A NOTE ON VANISHING PROPERTIES OF CERTAIN THETA CONSTANTS OF GENUS TWO Eyal Z. Goren* Preprint No. 15 1995/96

Abstract We study the vanishing of Riemann’s theta constants Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) for ε, ε′ ∈ \frac{1}{k} ℤ^{2}, k = 4, 6 and derive identities like

λ ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈A^{−}} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{2} = ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈E} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{96}

for some λ ∈ ℚ(ζ_{36})^{×}, where

A^{−} = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : −\frac{1}{2} ≤ ε_{i}, ε′_{i} < \frac{1}{2}, i = 1, 2,
[ \begin{matrix}ε \\ ε′ \end{matrix} ] is of exact order 6 in ℝ^{4}/ℤ^{4}, 36 ^{t}ε · ε′ ≡ 1 ( mod 2)\}

and

E = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : ε_{i}, ε′_{i} ∈ \{0, \frac{1}{2}\} , i = 1, 2, 4 ^{t}ε · ε′ ≡ 0 ( mod 2)\} .

Typeset by AMS-TEX

*Partially supported by the Edmund Landau Center for research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany).

Definition. Define Siegel’s upper half space of genus g, ℌ_{g}, by

ℌ_{g} = {τ ∈ Mat_{g×g}(ℂ)| ^{t}τ = τ, Im(τ) is positive definite}.

For ε, ε′ ∈ ℝ^{g}, z ∈ ℂ^{g}, τ ∈ ℌ_{g} define Riemann’s theta function with characteristic

[ \begin{matrix}ε \\ ε′ \end{matrix} ]
(1) Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ) = ∑_{N∈ℤ^{g}} exp \{2πi (\frac{1}{2} ^{t}(N + ε)τ(N + ε) + ^{t}(N + ε)(z + ε′))\} .

We call[ \begin{matrix}ε \\ ε′ \end{matrix} ] a rational characteristic if ε, ε′ ∈ ℚ^{g} and an integral characteristic if 2ε, 2ε′ ∈ ℤ^{g}.

It is well known that (1) defines a holomorphic function on ℂ^{g} × ℌ_{g} (see [FK] §VI.1. Note that our definition of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ) is slightly different from the def-

inition in [FK] which is also customary). Let Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) = Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (0, τ). The following transformation formulas are classical ([Mum1] p. 123):

(2) Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ) = exp \{2πi (\frac{1}{2} ^{t}ετε + ^{t}εz + ^{t}εε′)\} · Θ (z + (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ] , τ)

(3)  Θ [ \begin{matrix}ε + μ \\ ε′ + μ′ \end{matrix} ] (z, τ) = exp\{2πi( ^{t}εμ′)\} · Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ)  [ \begin{matrix}μ \\ μ′ \end{matrix} ] ∈ ℤ^{2g}

(4)

Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z+(τI) [ \begin{matrix}μ \\ μ′ \end{matrix} ] , τ) = exp \{2πi (−\frac{1}{2} ^{t}μ(τμ + 2z + 2ε′) + ^{t}εμ′)\}·Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ).

If [ \begin{matrix}ε \\ ε′ \end{matrix} ] is an integral characteristic then

(5)  Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (−z, τ) = exp\{2πi(2^{t}εε′)\} · Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ).

Remark. One often thinks about the functions Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ) and Θ [ \begin{matrix}ε + μ \\ ε′ + μ′ \end{matrix} ] (z, τ)

as the same if μ, μ′ ∈ ℤ^{g}. Indeed, by (3), Θ [ \begin{matrix}ε + μ \\ ε′ + μ′ \end{matrix} ] (z, τ) = exp(2πi ^{t}ε ·

μ′)Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ) and therefore, at least for questions concerning their divisors, these two theta functions could be identified. In this context we remark that if we associate to [ \begin{matrix}δ \\ δ′ \end{matrix} ] the point (τI) [ \begin{matrix}δ \\ δ′ \end{matrix} ] on ℂ^{g}/(τI)ℤ^{2g}, then the points associated to

[ \begin{matrix}ε \\ ε′ \end{matrix} ] and [ \begin{matrix}ε + μ \\ ε′ + μ′ \end{matrix} ] are the same.

Definition. Let J = ( \begin{matrix}0 & I_{g} \\ −I_{g} & 0 \end{matrix} ) where I_{g} is the g × g identity matrix. We define the symplectic group

Sp(2g, ℚ) = {M ∈ Mat_{2g×2g}(ℚ)| ^{t}MJM = J}.

We will denote Sp(2g, ℤ) also by Γ_{g}(1). For any matrix M = (m_{ij})_{1≤i,j≤g} we let Δ(M) = ^{t}(m_{11}, …, m_{gg}) and for a positive integer n we put Γ_{g}(n) = {M ∈ Γ_{g}(1)| M ≡ I_{2g} mod n}.

The group Sp(2g, ℝ) acts as a group of holomorphic automorphisms of ℌ_{g}: ( \begin{matrix}A & B \\ C & D \end{matrix} ) τ = (Aτ+B)(Cτ+D)^{−1}. This action extends to ℂ^{g}×ℌ_{g} by ( \begin{matrix}A & B \\ C & D \end{matrix} ) (z, τ) = (^{t}(Cτ + D)^{−1}z, (Aτ + B)(Cτ + D)^{−1})).

Definition. Let γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Sp(2g, ℤ). Define for [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ ℝ^{2g}:

(6) γ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ] = [ ^{t}γ^{−1} [ \begin{matrix}ε \\ ε′ \end{matrix} ] + \frac{1}{2} [ \begin{matrix}Δ(C^{t}D) \\ Δ(A^{t}B) \end{matrix} ]] = [ \begin{matrix}Dε − Cε′ + \frac{1}{2}Δ(C^{t}D) \\ −Bε + Aε′ + \frac{1}{2}Δ(A^{t}B) \end{matrix} ] .

It is easy to check that this is a group action on ℝ^{2g}/ℤ^{2g}. Let Ch(k) (k ∈ ℕ) be the set of characteristics [ \begin{matrix}ε \\ ε′ \end{matrix} ] such that kε, kε′ ∈ ℤ^{g}. If k is even, then for [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(k)

put sgn_{k} [ \begin{matrix}ε \\ ε′ \end{matrix} ] = sgn [ \begin{matrix}ε \\ ε′ \end{matrix} ] = (k^{2} ^{t}ε · ε′) mod 2. We call [ \begin{matrix}ε \\ ε′ \end{matrix} ] even if sgn [ \begin{matrix}ε \\ ε′ \end{matrix} ] = 0 and odd otherwise (if needed we will make the dependence on k explicit by saying k-even, k-odd). Note that since k is even, [ \begin{matrix}ε \\ ε′ \end{matrix} ] is k-even (k-odd) if and only if

[ \begin{matrix}ε + μ \\ ε′ + μ′ \end{matrix} ] is k-even (k-odd) for any [ \begin{matrix}μ \\ μ′ \end{matrix} ] ∈ ℤ^{2g}. The action of Sp(2g, ℤ) preserves both Ch(2) and the sgn function on it. Moreover it acts transitively on the even (odd) integral characteristics mod ℤ^{2g}. We have the important transformation formula ([FC] p. 161): (7)

Θγ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ] (γ(z, τ)) = K(γ)·ζ (γ, [ \begin{matrix}ε \\ ε′ \end{matrix} ]) det(Cτ+D)^{1/2} exp(πi^{t}z(Cτ+D)^{−1}Cz)Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ)

where K(γ) ∈ μ_{8} depends only on γ and the choice of the square root and (8)

ζ (γ, [ \begin{matrix}ε \\ ε′ \end{matrix} ]) = exp(−πi\{ ^{t}ε^{t}BDε′ + ^{t}ε′ ^{t}ACε′ −2^{t}ε^{t}BCε′ − ^{t}Δ(A^{t}B)(Dε−Cε′)\}).

Definition. 1) Let C be a Riemann surface of genus g. An Abel-Jacobi structure on C is a triple (B, Ω, c) where B = {γ_{1}, …, γ_{g}, δ_{1}, …, δ_{g}} is an ordered symplectic basis for H_{1}(C, ℤ) (w.r.t. the intersection pairing)^{1}, Ω = {θ_{1}, …, θ_{g}} is a basis for the holomorphic 1-forms on C which is dual to {γ_{1}, …, γ_{g}} w.r.t. the de-Rham pairing (i.e. ∫_{γ_{j}} θ_{i} = δ_{ij}), c ∈ C is a point. We call (C, B, Ω, c) an Abel-Jacobi data (A.J. data).

Given an A.J. structure we have ∫_{δ_{j}} θ_{i} = π_{ij} and π = (π_{ij}) ∈ ℌ_{g}. We denote by φ_{c} : C → ℂ^{g}/(πI)ℤ^{2g} the Abel-Jacobi map with base point c. That is φ_{c}(P) = ∫_{c}^{P} ^{t}(θ_{1}, …, θ_{g}), P ∈ C. Given an A.J. structure, denote Riemann’s vector of constants by K_{c} (see [FK] §VI.2.4). If C is of genus 2 the map C → ℂ^{2}/(πI)ℤ^{4} given by P ↦ φ_{c}(P) + K_{c} is then independent of c (loc. cit. §VI.3.7).

2) Let H_{1}^{+} = \{Sp(4, ℤ) ( \begin{matrix}τ_{1} & 0 \\ 0 & τ_{2} \end{matrix} ) : τ_{1}, τ_{2} ∈ ℌ\}. It is known that π ∉ H_{1}^{+} if and only if π is obtained from an A.J. structure on some Riemann surface C of genus 2 (see [OU]). The following theorem is well known:

Theorem 0. Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ \frac{1}{2}ℤ^{4} be an integral characteristic. The function Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (z, τ),

τ ∈ ℌ_{2} where τ corresponds to an A.J. data, vanishes at z = 0 if and only if [ \begin{matrix}ε \\ ε′ \end{matrix} ] is

an odd characteristic. Moreover, the six points (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ], [ \begin{matrix}ε \\ ε′ \end{matrix} ] odd, are precisely the images of the six Weierstrass points of C under the Abel-Jacobi embedding P ↦ φ_{c}(P) + K_{c} associated with the A.J. data.

Theta functions whose divisor is contained in H_{1}^{+} are (possibly) of particular importance for the construction of units in abelian extensions of quartic C.M. fields (see [DSG]). This explains our interest in the theorems below. We remark that these theorems are also of a certain Diophantine interest because of Riemann’s vanishing theorem: The vanishing of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] at τ, which does not lie on H_{1}^{+}, is equivalent to saying that under the Abel-Jacobi embedding P ↦ φ_{c}(P) + K_{c} (which is independent of c) the curve C passes through the torsion point (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ] in Jac(C) ≅ ℂ^{2}/(τI)ℤ^{4}. Since such Abel-Jacobi embeddings have a canonical definition (they are the embeddings such that L = O_{Jac(C)}(C) is symmetric and e_{L}^{∗}(0) = 1 (see [Mum2] for the definition of e_{L}^{∗})), it makes sense to study the set C ∩ Jac(C)_{tor} for these particular embeddings. Theorem 1. Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(6) be a characteristic corresponding to a point of

1By that we mean: γ_{i}·δ_{j} = δ_{j} ·γ_{i} = 0 for i ≠ j; γ_{i}·γ_{j} = δ_{i}·δ_{j} = 0 for all i, j; γ_{i}·δ_{i} = −δ_{i}·γ_{i} = 1.

exact order 6. Assume that [ \begin{matrix}ε \\ ε′ \end{matrix} ] is 6-odd, i.e., 36 ^{t}ε · ε′ = 1( mod 2). Then the

divisor of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) is contained in H_{1}^{+}.

Proof. First note that we can uniquely write [ \begin{matrix}ε \\ ε′ \end{matrix} ] = [ \begin{matrix}μ \\ μ′ \end{matrix} ] + [ \begin{matrix}η \\ η′ \end{matrix} ] where [ \begin{matrix}μ \\ μ′ \end{matrix} ] is

of exact order 3 and [ \begin{matrix}η \\ η′ \end{matrix} ] is of exact order 2. We easily verify that [ \begin{matrix}ε \\ ε′ \end{matrix} ] is odd

if and only if [ \begin{matrix}η \\ η′ \end{matrix} ] is an odd characteristic. Let τ ∈ ℌ_{2} − H_{1}^{+} and let (C, B, Ω, c) be an A.J. data yielding τ. Choose P ∈ C a Weierstrass point. Suppose that

Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) = 0. Then

1) ∃ζ ∈ C such that φ_{P} (ζ) + K_{P} = (τI) [ \begin{matrix}η \\ η′ \end{matrix} ].

2) ∃ξ ∈ C such that φ_{P} (ξ) + K_{P} = (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ] = (τI) [ \begin{matrix}μ \\ μ′ \end{matrix} ] + (τI) [ \begin{matrix}η \\ η′ \end{matrix} ].

(The equalities hold on the abelian variety ℂ^{2}/(τI)ℤ^{4}). Therefore φ_{P} (ξ)−φ_{P} (ζ) = (τI) [ \begin{matrix}μ \\ μ′ \end{matrix} ] is of exact order 3. That is, there exists a function f on C such that (f) = 3ξ − 3ζ. However, ζ is a Weierstrass point (by Theorem 0) and 3 is a gap for Weierstrass points on curves of genus 2 ([FK III.5.3). Contradiction. Q.E.D. Theorem 2. Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(4) be a characteristic corresponding to a point of

exact order 4. Then the divisor of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) is contained in H_{1}^{+}. Proof. We start with the following lemma Lemma. Let C be a hyperelliptic curve of genus g ≥ 2. Denote the Weierstrass points of C by {P_{1}, …, P_{2g+2}}.

1) Every point in Jac(C)[2] is of the form P (J) = ∑_{j∈J} P_{j} − #J · P_{1} for some J ⊆ {2, 3, …, 2g + 2} (considered as an element of Div^{0}(C)/Pr(C) ≅ Jac(C)). This

way of writing is unique up to the relations P (J) = P ({2, 3, …, 2g + 2} − J).

2) Let g = 2, fix some A.J. structure on C and let (τI) be the resulting period matrix. Identify the integral characteristics with the points of order two of Jac(C) = ℂ^{2}/(τI)ℤ^{4} by [ \begin{matrix}ε \\ ε′ \end{matrix} ] ↔ (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ] and call a point of order 2 in Jac(C)

odd/even if the corresponding [ \begin{matrix}ε \\ ε′ \end{matrix} ] is. Then every point of exact order 2 is of the form e − K_{P_{j}} for some odd point of order two e and some Weierstrass point P_{j}.

Proof (of Lemma). Consider the homomorphism (ℤ/2ℤ)^{2g+1} → Jac(C)[2], given by (x_{1}, x_{2}, …, x_{2g+1}) ↦ ∑ x_{i}P_{i+1} − (∑ x_{i})P_{1}. By counting it is enough to show that the kernel is contained in {0, (1, 1, …, 1)}.

In order to show that any element different from (1, 1, …, 1) is not in the kernel, it is enough to prove (by renaming the points if necessary) that for any n < 2g + 1 the divisor P_{2} + ⋯ + P_{n+1} − nP_{1} is not principal. If n is odd we are done because any odd n < 2g is a gap for the Weierstrass point P_{1} ([FK] p. 88 Remark). Suppose now that n is even. Then nP_{1}−nP_{n+1} is a principal divisor. If P_{2}+⋯+P_{n+1}−nP_{1} is a principal divisor, so is P_{2} + ⋯ + P_{n+1} − nP_{1} + nP_{1} − nP_{n+1} = P_{2} + ⋯ + P_{n} − (n − 1)P_{n+1} This is a contradiction, because n − 1 is odd and < 2g and therefore a gap for the Weierstrass point P_{n+1}.

It is a consequence of the proof that (1, 1, …, 1) is in the kernel. Otherwise said, that in any hyperelliptic surface the divisor P_{1}+⋯+P_{2g+2}−(2g+2)P_{1} is a principal divisor. This is easy to verify: writing the surface as y^{2} = (x−λ_{1}) · · · . . .·(x−λ_{2g+1}), P_{1} =the point above ∞, then (y) = P_{2} + … + P_{2g+2} − (2g + 1)P_{1}.

2) We have to prove that every point of order 2 is of the form e−K_{Q} for some Q ∈ {P_{1}, …, P_{6}} and an odd point e. These e’s are precisely the points φ_{Q}(P_{i}) + K_{Q}, 1 ≤ i ≤ 6. Thus, it is enough to prove that every point of exact order 2 is of the form P_{i} − P_{j} for some i ≠ j. There are 15 such differences. They all give different two torsion points else we have P_{i}−P_{j} = P_{k}−P_{ℓ} ⟹ P_{i}+P_{ℓ}−2P_{r} = P_{j} +P_{k}−2P_{r}. Separating cases and using part 1) of the Lemma one concludes the proof.

Returning to the proof of the theorem, we assume the notation of the Lemma and let P = P_{1}, B = Jac(C)[4]−Jac(C)[2]. We need to prove that B∩(φ_{P} (C)+K_{P}) = ∅. If x ∈ B ∩ (φ_{P} (C) + K_{P}) we know that for some Q ∈ {P_{1}, …, P_{6}} and some odd point e of exact order 2:

(i) 2x = e − K_{Q}. (ii) x − K_{Q} ∈ φ_{Q}(C) (using φ_{P} (C) + K_{P} = φ_{Q}(C) + K_{Q} see [FK] VI3.7.).

Now, since K_{Q} is a point of order 2 we get 2(x − K_{Q}) = 2x = e − K_{Q} ∈ φ_{Q}(C) (using e odd ⇒ e ∈ φ_{Q}(C) + K_{Q}). Moreover, using the symmetry of the theta divisor φ_{Q}(C) + K_{Q} (as follows from (5)) we conclude that since x ∈ φ_{Q}(C) + K_{Q} we get also −x ∈ φ_{Q}(C) + K_{Q} = φ_{Q}(C) − K_{Q}. That is, −(x − K_{Q}) ∈ φ_{Q}(C).

We have shown that if x ∈ B ∩ (φ_{P} (C) + K_{P}) then φ_{Q}(C) ⊇ {0, x − K_{Q}, 2(x − K_{Q}), 3(x−K_{Q})}. However, this implies that the intersection φ_{Q}(C).T_{x−K_{Q}}(φ_{Q}(C)) ≥ 4 which contradicts the fact that the theta divisor defines a principal polarization and is, therefore, of self intersection 2 (it is easy to prove that φ_{Q}(C) ≠ T_{x−K_{Q}}(φ_{Q}(C)) and therefore the intersection is proper). Q.E.D.

Theorem 3. 1) There exists a constant λ ∈ ℚ(ζ_{36})^{×} such that

(*)  λ ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈A^{−}} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{2} = ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈E} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{96}

where A^{−} = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : −\frac{1}{2} ≤ ε_{i}, ε′_{i} < \frac{1}{2}, i = 1, 2, [ \begin{matrix}ε \\ ε′ \end{matrix} ] is of exact order 6 in ℝ^{4}/ℤ^{4}

and 36^{t}ε·ε′ ≡ 1 ( mod 2)\} and E = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : ε_{i}, ε′_{i} ∈ \{0, \frac{1}{2}\} , i = 1, 2, 4 ^{t}ε · ε′ ≡ 0 ( mod 2)\}.

2) There exists a constants μ ∈ ℚ(ζ_{16})^{×} such that

μ ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈B} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{2} = ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈E} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{48}

where B = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : −\frac{1}{2} ≤ ε_{i}, ε′_{i} < \frac{1}{2}, i = 1, 2, [ \begin{matrix}ε \\ ε′ \end{matrix} ] is of exact order 4 in ℝ^{4}/ℤ^{4}\}.

Proof. 1) By Theorem 1 we know that the support of the divisor of both sides is contained in H_{1}^{+}. We claim that Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{2}, [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(k), is a modular form of weight 1 and some high level. Indeed, the transformation formula (7) shows that we have to fulfill two conditions: (i) ζ (γ, [ \begin{matrix}ε \\ ε′ \end{matrix} ])^{2} = 1 (ii) K(γ)^{2} = 1. Clearly (i) will hold if γ ∈ Γ_{2}(k^{2}). One can prove that K(γ)^{2} is trivial on Γ_{2}(4) (see [Mum1] p. 197 Corollary (5.9)). Thus, (i), (ii) will hold γ ∈ Γ_{2}(4k^{2}). We conclude that both sides are modular forms of weight equal to 10 × 96 × 1/2 = 480 = (3^{4} − 1) · 6 × 2 × 1/2 = (number of [ \begin{matrix}ε \\ ε′ \end{matrix} ] of exact order 6 and odd) ×2 × 1/2.

Now, the action of Γ_{2}(1) on the integral characteristics takes odd/even characteristics to odd/even characteristics and it acts transitively on the set of odd/even integral characteristics. Under the isomorphism (\frac{1}{3}ℤ/ℤ)^{4} × (\frac{1}{2}ℤ/ℤ)^{4} → (\frac{1}{6}ℤ/ℤ)^{4},

([ \begin{matrix}μ \\ μ′ \end{matrix} ] , [ \begin{matrix}η \\ η′ \end{matrix} ]) ↦ [ \begin{matrix}μ \\ μ′ \end{matrix} ] + [ \begin{matrix}η \\ η′ \end{matrix} ] = [ \begin{matrix}ε \\ ε′ \end{matrix} ] we find that [ \begin{matrix}ε \\ ε′ \end{matrix} ] is odd if and only if

[ \begin{matrix}η \\ η′ \end{matrix} ] is. The inverse isomorphism (\frac{1}{6}ℤ/ℤ)^{4} → (\frac{1}{3}ℤ/ℤ)^{4} × (\frac{1}{2}ℤ/ℤ)^{4} is just [ \begin{matrix}ε \\ ε′ \end{matrix} ] ↦

([ \begin{matrix}−2ε \\ −2ε′ \end{matrix} ] , [ \begin{matrix}3ε \\ 3ε′ \end{matrix} ]). Now, under this isomorphism the action of γ ∈ Sp(4, ℤ),

[ \begin{matrix}ε \\ ε′ \end{matrix} ] ↦ γ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ] comes out ([ \begin{matrix}μ \\ μ′ \end{matrix} ] , [ \begin{matrix}η \\ η′ \end{matrix} ]) ↦ ( ^{t}γ^{−1} [ \begin{matrix}μ \\ μ′ \end{matrix} ] , γ^{∗} [ \begin{matrix}η \\ η′ \end{matrix} ]) (in the first

coordinate we have the usual linear action). If follows that the action of Γ_{2}(1) on Ch(6) preserves the even/odd characteristics. Therefore, we may conclude from the transformation formula that the divisor of both sides of (*) is Γ_{2}(1) invariant and thus has the same multiplicity on every component of H_{1}^{+}. Let

f(τ) = ( ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈A^{−}} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{2})/( ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈E} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ)^{96}).

Then one of the functions f(τ), 1/f(τ) is a holomorphic function on Γ_{2}(n)/ℌ_{2}. Therefore, by the Koecher principle ([FC] p. 146) f(τ) is constant. It is easy to see that both sides of (*) have q-expansions with coefficients in ℚ(ζ_{36}). Hence, we can choose λ such that the equality holds and λ is in ℚ(ζ_{36})^{×}.

2) The proof is the same (using Theorem 2). Q.E.D. Remark. In complete contrast with the high dimensional cases the one dimensional analogues of Theorem 3 exist in abundance. For example, let n be any odd number. Define

χ_{1}(n) = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : −\frac{1}{2} ≤ ε < \frac{1}{2}, −\frac{1}{2} ≤ ε′ < \frac{1}{2}, [ \begin{matrix}ε \\ ε′ \end{matrix} ] is 2n-even and of order n or 2n\}

and for n ≠ 1 define

χ_{−1}(n) = \{[ \begin{matrix}ε \\ ε′ \end{matrix} ] : −\frac{1}{2} ≤ ε < \frac{1}{2}, −\frac{1}{2} ≤ ε′ < \frac{1}{2}, [ \begin{matrix}ε \\ ε′ \end{matrix} ] is 2n-odd and of order n or 2n\}

Let a_{1}(n) = #χ_{1}(n) = 3ϕ(n)(2n − ϕ(n)), a_{−1}(n) = #χ_{−1}(n) = ϕ(n)(2n − ϕ(n)). Let

T_{1}(n)(τ) = ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈χ_{1}(n)} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ]^{2} (τ), T_{−1}(n)(τ) = ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈χ_{−1}(n)} Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ]^{2} (τ).

Both T_{1} and T_{−1} are modular forms (of weights a_{1}(n), a_{−1}(n) respectively) for high enough level. Proposition. Let α, β ∈ {−1, 1}. Let d = l.c.m.(a_{α}(n), a_{β}(m)) (a_{−1}(1) is not allowed). Then there exists a non-zero constant λ such that

λ · T_{α}(n)^{d/a_{α}(n)} = T_{β}(m)^{d/a_{β}(m)}.

In fact λ = ( ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈χ_{β}(m)} α(ε, ε′)^{2d/a_{β}(m)})/( ∏_{[ \begin{matrix}ε \\ ε′ \end{matrix} ]∈χ_{α}(n)} α(ε, ε′)^{2d/a_{α}(n)}), where α(ε, ε′) =

exp(2πi · εε′) if ε ≠ −1/2, and α(ε, ε′) = exp(πi · ε′) + exp(−πi · ε′) if ε = −1/2. Proof. Both sides are modular forms of high enough level, say N. They extend to modular forms over the compactified modular curve of level N, X(N) = Γ(N)\ℌ^{∗}. Moreover, both sides do not vanish on Γ(N)\ℌ and vanish to the same order at every cusp of X(N) (The last assertion follows from the fact that Γ(1) acts transitively on the cusps and that the divisors of both sides are Γ(1) invariant as follows from the transformation formula (7)). Therefore either (T_{α}(n)^{d/a_{α}(n)})/(T_{β}(m)^{d/a_{β}(m)}) or (T_{β}(m)^{d/a_{β}(m)})/(T_{α}(n)^{d/a_{α}(n)}) are holomorphic and non-vanishing on X(N), hence constant.

To find the precise form of λ we note that the first non-zero coefficient in the q-expansion of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) is precisely α(ε, ε′). This follows immediately from the formula

Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) = ∑_{n=−∞}^{∞} exp(2πi(n + ε) · ε′) · exp(πi(n + ε)^{2}τ).

Since the first non-zero coefficient of T_{α}(n)^{d/a_{α}(n)} is the product of the first nonzero coefficients of all the theta functions appearing in its definition, the assertion follows.

Q.E.D.

The following theorem complements Theorem 1:

Theorem 4. Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(6) be a 6-even non-integral characteristic. Then the

divisor of Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) is not contained in H_{1}^{+}.

Proof. Consider the covering y^{2} = x^{6} + 1 → y^{2} = x^{2} + 1 ≅ ℙ^{1} ((x, y) ↦ (x^{3}, y)). This map is really dividing by the automorphism (x, y) ↦ (e^{2πi/3}x, y), which has the two fixed points Q = (0, 1), P = (0, −1). This shows that image of Q − P in Jac(y^{2} = x^{6} + 1) is of order 3. On the other hand the covering y^{2} = x^{6} + 1 → y^{2} = x^{3} + 1 ((x, y) ↦ (x^{2}, y))which comes from the involution (x, y) ↦ (−x, y), shows that Q + P (= the ramification points) is a divisor of a holomorphic differential on y^{2} = x^{6} + 1 (the pull-back of a non-zero holomorphic differential on y^{2} = x^{3} + 1). Therefore, using [FK] VI.3.6, we conclude that −2K_{P} = φ_{P} (Q + P) = φ_{P} (Q). Hence 6(φ_{P} (Q) + K_{P}) = 6φ_{P} (Q) − 3φ_{P} (Q) = 0. Choose any A.J. structure on y^{2} = x^{6} + 1 and let (τI) be the period matrices obtained from this structure. Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] be a characteristic such that (τI) [ \begin{matrix}ε \\ ε′ \end{matrix} ] = φ_{P} (Q)+K_{P}. Note that φ_{P} (Q)+K_{P}

is not of order 2, hence [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(6), non-integral and even (using Theorem 1).

We conclude that Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) = 0 and we have τ ∉ H_{1}^{+}.

Let us denote the set of even non-integral characteristics [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ Ch(6) by A^{+}.

Now, to get that for every [ \begin{matrix}ε \\ ε′ \end{matrix} ] ∈ A^{+} we have Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) = 0 for some τ ∉ H_{1}^{+}, it is enough, by the transformation formula (7), to prove that Γ_{2}(1) acts transitively on the set A^{+}(modℤ^{4}). We have already seen in the proof of Theorem 3 that Γ_{2}(1) acts on A^{+}(modℤ^{4}). In fact, recall that under the isomorphism

(\frac{1}{6}ℤ/ℤ)^{4} → (\frac{1}{3}ℤ/ℤ)^{4} × (\frac{1}{2}ℤ/ℤ)^{4}, [ \begin{matrix}ε \\ ε′ \end{matrix} ] ↦ ([ \begin{matrix}μ \\ μ′ \end{matrix} ] , [ \begin{matrix}η \\ η′ \end{matrix} ]) we find that [ \begin{matrix}ε \\ ε′ \end{matrix} ] is

odd if and only if [ \begin{matrix}η \\ η′ \end{matrix} ] is and that the action on γ ∈ Sp(4, ℤ), [ \begin{matrix}ε \\ ε′ \end{matrix} ] ↦ γ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ]

comes out ([ \begin{matrix}μ \\ μ′ \end{matrix} ] , [ \begin{matrix}η \\ η′ \end{matrix} ]) ↦ ( ^{t}γ^{−1} [ \begin{matrix}μ \\ μ′ \end{matrix} ] , γ^{∗} [ \begin{matrix}η \\ η′ \end{matrix} ]). Let [ \begin{matrix}ε \\ ε′ \end{matrix} ] = ([ \begin{matrix}μ \\ μ′ \end{matrix} ] , [ \begin{matrix}η \\ η′ \end{matrix} ])

and [ \begin{matrix}α \\ α′ \end{matrix} ] = ([ \begin{matrix}β \\ β′ \end{matrix} ] , [ \begin{matrix}ζ \\ ζ′ \end{matrix} ]) be in A^{+}. Since, by Witt’s Theorem, Sp(4, ℤ) acts

(linearly) transitively on (ℤ/3ℤ)^{4}, we can find γ ∈ Sp(4, ℤ) such that γ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ] =

([ \begin{matrix}β \\ β′ \end{matrix} ] , [ \begin{matrix}ξ \\ ξ′ \end{matrix} ]) and [ \begin{matrix}ξ \\ ξ′ \end{matrix} ] is necessarily an odd integral characteristic. We can act now by δ ∈ Γ_{2}(3) on γ^{∗} [ \begin{matrix}ε \\ ε′ \end{matrix} ] and get ([ \begin{matrix}β \\ β′ \end{matrix} ] , δ^{∗} [ \begin{matrix}ξ \\ ξ′ \end{matrix} ]). One has only to verify that Γ_{2}(3) acts transitively on the set of the odd integral characteristics (mod ℤ^{4}). There are six odd characteristic represented by:

[ \begin{matrix}0 & 1/2 \\ 0 & 1/2 \end{matrix} ] , [ \begin{matrix}0 & 1/2 \\ 1/2 & 1/2 \end{matrix} ] , [ \begin{matrix}1/2 & 0 \\ 1/2 & 0 \end{matrix} ] , [ \begin{matrix}1/2 & 0 \\ 1/2 & 1/2 \end{matrix} ] , [ \begin{matrix}1/2 & 1/2 \\ 0 & 1/2 \end{matrix} ] , [ \begin{matrix}1/2 & 1/2 \\ 1/2 & 0 \end{matrix} ] .

Denoting them by 1, 2, 3, 4, 5, 6, respectively, one checks that ( \begin{matrix}I & 3I \\ 0 & I \end{matrix} ) acts as the

permutation (12)(34), ( \begin{matrix}I & 0 \\ 3I & I \end{matrix} ) acts as (15)(36) and ( \begin{matrix}α & 0 \\ 0 & ^{t}α^{−1} \end{matrix} ), α = ( \begin{matrix}1 & 3 \\ 0 & 1 \end{matrix} ), acts as (12)(36)(45). Q.E.D.

Final Remark. Using ramified coverings of elliptic curves gives one a systematic approach for proving vanishing of theta constants Θ [ \begin{matrix}ε \\ ε′ \end{matrix} ] (τ) for ε, ε′ ∈ ℚ^{2}, τ ∈ ℌ_{2}, at points τ ∉ H_{1}^{+}. This will be explained in [Gor]. It seems to suggest that one can not expect theorems like Theorems 1,2 for other characteristics.

References

[DSG] DeShalit, E. and Goren, E.Z.: On special values of theta functions of genus two, Preprint, June 1996.

[Gor] Goren, E.Z.: On curves of genus 2 with non-hyperelliptic involution, in preparation.

[FC] Faltings, G. and Chai, C.-L.: Degenerations of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge Band 22, Springer-Verlag, Berlin Heidelberg, 1990.

[FK] Farkas, H.M. and Kra, I.: Riemann Surfaces, Second edition, Graduate Texts in Mathematics vol. 71, Springer-Verlag, New-York, 1992.

[Mum1] Mumford, D.: Tata Lectures on Theta I, Progress in Mathematics vol. 28, Birkhäuser, Boston, 1983.

[Mum2] Mumford, D.: On the Equations Defining Abelian Varieties I, Inventions Math. 1(1966), 287–357.

[OU] Oort, F. and Ueno, K.: Principally polarized abelian varieties of dimension two or three are Jacobian varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20(1973), 377–381