On the Torelli problem and Jacobian Nullwerte in genus three

0.18 MB·English

by  Jordi Guardia

ON THE TORELLI PROBLEM AND JACOBIAN NULLWERTE
IN GENUS THREE
9
0 JORDIGUA`RDIA
0
2

n
a
J Abstract. Wegiveaclosedformulaforrecoveringanon-hyperellipticgenus

8 threecurvefromitsperiodmatrix,andderivesomeidentitiesbetweenJacobian
2 Nullwerteindimensionthree.

]
G
1. Introduction
A
Itisknownthatthesetofbitangentlinesofanon-hyperellipticgenusthreecurve
.
h
determines completely the curve, since it admits a unique symplectic structure
t
a ([CS 03], [Le 05]). Given this structure, one can recover an equation for the curve
m followingamethodofRiemann([R 1876],[Ri 06]): onetakesanAronholdsystemof
[ bitangent lines, determines some parameters by means of some linear systems and

then writes down a Riemann model of the curve. Unfortunately, the parameters
1
involved in this construction are not defined in general over the field of definition
v
6 of the curve, but over the field of definition of the bitangent lines. This is rather
7 inconvenientfor arithmetical applications concernedwith rationality questions (cf.
3 [Oy 08] for instance).
4
We propose an alternative construction, giving a model of the curve directly
.
1 from a certain set of bitangent lines; this model is already defined over the field of
0
definition of the curve. In the particular case of complex curves, our construction
9
provides a closedsolution for the non-hyperelliptic Torelli problem on genus three:
0
:
v Theorem 1.1. Let C will be a non-hyperelliptic genus three curve defined over
Xi a field K ⊂ C, and let ω1,ω2,ω3 be a K-basis of H0(C,Ω1/K), and γ1,...,γ6 a
symplectic basis of H (C,Z). We denote by Ω = (Ω |Ω ) = ( ω ) the period
ar matrix of C with resp1ect to this bases and by Z = Ω1−12.Ω thReγjnorkmj,aklized period
1 2
matrix. A model of C defined (up to normalization) over K is:
[w w w ][w w′w′] [w w w ][w′w w′] [w w w ][w′w′w ] 2
7 2 3 7 2 3 X Y + 1 7 3 1 7 3 X Y − 1 2 7 1 2 7 X Y
(cid:18)[w w w ][w′w′w′] 1 1 [w w w ][w′w′w′] 2 2 [w w w ][w′w′w′] 3 3(cid:19)
1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
[w w w ][w w′w′][w w w ][w′w w′]
−4 7 2 3 7 2 3 1 7 3 1 7 3 X Y X Y =0,
[w w w ][w′w′w′][w w w ][w′w′w′] 1 1 2 2
1 2 3 1 2 3 1 2 3 1 2 3
Date:January28,2009.
2000 Mathematics Subject Classification. 14H45,14C34,14H42.
Key words and phrases. Torelliproblem,JacobianNullwerte,Frobeniusformula.
PartiallysupportedbyMTM2006-15038-C02-02.
1
2 JORDIGUA`RDIA
where
w =t(1,0,1)+t(0,0,1).Z w′ =t(0,0,1)+t(0,0,1).Z
1 2 2 2 1 2 2
w =t(1,1,0)+t(0,1,1).Z w′ =t(0,1,0)+t(0,1,1).Z
2 2 2 2 2 2 2 2 2
w =t(1,1,1)+t(0,1,0).Z w′ =t(0,1,1)+t(0,1,0).Z
3 2 2 2 2 3 2 2 2
w =t(0,0,1)+t(1,1,1).Z.
7 2 2 2 2
X =grad θ[w ](z;Z).Ω−1.(X,Y,Z)t,
j z=0 j 1
Y =grad θ[w′](z;Z).Ω−1.(X,Y,Z)t,
j z=0 j 1
and [u,v,w] denotes the Jacobian Nullwert given by u,v,w.
Of course, one could simplify the denominators in the equation above to obtain
a simpler formula. The advantage of writing the fractions is that their values are
algebraic over the field of definition of the curve.
Theformulaabovecanbeinterpretedasauniversalcurveoverthemodulispace
of complex non-hyperelliptic curves of genus three. Moreover, using the Frobenius
formula,wecanexpresstheequationofthecurveintermsofThetanullwerteinstead
of Jacobian Nullwerte (cf. section 7).
As it will become clear along the paper, many different choices for the w are
k
possible. They are only restrictedto some geometric conditions expressedin terms
of the associated bitangent lines (cf. section 4). Every election of the w leads
k
to a model for the curve, though all these models agree up to a proportionality
constant. The comparison of the models given for different elections provides a
number of identities between JacobianNullwerte in dimension three. For instance,
we prove:
Theorem 1.2. Take w ,w ,w ,w′,w′,w′ as before, and write
1 2 3 1 2 3
w =t(1,0,0)+t(1,1,1).Z, w′ =t(0,0,0)+t(1,1,1),
4 2 2 2 2 4 2 2 2
w′ =t(1,1,0)+t(1,0,1).Z.
7 2 2 2 2
The following equalities hold on the space H :
3
′ ′ ′ ′
[w w w ](Z)[w w w ](Z)=[w w w ](Z)[w w w ](Z),
2 3 7 1 3 7 1 3 7 2 3 7
[w′w w′](Z)[w′w′w ](Z)=[w′w w′](Z)[w′w′w ](Z),
2 3 7 1 3 7 1 3 7 2 3 7
′ ′ ′ ′
[w w w ](Z)[w w w ](Z)[w w w ](Z)[w w w ](Z)
3 1 2 3 1 2 4 1 2 4 1 2
q
[w w w ](Z)[w′w w ](Z)[w w w′](Z)[w′w w′](Z),
4 1 2 4 1 2 3 1 2 3 1 2
ON THE TORELLI PROBLEM AND JACOBIAN NULLWERTE IN GENUS THREE 3
′ ′ ′ ′
[w w w ](Z)[w w w ](Z)[w w w ](Z)[w w w ](Z)
3 1 2 3 1 2 4 2 1 4 2 1
q
[w w w ](Z)[w′w w ](Z)[w w w′](Z)[w′w w′](Z),
4 1 2 4 1 2 3 2 1 3 2 1
′ ′ ′ ′ ′ ′
[w w w ](Z)[w w w ](Z)[w w w ](Z)[w w w ](Z)
3 1 2 3 1 2 4 1 2 4 1 2
q
[w w w ](Z)[w′w w ](Z)[w w′w′](Z)[w′w′w′](Z).
4 1 2 4 1 2 3 1 2 3 1 2
Again, there are many possible elections of the w , each one leading to a set of
k
identities between Jacobian Nullwerte.
Apart from its intrinsical theoretical interest, these results have different appli-
cations. Fromthe computationalviewpoint, for instance, theorem1.1(or corollary
7.2) can be used to determine equations for modular curves or to present three-
dimensional factors of modular jacobians as jacobians of curves (thus improving
theresultsin[Oy 08]). Inamoretheoreticalframe,theidentities in(1.2)maylead
to simplified expressions for the discriminant of genus three curves (cf. [Ri 06]).
The results stated above are the analytic version of the corresponding results
in an algebraic context. These results are proved in the first part of the paper.
We start recalling Riemann construction for curves of genus three, and the basic
concepts regarding the symplectic structure of the set of bitangent lines of these
curves. The relation between different Steiner complexes is described explicitly in
section 3. In sections 4 and 5 we combine the ideas in previous sections to provide
the algebraic versions of our theorems. The second part of the paper contains the
resultsintheanalyticcontext. Theproofoftheorems4.1and1.2isgiveninsection
6. Finally, we give the Thetanullwerte version of theorem 4.1 in section 7, which
also contains a geometric description of the fundamental systems appearing in the
Frobenius formula.
The authoris muchindebtedto C.Ritzenthalerforpointing hisattentiontothe
Torelli problem for non-hyperelliptic genus three curves, and to E. Nart for many
fruitful discussions.
Notation and conventions: We will work with a non-singular genus three
curve C, defined over a field K with char(K) 6=2, and given by a quartic equation
Q=0. We assumethatthecurveisembeddedintheprojectiveplaneP2(K)byits
canonical map. Given two polynomials Q ,Q , we will write Q ∼ Q to express
1 2 1 2
that they agree up to a constant proportionality factor. A number of geometric
concepts and facts about genus three curves will be used along the paper. We will
take [Do 07] as basic reference, following the notations introduced there.
Part 1. ALGEBRAIC IDENTITIES
2. Riemann model and Steiner complexes
A triplet or tetrad of bitangent lines is called syzygetic whenever their contact
points with the curve lie on a conic. Any pair of bitangent lines to C can be
completed in five different wayswith another pair to a syzygetic tetrad. Moreover,
any two of the six pairs form a syzygetic tetrad. Such a set of of six pairs of
bitangent lines is called Steiner complex (cf. [Do 07]).
4 JORDIGUA`RDIA
Riemann showed that, given three pairs of bitangent lines in a Steiner complex,
one can determine proper equations {X = 0,Y = 0},{X = 0,Y = 0}, {X =
1 1 2 2 3
0,Y = 0} of these lines so that the quartic equation Q = 0 defining the curve C
3
can be written as:
Q∼(X Y +X Y −X Y )2−4X Y X Y =0,
1 1 2 2 3 3 1 1 2 2
and moreover they satisfy the relation:
X +X +X =Y +Y +Y .
1 2 3 1 2 3
To find the proper equations of the bitangent lines, one takes arbitrary equations
forthemandsolvescertainlinearsystemstocomputescalingfactorsleadingtothe
well adjusted equations.
Reciprocally, whenever we take three pairs {X ,Y },{X ,Y }{X ,Y } of linear
1 1 2 2 3 3
polynomials over K, such that no triplet formed by a polynomial in each pair is
linearly dependent and X +X +X = Y +Y +Y , the equation above gives a
1 2 3 1 2 3
non-singular genus three curve over K, with {X = 0,Y = 0}, {X = 0,Y = 0},
1 1 2 2
{X =0,Y =0}beingthreepairsofbitangentlinesinacommonSteinercomplex.
3 3
Let us define W =X +X +X =0, Z =X +Y +Y , Z =Y +X +Y ,
1 2 3 1 1 2 3 2 1 2 3
Z =Y +Y +X . The trivial equalities:
3 1 2 3
Q∼ (X Y +X Y −WZ )2−4X X Y Y
1 2 2 1 3 1 2 1 2
= (X Y +X Y −WZ )2−4X X Y Y
1 3 3 1 2 1 3 1 3
= (X Y +X Y −WZ )2−4X X Y Y
2 3 3 2 1 2 3 2 3
= (X X +Y Y −Z Z )2−4X X Y Y
1 2 1 2 1 2 1 2 1 2
= (X X +Y Y −Z Z )2−4X X Y Y
1 3 1 3 1 3 1 3 1 3
= (X X +Y Y −Z Z )2−4X X Y Y
2 3 2 3 2 3 1 3 1 3
= (X W +Y Z −Y Z )2−4X WY Z
1 2 3 3 2 1 2 3
= (X W +Y Z −Y Z )2−4X WY Z
2 1 3 3 1 2 1 3
= (X W +Y Z −Y Z )2−4X WY Z
3 1 2 2 1 3 1 2
= (X Z +Y W −X Z )2−4X Z Y W
1 3 2 3 1 1 3 2
= (X Z +Y W −X Z )2−4X Z Y W
1 2 3 2 1 1 2 3
(1)
= (X Z +X Z −X Z )2−4X Z X Z
1 1 2 2 3 3 1 1 2 2
= (Y Z +Y Z −Y Z )2−4X Y Y Z
1 1 2 2 3 3 1 1 2 2
show that W = 0,Z = 0,Z = 0,Z = 0 are also bitangent lines to C, and they
1 2 3
make apparent a number of different Steiner complexes on C.
Onthe otherhand,onecancheckeasilythatinthissituationanytripletformed
pickingaline fromeachpair{X =0,Y =0}is asyzygetic. Analogousconclusions
i i
can be derived from each of the equations above. For instance, we mention for
our later convenience that {X = 0,X = 0,W = 0}, {X = 0,X = 0,W = 0},
1 2 1 3
{X =0,Y =0,Z =0} and {X =0,Y =0,Z =0} are asyzygetic triplets.
1 2 3 1 3 3
ON THE TORELLI PROBLEM AND JACOBIAN NULLWERTE IN GENUS THREE 5
3. Relations between Steiner complexes
EverySteinercomplexShasanassociatedtwo-torsionelementO(D )∈Pic0(C);
S
a divisor D defining it is given by the difference of the contact points of the two
S
bitangentlines in anypair in S. We willcall sucha divisor anassociated divisor of
S. Indeed, the map S 7→ O(D ) establishes a bijection between the set of Steiner
S
complexes on C and Pic0(C)[2] (cf. [Do 07]).
TwodifferentSteinercomplexessharefourorsixbitangentlines;they arecalled
syzygetic or asyzygetic accordingly. There is a simple criterion to check whether
two Steiner complexes are syzygetic:
Lemma 3.1. [Do 07,p. 96]Let S ,S be two Steiner complexes, and let D ,D be
1 2 1 2
associated divisors to them. Then ♯S ∩S =4 if e (D ,D )=0 and ♯S ∩S =6
1 2 2 1 2 1 2
if e (D ,D )=1, where e denotes the Weil pairing on Pic0(C)[2].
2 1 2 2
We note that the Weil pairing can be computed by means of the Riemann-
Mumford relation ([ACGH 84, p.290]): for any semicanonical divisor D, we have
(2) e (D ,D )=h0(D)+h0(D+D +D )+h0(D+D )+h0(D+D ) (mod 2).
2 1 2 1 2 1 2
Given a pair {X = 0,Y = 0} of bitangent lines, we shall denote by S
XY
the Steiner complex determined by them. For any set S = {{X = 0,Y =
1 1
0},...,{X = 0,Y = 0}} of pairs of bitangent lines, the subjacent set of lines
r r
will be denoted by S ={X =0,Y =0,...,X =0,Y =0}.
1 1 r r
A priori, the bitangent lines which form a pair in a Steiner complex, are com-
pletely indistinguishable. But if we consider the Steiner complex with relation to
others, there appears a individualization of every line. This idea is made explicit
in corollary 3.4 and proposition 3.7.
It is evident that whenever {X = 0,Y = 0},{X = 0,Y = 0} is a Steiner
1 1 2 2
couple, the pairs {X = 0,Y = 0},{X = 0,Y = 0} form also a Steiner couple,
1 2 2 1
and also the pairs {X = 0,X = 0},{Y = 0,Y = 0} do. The corresponding
1 2 1 2
Steiner complexes are tightly related:
Proposition 3.2. Let S = {{X = 0,Y = 0},...,{X = 0,Y = 0}} be a
X1Y1 1 1 6 6
Steiner complex.
a) The Steiner complexes S and S are syzygetic and they share the
X1Y1 X1Yj
four lines X =0,Y =0,X =0,Y =0, i.e., S ∩S ={X =0,
1 1 j j X1Y1 X1Yj 1
Y =0,X =0,Y =0}.
1 j j
b) TheSteinercomplexesS andS aresyzygeticandS ∩S =
X1Y1 X1X2 X1Y1 X1X2
{X =0, Y =0,X =0,Y =0}.
1 1 2 2
c) TheSteinercomplexesS andS aresyzygeticandS ∩S =
X1X2 X1Y2 X1Y1 X1X2
{X =0, Y =0,X =0,Y =0}.
1 1 2 2
d) For j 6=k, j,k 6=1 the Steiner complexes S and S are asyzygetic.
X1Yj X1Yk
Proof: Let us write div(X ) := 2P +2Q , div(Y ) := 2R +2S . We apply the
i i i i i i
criterionoflemma3.1,using formula2to computethe WeilpairingofdivisorsD
11
and D associated to S and S respectively. We take D = P +Q and
1j X1Y1 X1Yj 1 1
6 JORDIGUA`RDIA
find
e (D ,D ) =h0(3P +3Q −R −S −R −S )−h0(2P +2Q −R −S )
2 11 1j 1 1 1 1 j j 1 1 1 1
−h0(2P +2Q −R −S )+h0(P +Q )=
1 1 j j 1 1
= h0(KC +P1+Q1−R1−S1−Rj −Sj)−h0(KC−R1−S1)
−h0(KC−Rj −Sj)+1=
= h0(P +Q +R +S −R −S )−h0(R +S )−h0(R +S )+1
1 1 1 1 j j 1 1 j j
= h0(P +Q +R +S −R −S )−1=0,
1 1 1 1 j j
since {X = 0,Y = 0,X = 0,Y = 0} form a syzygetic tetrad. This proves part
1 1 j j
a). Remaining parts are proved analogously. (cid:4)
We derive from here a more explicit version of theorem 6.1.8 in [Do 07]:
Corollary 3.3. Let {{X =0,Y =0},{X = 0,Y =0}} be a syzygetic tetrad of
1 1 2 2
bitangent lines.. The Steiner complexes S , S , S satisfy:
X1Y1 X1Y2 X1X2
S ∪S ∪S =Bit(C).
X1Y1 X1Y2 X1X2
Some immediate consequences of the above proposition are:
Corollary 3.4. Let S = {{X = 0,Y = 0},...,{X = 0,Y = 0}} be a Steiner
1 1 6 6
complex.
a) Any triplet {X =0,Y =0,X =0} is syzygetic.
i i j
b) Any triplet {X = 0,X = 0,X = 0} formed picking a line from three
i j k
different pairs of S is asyzygetic.
c) Let {U = 0,V = 0} be an arbitrary pair in the Steiner complex S .
X1Y2
Then U = 0 ∈ belongs to S but not to S and V = 0 belongs to
X1X3 X1Y3
S but not to S (or the other way round).
X1Y3 X1X3
We now derive a geometrical property of asyzygetic triplets of bitangent lines
that we will need later.
Corollary 3.5. Every three asyzygetic bitangent lines X =0,X =0,X =0 can
1 2 3
be paired with other three bitangent lines Y = 0, Y = 0, Y = 0 so that the three
1 2 3
pairs {X =0,Y =0} belong to a common Steiner complex.
i i
Proposition 3.6. Three asyzygetic bitangent lines do not cross in a point.
Proof: Complete the lines to three pairs {{X = 0,Y = 0},{X = 0,Y =
1 1 2 2
0},{X =0,Y =0}} in a common Steiner complex, and re-scale the equations to
3 3
have a Riemann model for C: (X Y +X Y −X Y )2 −4X Y X Y = 0. If the
1 1 2 2 3 3 1 1 2 2
lines X =0, X =0, X =0 crossed in a point, this should be a singular point of
1 2 3
C, which is non-singular by hypothesis. (cid:4)
We end this sectionwith a explicit descriptionof triplets of mutually asyzygetic
Steiner complexes:
Proposition 3.7. Let X = 0,X =0,X =0 be three asyzygetic bitangent lines.
1 2 3
After a proper labeling of the bitangent lines, the Steiner complexes S , S
X1X2 X2,X3
ON THE TORELLI PROBLEM AND JACOBIAN NULLWERTE IN GENUS THREE 7
and S have the following shape:
X3X1
X X X X X X
1 2 2 3 3 1
X4 X9 X9 X14 X14 X4
X X X X X X
S = 5 10, S = 10 15, S = 15 5.
X1X2 X6 X11 X2X3 X11 X16 X3X1 X16 X6
     
X7 X12 X12 X17 X17 X7
     
X X  X X  X X 
8 13 13 18 18 8
     
In particular, the three complexes are asyzygetic.
Proof: Takeasecondpairofbitangentlines{A=0,B =0}inS . Aftercorol-
X1X2
lary 3.3, any other bitangent line must lie in exactly one of the Steiner complexes
S ={{X =0,X =0},{A=0,B =0},...},
X1X2 1 2
S ={{X =0,A=0},{X =0,B =0},...},
X1A 1 2
S ={{X =0,B =0},{X =0,A=0},...}.
X1B 1 2
Since X = 0 cannot be in S by hypothesis, we may suppose that we have a
3 X1X2
pair {X =0,C =0} in S , so that X =0,A=0,X =0,C =0 is a syzygetic
3 X1A 1 3
tetrad, and thus {A = 0,C = 0} belongs to S ; but also X = 0,B = 0,X =
X1X3 2 3
0,C =0 is a syzygetic tetrad and hence {B =0,C =0} belongs to S . (cid:4)
X2X3
4. From bitangent lines to equations
The basic tool for the proof of theorem 1.1 is an algebraicversionof it, giving a
general equation for a curve of genus three in terms of some of its bitangent lines:
Theorem 4.1. Let C : Q = 0 be a non-singular genus three plane curve defined
over a field K with char(K) 6=2. Let {X = 0,Y = 0}, {X = 0,Y = 0},
1 1 2 2
{X = 0,Y = 0} be three pairs of bitangent lines from a given Steiner complex.
3 3
Let {X = 0,Y = 0} be a fourth pair of bitangent lines from the Steiner complex
7 7
given by the pairs {{X = 0,Y = 0},{X = 0,Y = 0}}, ordered so that {X =
1 2 2 1 1
0,X =0,X =0} and {X =0,Y =0,Y =0} are asyzygetic.
3 7 1 3 7
The equation of C can be presented as
(3)
2
Q∼ (X7X2X3)(X7Y2Y3)X Y + (X1X7X3)(Y1X7Y3)X Y − (X1X2X7)(Y1Y2X7)X Y
(cid:16)(X1X2X3)(Y1Y2Y3) 1 1 (X1X2X3)(Y1Y2Y3) 2 2 (X1X2X3)(Y1Y2Y3) 3 3(cid:17)
−4(X7X2X3)(X7Y2Y3)(X1X7X3)(Y1X7Y3)X Y X Y =0.
(X1X2X3)(Y1Y2Y3) (X1X2X3)(Y1Y2Y3) 1 1 2 2
Here (ABC) denotes the determinant of the matrix formed by the coefficients of
the homogeneous linear polynomials A,B,C.
Remark: All the determinants appearing in the theorem are non-zero, since they
areformed with asyzygetic triplets ofbitangent lines, which are non-concurrentby
proposition 3.6.
Proof: We know that for a proper re-scaling X = α X , Y = β X , we will have
i i i i i i
the following Riemann model for C:
Q=(X Y +X Y −X Y )2−4X Y X Y =0,
1 1 2 2 3 3 1 1 2 2
8 JORDIGUA`RDIA
with X =X +X +X =−Y −Y −Y =0. We look at these two equalities
7 1 2 3 1 2 3
as equations in the scaling factors α ,β to determine them. We find:
i i
(X X X ) (X X X ) (X X X )
7 2 3 1 7 3 1 2 7
α =α , α =α , α =α ,
1 7 2 7 3 7
(X X X ) (X X X ) (X X X )
1 2 3 1 2 3 1 2 3
(X Y Y ) (Y X Y ) (Y Y X )
7 2 3 1 7 3 1 2 7
β =−α , β =−α , β =−α .
1 7 2 7 3 7
(Y Y Y ) (Y Y Y ) (Y Y Y )
1 2 3 1 2 3 1 2 3
We substitute the X ,Y on the Riemann model by his values, and simplify the
i i
constant α to get the desired equation. (cid:4)
7
5. Determinants of bitangent lines
Wehaveseenintheorem4.1howtoconstructapresentationofthecurveC from
certain set of bitangent lines, which we can choose in many different ways. If we
make different elections, the comparisonof the corresponding presentations, which
all agree up to a constant, leads us to a number of equalities between the involved
determinants. It turns out that the identities obtained in this way can be easily
proved independently of their deduction.
5.1. Relations between pairs in the same Steiner complex.
Theorem 5.1. Let {X = 0,Y = 0},...,{X =0,Y =0} be four different pairs
1 1 4 4
in the Steiner complex S =S . Then
XiYi
(X X X )(Y X X ) (X X Y )(Y X Y )
3 1 2 3 1 2 3 1 2 3 1 2
=
(X X X )(Y X X ) (X X Y )(Y X Y )
4 1 2 4 1 2 4 1 2 4 1 2
(X X Y )(Y X Y ) (X Y Y )(Y Y Y )
3 2 1 3 2 1 3 1 2 3 1 2
= = .
(X X Y )(Y X Y ) (X Y Y )(Y Y Y )
4 2 1 4 2 1 4 1 2 4 1 2
Proof: It is known (cf. [Do 07, p. 94]) that all the pairs in a Steiner complex can
be seen as one of the degenerate conics on a fixed pencil of conics. In particular,
we must have a relation
λ2X Y +X Y +λX Y =X Y ,
1 1 2 2 3 3 4 4
∗
for certain λ ∈ K . Let P ,P ,P ,P be the points of intersection of the pairs of
1 2 3 4
lines {X = X = 0}, {X = Y = 0}, {Y = X = 0}, {Y = Y = 0}. If we
1 2 1 2 1 2 1 2
substitute these points in the equality above we find:
λX (P )Y (P )=X (P )Y (P ),j =1,...,4,
3 j 3 j 4 j 4 j
so that for j 6=k we have
X (P )Y (P ) X (P )Y (P )
3 j 3 j 3 k 3 k
(4) = .
X (P )Y (P ) X (P )Y (P )
4 j 4 j 4 k 4 k
An elementary exercise in linear algebrashows that for any two lines U =0,V =0
we have the equalities
U(P ) (UX X ) U(P ) (UX Y ) U(P ) (UY X ) U(P ) (UY Y )
1 1 2 2 1 2 3 1 2 4 1 2
= , = , = , = .
V(P ) (VX X ) V(P ) (VX Y ) V(P ) (VY X ) V(P ) (VY Y )
1 1 2 2 1 2 3 1 2 4 1 2
The theorem follows if we apply these relations to the two pairs of lines X =
3
0,X =0 and Y =0, Y =0 and substitute the resulting relations in (4). (cid:4)
4 3 4
ON THE TORELLI PROBLEM AND JACOBIAN NULLWERTE IN GENUS THREE 9
5.2. Relations between pairs in syzygetic Steiner complexes.
Theorem 5.2. Let {X = 0,Y = 0}, {X = 0,Y = 0}, {X = 0,Y = 0} be
1 1 2 2 3 3
three pairs of bitangent lines from a given Steiner complex. Let {X = 0,Y = 0}
7 7
be a fourth pair of bitangent lines from the Steiner complex given by the pairs
{{X = 0,Y = 0},{X = 0,Y = 0}}, ordered so that {X = 0,X = 0,X = 0}
1 2 2 1 1 3 7
and {X =0,Y =0,Y =0} are asyzygetic. The following relations hold:
1 3 7
(X X X ) (X Y Y ) (Y X Y ) (Y Y X )
2 3 7 2 3 7 2 3 7 2 3 7
= , = .
(X X X ) (X Y Y ) (Y X Y ) (Y Y X )
1 3 7 1 3 7 1 3 7 1 3 7
Proof: The validity of the equalities is not affected by a re-scaling of the involved
lines, so that we can assume that X = X + X +X , Y = Y +Y +X =
7 1 2 3 7 1 2 3
−X − X −Y . The result follows from the very elementary properties of the
1 2 3
determinants. (cid:4)
Part 2. ANALYTIC IDENTITIES
We nowprovethe theorems statedin the introduction,whichturnoutto be the
translation of the results in the first part of the paper to the context of complex
geometry.
From now on, we suppose that C is a non-hyperelliptic plane curve of genus
three defined over a field K ⊂ C. We choose the basis ω ,ω ,ω ∈ H0(C,Ω1 ) of
1 2 3 /K
holomorphicdifferentialformssuchthatthewecanidentifyC withtheimageofthe
associatedcanonicalmapι:C −→P2(K). We alsofix asymplectic basisγ ,...,γ
1 6
of the singular homology H (C,Z). We denote by Ω = (Ω |Ω ) = ( ω ) the
1 1 2 γj k j,k
periodmatrix of C with respectto these bases, andby Z =Ω−1.Ω thRe normalized
1 2
period matrix. We consider the Jacobian variety JC, represented by the complex
torus C3/(1|Z), with the Abel-Jacobi map:
C2 −Π→ JC
(5)
D
D −→ (ω ,ω ,ω ),
κ 1 2 3
R
whereκ isthe Riemannvector,whichguaranteesthatΘ=Π(C2) isthe divisorcut
by the Riemann theta function θ(z;Z), and that Π(KC −D)=−Π(D).
Weshalldescribe,ascustomary,theelementsofJC[2]bymeansofcharacteristics:
every w ∈ JC[2] is determined uniquely by a six-dimensional vector ǫ = (ǫ′,ǫ′′) ∈
{0,1/2}6bytherelationw =ǫ+Zǫ′′. Withthisnotation,theWeilpairingonJC[2]
is given by:
′ ′′ ′′ ′
(6) e˜ (w ,w ):=e˜ (ǫ ,ǫ ):=ǫ ·ǫ +ǫ ǫ (mod 2).
2 1 2 2 1 2 1 2 1 2
Every w ∈JC[2] defines a translate of the Riemann theta function:
θ[w](z;Z):= eπitǫ′.Z.ǫ′+2πitǫ′.(z+ǫ′′)θ(z+Zǫ′+ǫ′′)
= eπit(n+ǫ′).Z.(n+ǫ′)+2πit(n+ǫ′).(z+ǫ′′).
nX∈Zg
The values θ[w](0,Z) are usually called Thetanullwerte, and denoted shortly by
θ[w](Z). Forasequenceofthreepointsw1,w2,w3 ∈JC[2]themodifieddeterminant
10 JORDIGUA`RDIA
[w ,w ,w ](Z):=π3detJ[w ,w ,w ](Z) of the matrix
1 2 3 1 2 3
∂θ[w ] ∂θ[w ] ∂θ[w ]
1 1 1
(0;Z) (0;Z) (0;Z)
 ∂z1 ∂z2 ∂z3 
∂θ[w ] ∂θ[w ] ∂θ[w ]
J[w ,w ,w ](Z):= 2 (0;Z) 2 (0;Z) 2 (0;Z)
1 2 3  
 ∂z1 ∂z2 ∂z3 
 ∂θ[w ] ∂θ[w ] ∂θ[w ] 
 3 (0;Z) 3 (0;Z) 3 (0;Z) 
 ∂z ∂z ∂z 
 1 2 3 
is called Jacobian Nullwert. (These definitions can be given for a g-dimensional
complextorusabelianvarieties,butwerestrictthemtoourcaseofdimensionthree
for brevity).
6. Proof of main results
It is well-knownthat the Abel-Jacobi map establishes a bijection between semi-
canonicaldivisorsonCandthesetJC[2]. Inourparticularcaseofanon-hyperelliptic
genus three curve, the odd semicanonical divisors are given by the bitangent lines:
given a bitangent X =0, then div(X)=2D with D a semicanonicaldivisor which
goes to a w :=Π(D)∈JC[2]odd. Reciprocally, given w ∈JC[2]odd, the line
X
∂θ[w] ∂θ[w] ∂θ[w]
Hw :=(cid:18) ∂z (0;Z), ∂z (0;Z), ∂z (0;Z)(cid:19).Ω−11. Y =0,
1 2 3 Z
 
is a bitangent line to C, whose corresponding semicanonical divisor D satisfies
Π(D)=w. Moreover,we have
e (Π(w ),Π(w ))=e˜ (w ,w ).
2 1 2 2 1 2
Note also that, given wi ∈JC[2]odd, we have
[w ,w ,w ](Z) (H H H )
(7) 1 2 3 = w1 w2 w3 .
[w ,w ,w ](Z) (H H H )
4 5 6 w4 w5 w6
The theorems stated in the introduction are the translationto the analytic con-
text of theorems 4.1, 5.1 and 5.2. In order to proof them, we have only to check
that the bitangent lines associated to the odd two-torsion points wk ∈ JC[2]odd
appearing in the theorems satisfy the hypothesis of the mentioned theorems. But
this is immediate, using the analytic description (6) of the Weil pairing.
7. Some geometry around the Frobenius formula
While theclosedsolutiontotheTorelliproblemgivenbytheorem1.1issatisfac-
tory from the geometrical viewpoint, it would be desirable to have also a formula
involving only Thetanullwerte instead of Jacobian Nullwerte. We can derive this
formula from our theorem just taking into account Frobenius formula, which we
recall briefly.
BoththeThetanullwerteandtheJacobianNullwertecanbeinterpretedasfunc-
tions defined on the Siegel upper half space H : the idea consists in fixing the
3
necessary characteristics and letting Z run through H . Frobenius formula relates
3
certain Jacobian Nullwerte with products of Thetanullwerte.
Three characteristics ǫ ,ǫ ,ǫ are called asyzygetic if e (ǫ −ǫ ,ǫ −ǫ ) = −1.
1 2 3 2 1 2 1 3
A sequence of characteristics is called asyzygetic if every triplet contained in it
is asyzygetic. A fundamental system of characteristics is an asyzygetic sequence
S ={ǫ ,...,ǫ } of characteristics, with ǫ ,ǫ ,ǫ odd and ǫ ,...,ǫ even.
1 8 1 2 3 4 8