Theta constants identities for Jacobians of cyclic 3-sheeted covers of
  the sphere and representations of the symmetric group
THETA CONSTANTS IDENTITIES FOR JACOBIANS OF
CYCLIC 3-SHEETED COVERS OF THE SPHERE AND
REPRESENTATIONS OF THE SYMMETRIC GROUP
BY YAACOV KOPELIOVICH
To my friend Elizabeth Drake
Abstract. We find identities between theta constants with rational charac-
teristics evaluated at period matrix of R, a cyclic 3 sheeted cover of the sphere
with 3k branch points λ1...λ3k. These identities follow from Thomae formula
[BR]. This formula expresses powers of theta constants as polynomials in
λ1...λ3k. We apply the representation of the symmetric group to find relations
between the polynomials and hence between the associated theta constants.
1. Introduction
Let R be a Riemann surface with the equation:
(z − λi)(∗).
We find relations that are satisfied by theta constants with rational characteristics
evaluated at τR, the period matrix of R. Special type identities for period matrices
are known in the case of a general Riemann surface ( Schottky-Jung identities).
For hyperelliptic curves there are vanishing theta constants of even characteristics
that characterize the associated period matrix. According to Mumford, [Mu] spe-
cial relations of non vanishing of theta constants evaluated at period matrices of
hyperelliptic curves were obtained by Frobenius.
The original Schottky problem seeks special relations among theta constants that
characterize the entire moduli space of algebraic curves of genus g. In this note we
seek special relations that are satisfied by n-sheeted cyclic covers of the sphere.
When n = 2 cyclic covers are just hyperelliptic curves. The next case is n = 3 and
we find relations between theta constants with rational characteristics evaluated at
τR the period matrices of such curves .
These identities are a result of Thomae formula for cyclic n sheeted covers of
the sphere. This formula expresses powers of such theta constants evaluated at the
period matrix τR through polynomial expression of λi. A relation between these
polynomials produces a relation between associated theta constants. Applying the
representation theory of the symmetric group, S3m we produce a basis for the vector
space spanned by the polynomials and as a result relations between the associated
theta constants.
For the simplest case of 6 branch points our results overlap with results of Mat-
sumoto [Ma]. In his paper Matsumoto finds the explicit action of S6 on theta
Date: 04 April 2007.
http://arxiv.org/abs/0704.1032v1
2 KOPELIOVICH
constants evaluated at τR and expresses branch points λi as rational functions of
theta constants. As a result he writes identities between cubic powers of these
constants which essentially coincide with the identities obtained by us in the last
section of our note. Using the representation theory of S6 we see that the space
generated by theta constants is 5 dimensional. This seems to be a new result even
in this case. We note that the Algebraic dimension of this particular family of
curves is 3.
This work was partially done during a visit to the TAMU math department and
the author thanks the department for the invitation and kind hospitality. I thank
Samuel Grushevsky and Mike Fried for constructive remarks on this note.
2. Thomae formula for cyclic covers and relations between theta
constants
We explain the general Thomae formula following [Na] for an algebraic curve R
given by the equation:
(z − λi)(∗)
We denote f : R 7→ CP1 the projection (z, y) 7→ z. Define Qi = f
−1(λi), to be
the unique branch point on R that is the pre image of λi. Fix a homology basis
a1, a2...a3m−2, b1, b2, ..., b3m−2 on R such that the intersections are aiaj = 0 = bibj
and aibj = 1. Let v1...v3m−2 be a basis of standard holomorphic differentials dual
to the basis a1, a2...a3m−2, b1, b2, ..., b3m−2 i.e.
vj = 0,
vj = δij . Now fix an
ordering of λi. Let φ be the automorphism of order 3 defined by (z, y) 7→ (z, ωy) for
ω3 = 1.We write α ≡ β for linear equivalent of divisors, i.e. if there exists a function
g : R 7→ CP 1 and div(g) = α − β. The group Div0/ ≡ is Jac(R), the Jacobian of
R. (Div0 - divisors of degree 0.) Let ψ be the mapping ψ : Div 7→ Div/ ≡ . Then
the following lemma is true:
Lemma 2.1. Let P1, P2 ∈ R,P1 6= P2 and
Di = Pi + φ(Pi) + φ
2(Pi), i = 1, 2
then ψ(D1) ≡ ψ(D2).
Proof. Let f1(P ) =
f(P )−f(P1)
f(P )−f(P2)
, then div(f1) = D1 −D2. �
Define D = ψ
P + φ(Pi) + φ
2(Pi)
as the equivalence class in the Jacobian.
Lemma 2.2. Let K be the canonical divisor of R Then the following holds:
D ≡ 3Qi ≡ ∞1 +∞2 +∞3
K ≡ (2m− 2)D
Qi ≡ mD
Proof. The first item follows exactly as in the previous lemma. To show the rest,
note that z dz
is a holomorphic differential with the divisor Q6m−61 . �
THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 3
Now let Λ = {Λ1,Λ2,Λ3} be a partition of {1, 2, 3, 4, 5, ...3m} with |Λi| = m
for i = 1, 2, 3. We are interested in the following divisor eΛ associated with the
partition:
eΛ = XΛ1 + 2XΛ2 −D −∆
where for each subset S of {1, 2...3m} we set
Fix a point P0 ∈ R and let ΦP0 : R→ Jac(R) be given by ΦP0 (P ) =
v1...
v3m−2
Definition 2.3. Let Hg denote the set of g × g symmetric matrices, τ such that
the imaginary part of τ is positive definite. For ε, ε′,∈ Rg and τ ∈ Hg we denote
(τ) =
lεZ2g
exp 2πi
)t ε′
This series is uniformly and absolutely convergent on compact subsets of Cg×Hg.
To each w ∈ C3m−2 associate a unique w1, w2 ∈ R
g such that w = w1 + τw2.
[Na] proves the following formula for theta constants with characteristics associ-
ated to divisors eΛ. see [BR] as well:
Theorem 2.4. The divisor eΛ is a point of order 6 on the Jacobian and
(1) θ[eΛ]
6 (τR) = CΛ(detA)
3((Λ0Λ0)(Λ1Λ1)(Λ2Λ2))
(Λ0Λ1)(Λ1Λ2)(Λ0Λ2)
Here A is the matrix of certain differentials integrated with respect to ai. and if
Λi = {i1 < ... < im} ,Λj = {j1 < ... < jm}
(ΛiΛi) =
(λik − λil) , (ΛiΛj) =
k=1,l=1
(λik − λjl)
We apply the theorem to generate special relations between theta functions
with characteristics eΛ, evaluated at τR. For each partition Λ denote the poly-
nomial on the right hand side of the last equation by pΛ. To obtain identities
for θ[eΛ] we search for identities between pΛ. The key observation that allows us
to simplify the problem is the following form of the polynomials: choose Λ =
{{1, 2...,m}, {m+ 1, ..., 2m}, {2m, ..., 3m}}. Then by definition of pΛ the factor
ΛiΛj is the discriminant and a common factor for each pΛ which does not
depend on the partition Λ. Thus identities between θ6[eΛ] are equivalent to identi-
ties between the polynomials
((Λ0Λ0)(Λ1Λ1)(Λ2Λ2))
Consequently, identities between
θ6[eΛ] are equivalent to identities between the
polynomials:
((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) .
To get a hint for the result observe that the group S3m acts naturally on the
polynomials ((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) via its action on the partitions of {1...3m} .
Thus Span(((Λ0Λ0)(Λ1Λ1)(Λ2Λ2)) , is a vector space and has a representation of
S3m on it.
4 KOPELIOVICH
3. Explicit Basis
In this section we provide an explicit basis for the space of polynomials from the
previous section. We imitate the process described in [J] to construct a basis for
the irreducible representation of the symmetric group of Sn. For complex numbers
these representations are completely classified. We describe the construction for
any representation of the symmetric group and obtain the relevant case of cyclic
covers as an immediate corollary of the general case. We remind the reader some
facts from the representation theory of Sn.
Let n be a natural number and let k1...km be a partition of n. i.e.
i=1 ki = n
and k1 ≥ k2 ≥ k3... ≥ km.
Definition 3.1. A Young diagram associated to a partition consists of m rows
such that i’th row has ki elements.
Definition 3.2. Let Y be a Young diagram; a tableau is obtained by distributing
the numbers {1...n} within the m rows with the following properties
• Each row contains exactly ki elements
• The numbers in each row form an increasing sequence
Assume that Λ = {Λ0, ...,Λk} is a tableau of n. Define the polynomial:
(ΛiΛi) =
ik<il,{ik,il}∈Λi
(λik − λil ), where pΛ =
(ΛiΛi).
The symmetric group, Sn acts on Λ and therefore acts on the polynomials pΛ. To
find the basis for pΛ we use a modification of Garnier relation [J] (7.1) to construct
a basis for the polynomials.1 Arrange the tableau in columns ( i.e. the first column
will be elements of Λ1 the second column elements of Λ2 etc). Overall we have k
columns for Λ. Let X be a subset of the i − th column of Λ and Y is a subset of
the i+1− th column of Λ. Let σ1...σk be coset representatives for SX×Y in SX
Then we have the Garnier relations:
Theorem 3.3. Let µi denote the number of elements in the i− th column of Λ. if
Y | > µi then
sign(σm) (pσmΛ) = 0.
Proof. If |X
Y | > µi, by the pigeon hole principle there exists an involution δ
such that σmΛ is invariant under it. Thus
sign(σm) (pσmΛ) =
sign(σm) (pδσmΛ) =
signσm (pσmΛ) = 0
In order to exhibit an explicit basis we define a standard Young tableau
1We were not able to find a reference to our approach of constructing Specht modules though
we are confident its a folklore.
THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 5
Definition 3.4. A standard tableau is a tableau where the rows and the columns
are arranged in an increasing order.
Definition 3.5. We define an ordering on the set of tableaux by setting Λ1 < Λ2
if there is an i such that
• if j > i than j is in the same column of Λ1,Λ2
• i is in more left column in Λ1 than Λ2.
Theorem 3.6. Let Λ1...Λk be the collection of standard tableaux for a given parti-
tion. Then pΛ1 ...pΛk is a basis for the vector space spanned by Λ.
Proof. We follow [J] in the proof. We show that pΛk spans any other polynomial
corresponding to our partition. Let t be a tableau and suppose by induction that
the theorem is proved for each t1 tableau such that t1 < t. If t is non standard
there exists adjacent columns a1 < ... < aq < ... < ar and b1 < b2... < bq < ...bs
such that aq > bq. Apply Garnier relation for X = a1...ar, Y = b1...bq. For each σ
a representative in SX
Y in SX×Y we have that [tσ] < t by the definition of the
order < . The result follows immediately from the induction hypothesis. �
Definition 3.7. For an element k of the tableau t Let Ck, Rk be the unique column
and row k belongs to. The hook of k, hk is the number of elements beneath k in
Ck plus the number of elements to the right of k in Rk (include the element itself
in the row but not in the column.)
It is well known that the number of standard tableaux equals to
See [J].
4. The ideal of theta identities
We apply the theory of the previous paragraph to cyclic covers of order 3. Ac-
cording to the theory, the hooks of the partitions correspond to tableau with 3 rows
and m elements in each row. Our first corollary is
Corollary 4.1. The dimension of the polynomials pΛ (and hence the vector space
spanned by
θ6[eΛ] (τR) corresponding to them) is:
(3m)!×2
(m+2)!(m+1)!m!
Hence we can also give a basis for θ6[eΛ] (τR) that correspond to the different
partitions eΛ.
Corollary 4.2. The set of
θ6[eΛS ] (τR), ΛS is a standard partition is a basis for
a vector space spanned by
θ6[eΛ] (τR). In particular each
θ6[eΛ] (τR), Λ can be
written as a linear combination of elements from the set
θ6[eΛS ] (τR).
5. Example
Let us revisit the case when there are 6 branch points and the genus of the
surface is 4. In this case, by the formula for the dimension, the number of basis
functions, θ3[eΛ] is : 2 ×
4!3!2!
= 5. We enumerate the 15 partitions as well as the
the polynomials that correspond to them:
(1) Λ = {(1, 2), (3, 4), (5, 6)}pΛ = (λ1 − λ2)(λ3 − λ4)(λ5 − λ6)
(2) Λ = {(1, 2), (3, 5), (4, 6)}pΛ = (λ1 − λ2)(λ3 − λ5)(λ4 − λ6)
6 KOPELIOVICH
(3) Λ = {(1, 2), (3, 6), (4, 5)}pΛ = (λ1 − λ2)(λ3 − λ6)(λ4 − λ5)
(4) Λ = {(1, 3), (2, 4), (5, 6)}pΛ = (λ1 − λ3)(λ2 − λ4)(λ5 − λ6)
(5) Λ = {(1, 3), (2, 5), (4, 6)}pΛ = (λ1 − λ3)(λ2 − λ5)(λ4 − λ6)
(6) Λ = {(1, 3), (2, 6), (4, 5)}pΛ = (λ1 − λ3)(λ2 − λ6)(λ4 − λ5)
(7) Λ = {(1, 4), (2, 5), (3, 6)}pΛ = (λ1 − λ4)(λ2 − λ5)(λ3 − λ6)
(8) Λ = {(1, 4), (2, 6), (3, 5)}pΛ = (λ1 − λ4)(λ2 − λ6)(λ3 − λ5)
(9) Λ = {(1, 4), (2, 3), (5, 6)}pΛ = (λ1 − λ4)(λ2 − λ3)(λ5 − λ6)
(10) Λ = {(1, 5), (2, 3), (4, 6)}pΛ = (λ1 − λ5)(λ2 − λ3)(λ4 − λ6)
(11) Λ = {(1, 5), (2, 4), (3, 6)}pΛ = (λ1 − λ5)(λ2 − λ4)(λ3 − λ6)
(12) Λ = {(1, 5), (2, 6), (3, 4)}pΛ = (λ1 − λ5)(λ2 − λ6)(λ3 − λ4)
(13) Λ = {(1, 6), (2, 3), (4, 5)}pΛ = (λ1 − λ6)(λ2 − λ3)(λ4 − λ5)
(14) Λ = {(1, 6), (2, 4), (3, 5)}pΛ = (λ1 − λ6)(λ2 − λ4)(λ3 − λ5)
(15) Λ = {(1, 6), (2, 5), (3, 4)}pΛ = (λ1 − λ6)(λ2 − λ5)(λ3 − λ4)
The basis for the vector space of the polynomials corresponds to the following
standard tableaux:
(1) Λ = {(1, 2), (3, 4), (5, 6)}pΛ = (λ1 − λ2)(λ3 − λ4)(λ5 − λ6)
(2) Λ = {(1, 2), (3, 5), (4, 6)}pΛ = (λ1 − λ2)(λ3 − λ5)(λ4 − λ6)
(3) Λ = {(1, 3), (2, 4), (5, 6)}pΛ = (λ1 − λ3)(λ2 − λ4)(λ5 − λ6)
(4) Λ = {(1, 3), (2, 5), (4, 6)}pΛ = (λ1 − λ3)(λ2 − λ5)(λ4 − λ6)
(5) Λ = {(1, 4), (2, 5), (3, 6)}pΛ = (λ1 − λ4)(λ2 − λ5)(λ3 − λ6)
The rest of the 10 polynomials can be rewritten as a linear combination of the set
above applying Garnier’s algorithm as in Theorem 3.7. For example we have:
(λ1−λ2)(λ3−λ6)(λ4−λ5) = −(λ1−λ2)(λ3−λ4)(λ5−λ6)+(λ1−λ2)(λ3−λ5)(λ4−λ6)
(λ1−λ3)(λ2−λ6)(λ4−λ5) = −(λ1−λ3)(λ2−λ4)(λ5−λ6)+(λ1−λ3)(λ2−λ5)(λ4−λ6)
(λ1−λ6)(λ2−λ5)(λ3−λ4) = (λ1−λ4)(λ2−λ5)(λ3−λ6)−(λ1−λ3)(λ2−λ5)(λ4−λ6)
The others polynomials can be expressed in a similar way leading to identities
between
θ6[eΛ] (τR) in this case. Let us conclude with the following remarks on
the identities above: In the hyperelliptic curve case the identities between integral
characteristics of theta functions evaluated at period matrix of hyperelliptic curves
arise from vanishing properties of theta functions. In our case it is interesting to
investigate whether an analogous situation can arise. The only source of cubic theta
identities known to the author, is the following theorem in [Ko]:
Theorem 5.1. Let
be an odd integral theta characteristics in genus 3m− 2
Then for any τ ∈ H3m−2:
0≤νi≤3
µiνiθ3
µ′ + 2ν
(0, τ) = 0
It is plausible that the vanishing of theta constants with rational characteristics
of order 3 on τR will produce a new proof for the special identities obtained in
this note using Thomae formula. Finally note that for all the identities (4) the
coefficients are ±1 It is plausible that this a general phenomenon.
THETA CONSTANTS FOR 3-SHEETED COVERS OF THE SPHERE 7
6. conclusion
There exists an extensive literature on Schottky-Jung identities and on theta
constants for hyperelliptic curves. In this note we obtained special identities for
other classes of algebraic curves. In subsequent notes we plan to pursue and develop
further the themes touched in this note, especially applications of similar methods
to general Hurwitz spaces and their mapping class groups.
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tionen Crelle’s Journal 71(1870) 201-222
5736 Las Virgenes Rd. Calabasas CA 91302
	1. Introduction
	2. Thomae formula for cyclic covers and relations between theta constants
	3. Explicit Basis
	4. The ideal of theta identities
	5. Example
	6. conclusion
	References
