(2)](also[36,4.1.2])
29 3 X3+X2−10X− 8 31 33 ·314 1 6 P [31,6.1(4)](also[36,4.1.4])
30 3 X3+X2−14X−+8 43 33·434 1 6 P [31,6.1(5)](also[36,4.1.5])
31 3 X3+3X2− 18X+8 32 7 39· 74 1 6 P [36,4.2.1.1]
32 3 X3+6X2−9X+1 32·7 39·74 1 6 P [36,4.2.1.2]
33 3 X3+3X2−36X 64 32 ·13 39 ·134 1 6 P [36,4.2.1.3]
34 3 X3+4X2−15X−27 6·1 33·614 4 6 P [36,4.3.1]
35 3 X3+2X2−21X−27 67 33·674 4 6 P [36,4.3.3]
36 7 X3+X2 −2X 1− 7 7·5 1 14 H y2=x7 1
37 3 X3 3X−1 − 32 39 1 18 P y3=x4−x
− − −
Table 1. CM fields in genus 3 whose maximal orders give rise
to CM curves with field of moduli Q, sortedby the order# of the
group of roots of unity.
In the hyperelliptic cases, curves can be reconstructed by applying the Sage-
Math [62] code of Balakrishnan, Ionica, Lauter and Vincent [3] (based on [67, 4])
PLANE QUARTICS OVER Q WITH COMPLEX MULTIPLICATION 5
and Magma [5] functionality due to Lercier and Ritzenthaler for hyperelliptic re-
construction in genus 3 [41]. Some of these curves were already computed by
Weng [67]. The final cases 4, 25, 26 were found by Balakrishnan, Ionica, Kılıçer,
Lauter,Somoza,StrengandVincentandwillappearonlinesoon. ThePicardcurves
can be obtained as a special case of our construction, but are more efficiently ob-
tainedusingthe methods ofKoike–Weng[31]andLario–Somoza[36]. The rational
models in [67, 31, 36] as well as those that can be obtained with [4, 3, 41] are
correctuptosomeprecisionoverC. Incase23,thehyperelliptic modelwasproved
to be correct in Tautz–Top–Verberkmoes [61, Proposition 4]. The hyperelliptic
model y2 = x7 1 for case 36 is a classical result (see Example (II) on page 76
−
in Shimura [53]) and the Picard model y3 = x4 x for case 37 is similar (e.g.
−
Bouw–Cooley–Lauter–Lorenzo–Manes–Newton–Ozman[6, Lemma 5.1]); both can
be proven by exploiting the large automorphism group of the curve.
Remark 1.2. InfactthecurveinCase4alsoadmitsahyperellipticdefiningequation
overQ,whichisnotautomatic;apriori itisadegree2coverofconicthatwedonot
knowtobeisomorphictoP1. However,inthiscasethealgorithmsin[10]showthat
the conjectural model obtained is correct, so that also in this case a hyperelliptic
model exists over the field of moduli Q.
In this paper, we construct models for the generic plane quartic cases.
1.2. Obtaining Riemann matrices from CM fields. Let be a lattice of full
L
rank2g inacomplexg-dimensionalvectorspaceV. ThequotientV/ isacomplex
L
Lie group, called a complex torus. This complex manifold is an abelian variety if
andonly if it is projective,whichis true if and only if there exists a Riemann form
for , thatis, anR-bilinear formE :V V R suchthat E( , ) Z andsuch
L × −→ L L ⊂
that the form
V V R (1.3)
× −→
(u,v) E(u,iv)
7−→
is symmetric and positive definite. The Riemann form is called a principal polar-
ization if and only if the form E on has determinant equal to 1. We call a basis
L
(λ ,...,λ )of symplectic if the matrixofE with respectto the basisis givenin
1 2g
L
terms of g g blocks as
×
0 I
Ω = g . (1.4)
g (cid:18) Ig 0(cid:19)
−
Foreveryprincipalpolarization,thereexistsasymplecticbasis. Ifwewriteoutthe
elements of a symplectic basis as column vectors in terms of a C-basis of V, then
we get a g 2g period matrix.
×
The final g elements of a symplectic basis of for E form a C-basis of V, so we
L
use this as our basis of V. Then the period matrix takes the form (τ I ), where
g
|
the g g complex matrix τ has the following properties:
×
(1) τ is symmetric,
(2) Im(τ) is positive definite.
Wecallamatrixsatisfying(1)and(2)aRiemann matrix. Thesetofsuchmatrices
is called the Siegel upper half space and denoted by . Conversely, from every
g
H
Riemann matrix τ, we get the complex abelian variety
Cg/(τZg +Zg) (1.5)
6 KILICER,LABRANDE,LERCIER,RITZENTHALER,SIJSLING,ANDSTRENG
which we can equip with a Riemann form given by Ω with respect to the basis
g
given by the columns of (τ I ).
g
|
Given a CM field K, Algorithm 1 of Van Wamelen [64] (based on the theory
of Shimura–Taniyama [54]) computes at least one Riemann matrix for each iso-
morphism class of principally polarized abelian variety with CM by the maximal
orderofK. Fordetails,andanimprovementwhichcomputesexactly oneRiemann
matrix for each isomorphismclass, see also Streng [59]. In our implementation, we
couldsimplify the algorithmslightly,because the groupappearingin Step 2 of[64,
Algorithm 1] is computed by Kılıçer [26, Lemma 4.3.4] for the fields in Table 1.
1.3. Reduction of Riemann matrices. Once we have Riemann matrices τ, we
change them into Sp (Z)-equivalent matrices on which the theta constants have
6
faster convergence. For this, we want the imaginary part Y of τ, interpreted as a
ternary quadratic form, to be as “nice” as possible, in the sense that its shortest
vectors are large and that the standard basis vectors are reasonably Y-orthogonal
and close to being the Y-shortest vectors.
In any genus g, there is an action on the Siegel upper half space by the
g
H
symplectic group
Sp (Z)= M GL (Z):MtΩ M =Ω GL (Z), (1.6)
2g { ∈ 2g g g}⊂ 2g
given by
A B
(τ)=(Aτ +B)(Cτ +D)−1. (1.7)
(cid:18)C D(cid:19)
The association τ (Cg/τZg +Zg,Ω ) gives a bijection between Sp (Z)
7→ g g \Hg
and the set of principally polarized abelian varieties over C up to isomorphism.
FollowingKlingen[30,Section3],wedefinetheSiegelfundamentaldomain
g
F ⊂
as the set of τ =X +iY that satisfy
g g
H ∈H
(i) for every M =(A B) Sp (Z), we have det(Cτ +D) 1.
C D ∈ 2g | |≥
(ii) Y is Minkowski-reduced, that is,
(a) for all k =1,...,g, we have tvYv Y for all v =(v ,...,v ) Zg
kk 1 g
≥ ∈
with gcd(v ,...,v )=1 and
k g
(b) for all k =1,...,g 1, we have Y 0.
kk+1
− ≥
(iii) X 1/2 for all j and k,
jk
| |≤
Here part (i) corresponds to the determinant of Y being maximal for the Sp (Z)-
2g
orbit,because of the relation(see [30, (1.6)on page8 andthe definition onpage 3]
or [59, Lemma 6.9])
detIm(τ)
detIm(Mτ)= . (1.8)
det(Cτ +D)2
| |
In theory, one reduces a Riemann matrix to as follows.
g
F
Algorithm 1.9 (Theoretical sketch of algorithm for reduction to .).
g
F
(i) First take M such that det(Cτ +D) is minimal and replace τ by Mτ.
| |
After this step, the determinant of Y is maximal for the orbit and the
resulting τ satisfies (i) by (1.8).
(ii) Next, using only transformations of the form
U 0 τ =UτtU for some U GL (Z), (1.10)
0 tU−1 ∈ 2
(cid:0) (cid:1)
PLANE QUARTICS OVER Q WITH COMPLEX MULTIPLICATION 7
whichpreservemaximalityofdetY,make Y =Im(τ) Minkowski-reduced.
Algorithms for Minkowski-reduction are well-known; see for example [1,
24, 48].
(iii) Finally, using only translations
Ig B τ =τ +B, (1.11)
(cid:16) 0 Ig(cid:17)
make sure that (iii) is satisfied without changing Y.
In practice, however, it is hard to find M such that det(Cτ +D) is minimal,
| |
so an iterative algorithm is used. Let S be a finite subset S Sp (Z) and define
⊂ 2g
condition (i)(S) by
(i)(S) for every M =(A B) S, we have det(Cτ +D) 1.
C D ∈ | |≥
Let (S) be defined by (i)(S), (ii) and (iii).
g g
F ⊂H
Algorithm 1.12 (Reduction to (S)). Repeat the following steps until one gets
g
F
τ (S).
g
∈F
(i)(S) If condition (i)(S) is not satisfied, take a matrix M S that violates it
∈
and replace τ by Mτ.
(ii) As in Algorithm 1.9.
(iii) As in Algorithm 1.9.
This algorithm terminates by [30, Lemma 3.1 on page 29].
Foreveryg,thereexistsafinitesetS suchthat = (S)holds([30,V above
g g n
F F
Proposition 3.3]). For such S, Algorithm 1.12 gives a reduction to .
g
F
In case g =1, the set S = 0 1 suffices. Moreover, in that case, step (i)(S)
{ −10 }
is simply (cid:0) (cid:1)
“if τ <1, then replace τ by 1/τ”,
| | −
andAlgorithm1.12is the usualreductionalgorithmto the fundamentaldomainin
the upper half plane.
In case g = 2, a suitable set S of 19 matrices is given by Gottschling [21], and
Algorithm 1.12 is written out in detail in Dupont’s PhD thesis [13, Algorithme 10]
and analysed further in Streng [59, Sections 6.3 and 6.4].
In case g 3, we know no explicit finite set S satisfying = (S), so we use
g g
≥ F F
a weaker condition instead of (i). We use (i)(S) for the one matrix set defined by
S = N =(n1 n2) , where
{ n3 n4 }
n1 =diag(0,1,1,...,1), n2 =diag(1,0,0,...,0), n3 =−n2, n4 =−n1. (1.13)
Note det(n τ +n )= τ , so that the condition is
3 4 11
−
(i)( N ) τ 1.
11
{ } | |≥
This defines a set ( N ) which is too large to be a fundamental domain, but
g
F { }
for which we do have an algorithm (Algorithm 1.12). As we are already relaxing
some conditions and making the set larger, we may as well relax the conditions a
little bit more. In our implementation, we replace (i)( N ) by
{ }
(i’) τ 0.99
1,1
| |≥
to prevent infinite loops coming from numerical instability.
8 KILICER,LABRANDE,LERCIER,RITZENTHALER,SIJSLING,ANDSTRENG
1.3.1. LLL-reduction. Instead of using Minkowski-reduction, one could also use
LLL-reduction [39]. The main advantage being that it is fast for every genus,
instead of only for small genera.
Let v ,...,v be column vectors that form a basis of a lattice in Rg and
1 n
{ }
let v∗,...,v∗ be the Gram-Schmidt orthogonalisation. The basis (v ,...,v ) is
{ 1 g} 1 g
LLL-reduced if
(1) µ 1/2 for 1 j <k g, where µ = v ,v∗ / v∗ 2, and
| kj|≤ ≤ ≤ kj h k ji || k||
(2) v∗ 2 (3/4 µ 2) v∗ 2 for 1<k g.
|| k|| ≥ −| kk−1| || k−1|| ≤
The domain is the set of τ =X +iY satisfying
g g g
B ⊂H ∈H
(i’) τ 0.99.
11
| |≥
(ii’) (a) Y is the Gram matrix of an LLL-reduced basis,
(b) as before: Y 0 for all k =1,2,...,g 1.
kk+1
≥ −
(iii) as before: X 1/2.
jk
| |≤
Algorithm 1.14 (Reduction to ). Repeat until τ .
g g
B ∈B
(i’) If condition (i’) is not satisfied, then replace τ by Nτ.
(ii’) Apply the LLL-algorithm to Y and flip the signs of some basis elements
to also get (ii’)(b). Then apply the corresponding transformation to τ.
(iii) As in Algorithm 1.9.
Lemma 1.15. Algorithm 1.14 terminates, and every Sp (Z)-orbit in contains
2g Hg
only finitely many elements of .
g
B
Proof. Given any τ F and any constant c > 0, by Lemma 3.1 of [30] and
0 g
∈
(1.8), there are only finitely many values for detIm(τ) c for τ in the orbit of τ .
0
≥
In particular, step (i) can only be run finitely many times. This proves that the
algorithm terminates.
It remains to prove the finiteness statement. We first show that for the matrix
τ = X +iY , there is a lower bound (depending only on g) on det(Y). Note
g
∈ B
X2 +Y2 0.99 and X 1/2,hence Y √0.99 0.25>0.86. Using a result
11 11 ≥ | 11|≤ 11 ≥ −
on LLL-reduction, Cohen [9, Theorem 2.6.2-(3)], we get
detY 2−g(g−1)/4Yg/2 >2−g(g+1)/4 =:c.
≥ 11
Given an orbit Sp (Z) τ , let m c be the infimum of detIm(τ) for τ
2g · 0 ≥ ∈ Bg
in the orbit. Note that this infimum is attained because (as noted in the first
paragraphof this proof) there are only finitely many values ofdetIm(τ) m for τ
≥
in the orbit. So let τ be such that it attains this infimum.
0 g
∈B
We now show that there are only finitely many
AB
τ =X +Yi=Mτ = τ .
0 g
(cid:18)C D(cid:19) ∈B
By (1.8) and the choice of τ , we get
0
det(Cτ +D)<1. (1.16)
| |
By Lemma 3.1 of [30], this implies that (C,D) is among a finite set (depending
only on g) of candidates up to left-multiplication by U GL (Z). In other words,
g
∈
it implies that we can write
1S U 0
M = M (1.17)
(cid:18)0 0(cid:19)(cid:18)0 tU−1(cid:19) 0
PLANE QUARTICS OVER Q WITH COMPLEX MULTIPLICATION 9
with S symmetric, U invertible,and M in a finite set of matrices (depending only
0
on g). Let τ =M τ and Y =Imτ .
1 0 0 1
NotethatthereareonlyfinitelymanyLLL-reducedbasesforanylattice. Indeed,
the product of the lengths of the vectors in an LLL-reduced basis satisfies the
upper bound [9, 2.6.2(1)], while the individual basis vectors are lower-bounded by
the length of the shortest vector, hence all vectors in an LLL-reduced basis are
bounded in terms of the lattice. A more precise analysis1 even leads to a uniform
bound 2O(g3) which depends only on the dimension g of the lattice. In particular,
there are finitely many choices for an LLL-reduced basis. It follows that there are
only finitely many many U such that UYtU is an LLL-reducedGrammatrix. And
thenforeachU,thereareatmost2g(g+1)/2 matricesS forwhichtherealpartsX
ij
are in the interval [ 1/2,1/2]. (cid:3)
−
1.4. Conclusion and efficiency. Without reduction,wewereunable tocompute
Dixmier-Ohno invariants (as in Section 2) to sufficient precision. It takes only a
minute to compute the reduction to (with Algorithm 1.14) or ( N ) (with
3 g
B F { }
Algorithm1.12)forallourRiemannmatrices. Thencomputing the Dixmier–Ohno
invariantswith the fastalgorithmof Section2 below takes aboutthe same amount
of time for as for ( N ). We conclude that for g = 3, there is no reason to
3 3
B F { }
preferoneofthesealgorithmsovertheother,butitisveryimportanttouseatleast
one of them. We do advise caution with the LLL-reduced version, as the analysis
inSection2 below is validonly forMinkowski-reducedmatrices. Still, we managed
to compute all our curves using the LLL-reduced version.
2. Computing the Dixmier–Ohno invariants
Inthissection,weshowhowgivenaRiemannmatrixτ wecanobtainanapproxi-
mationoftheDixmier–Ohnoinvariantsofacorrespondingplanequarticcurve. One
procedurehasbeendescribedin[23]andreliesonthecomputationofderivativesof
odd theta functions. Here we take advantage of the existence of fast strategies to
computetheThetanullwertetoemulatetheusualstrategyforsuchcomputationsin
the hyperelliptic case[67,4]: weuse ananalogueofthe Rosenhainformulato com-
pute a special Riemann model for the curve from the Thetanullwerte, from which
we then calculate anapproximationof the Dixmier–Ohnoinvariants. By normaliz-
ing these, we find an explicit representative of the Dixmier–Ohno invariants as an
element of a weighted projective space over Q.
2.1. Fast computation of the Thetanullwerte from a Riemann matrix.
Definition 2.1. The Thetanullwerte or theta-constants of a Riemann matrix τ
in (that is, a g g complex matrix with positive definite imaginary part) are
g
H ×
defined as
ϑ (0,τ)= eiπ(t(n+a)τ(n+a)+2t(n+a)b), (2.2)
[a;b]
nX∈Zg
wherea,b 0,1/2 g. Wedefinethefundamental Thetanullwerte tobethoseϑ
[a;b]
∈{ }
with a=0; there are 8 of them. We simplify notation by writing
ϑ =ϑ , i=2(b +2b + +2g−1b )+2g+1(a +2a + +2g−1a ) (2.3)
[a;b] i 0 1 g−1 0 1 g−1
··· ···
1seehttp://mathoverflow.net/questions/57021/how-many-lll-reduced-bases-are-there.
10 KILICER,LABRANDE,LERCIER,RITZENTHALER,SIJSLING,ANDSTRENG
Inotherwords,wenumber theThetanullwertebyinterpretingthe reverseofthe
sequence(2b 2a)asabinaryexpansion. Thisisthenumberingusedin,e.g.,[13,34].
||
For notational convenience, we write ϑ for the k-tuple ϑ ,...,ϑ . In this
n1,...,nk n1 nk
section, we describe a fast algorithm to compute the Thetanullwerte with high
precision. Note thatitis sufficienttodescribeanalgorithmthatcomputesthe fun-
damentalThetanullwerte;wecanthencomputethesquaresofall64Thetanullwerte
by computing the fundamental ones at τ/2, then use the following τ-duplication
formula [25, Chap. IV]:
1 τ τ
ϑ (0,τ)2 = e−4iπtaβϑ 0, ϑ 0, . (2.4)
[a;b] 2g [0;b+β] 2 [0;β] 2
β∈21XZg/Zg (cid:16) (cid:17) (cid:16) (cid:17)
We can then recover the 64 Thetanullwerte from their square, by using a low-
precision approximation of their value to decide on the appropriate square root.
BothalgorithmsdescribedinthissubsectionhavebeenimplementedinMagma[33].
2.1.1. Naive algorithm for the Thetanullwerte. A (somewhat) naive algorithm to
compute the Thetanullwerte consists in computing the sum in Definition 2.1 until
the remainder is too small to make a difference at the required precision. We
show in this section that it is possible to compute the genus 3 Thetanullwerte
up to 10−P using O( (P)P1.5) bit operations, where (P) is the number of
M M
bit operations needed for one multiplication of P-bit integers; the strategy is to
determine a suitable summation domain, and use induction relations to compute
the terms. The running time is the same as the general strategy given in [11], as
analyzed in [34, Section 5.3].
We do an analysis similar to the ones in [13, 34]. We can use the Minkowski
reduction of Section 1.3 and hence assume without loss of generality that Im(τ) is
Minkowski-reduced.
Lemma 2.5. Let Y = (Y ) be a Minkowski-reduced 3 3 positive definite sym-
ij ij
×
metricreal matrix. Let d=1/100. Then, for all n R3 we have tnYn dY tnn.
11
∈ ≥
Proof. Suppose not. Let n be a counterexample. Let I 1,2,3 be such that
∈ { }
n2 = max n2. Let I,J,K = 1,2,3 . Without loss of generality, we have
I i i { } { }
Y =1 (scale Y) and n =1 (scale n). Minkowski-reducedness gives Y Y for
II I ii 11
≥
all i and 2Y Y for all i=j.
ii ij
≥| | 6
Let s = 1 if Y 0 and s = 1 if Y < 0. In particular, we have s = s
ij ij ij ij ji ij
≥ −
and by Minkowski-reducedness also s =s =1.
12 23
Note that we now have
3d dY tnn>tnYn= n2(Y Y )+ (n +s n )2 Y . (2.6)
≥ 11 i ii− | ij| i ij j | ij|
Xi Xj6=i {Xi,j}
s.t. i6=j
Notethatalltermsontherighthandsidearenon-negativeasMinkowski-reduction
implies Y 2 1Y Y . As all terms on the right hand side are non-
ii ≥ · 2 ii ≥ j6=i| ij|
negative, we find that allPterms are less than 3d.
Let α=3/4.
We distinguish between two cases.
Case I: there exists a j with s n <α.
Ij j
−
Case II: for all j, we have s n α.
Ij j
− ≥
Proof in case I.
We get 3d (1+s n )2 Y Y (1 α)2, hence Y 3d/(1 α)2.
Ij j Ij Ij Ij
≥ | |≥| | − | |≤ −