ON THE DIMENSION OF THE SPACE OF THETA FUNCTIONS

DANIEL BUMP AND ALEXANDER PEKKER

Abstract. We compute the dimension of the space of theta functions of a given type using a variant of the Selberg trace formula.

1. Main Result

Theta functions play an important role in the theory of abelian varieties; for instance, theta functions are used to construct meromorphic functions on a multidimensional torus and to embed a multidimensional torus into projective space (see, for instance, [8], [10], [14]).

Frobenius [4] computed the dimensions of spaces of theta functions in several variables. As far as we know other expositions of this theorem ([8], [7], [14]) more or less follow Frobenius in making Fourier expansions and discussiong the relations between the Fourier coefficients.

Selberg [12] and Eichler [3] invented a method of computing the dimensions of spaces of modular forms which has proved important. One constructs an automorphic reproducing kernel by averaging the Bergman kernel function, then computes the dimension of a space of modular forms as the trace of the kernel. See [13] for an accessible account of this technique, and Hejhal [5] for a recent variation.

In these examples, the Bergman kernel function is a function on a Hermitian symmetric space, conformally equivalent to a bounded domain in ℂ^{n}. In a different domain—ℂ^{n} itself—Bargman [1] and independently Newman and Shapiro [9] introduced a Bergman kernel function. This is a reproducing kernel for the Fock space, a Hilbert space of entire functions. The relevance of the Fock space and the reproducing aspect of this kernel to the theory of theta functions was revealed by Cartier [2] and Satake [11] and (implicitly, earlier) by Weil [15]. See also Hurt [6], Chapter 8.

We imitate Eichler and Selberg in constructing an automorphic reproducing kernel for theta functions by averaging the Bergman kernel function on the Fock space, and we recover Frobenius’ dimension formula by computing the trace of this reproducing kernel.

Let Λ be a lattice in ℂ^{n} (i.e., Λ is a ℤ-module of dimension 2n). A theta function of (L, M) type on ℂ^{n} with respect to Λ is a meromorphic function, not identically zero, satisfying

θ(z + λ) = θ(z)J(zλ) = θ(z) exp [2πi(L(z, λ) + M(λ))] (1.1)

for every z ∈ ℂ^{n} and λ ∈ Λ, where L(z, λ) is linear in z. It is easily checked that all entire theta function of a given (L, M) type together with the zero function form a complex vector space, which we denote by T_{L,M}. Further, define

1991 Mathematics Subject Classification. Primary: 14K25.

We would like to thank Dennis Hejhal for help with the references.

E(z, w) = L(z, w) − L(w, z) and H(z, w) = E(iz, w) + iE(z, w). (1.2)

Classical results show that E is a Riemann form and H is a positive Hermitian form. Moreover, H is positive definite if and only if E is nondegenerate. If we select a ℤ-basis for Λ, then it is also a basis for ℂ^{n}, considered as a real vector space of dimension 2n. In such a basis, the \sqrt{det E} is an integer independent of the choice of ℤ-basis for Λ and is called the Pfaffian of E.

Theorem 1.1. (Frobenius). Let Λ be a lattice in ℂ^{n}, and let (L, M) be a type of theta functions. Let E and H be as in (1.2). Supposing that E is nondegenerate, let P f denote the Pfaffian of E. Then,

dim T_{L,M} = det(H) Vol(ℂ^{n}/Λ) = P f.

When E is degenerate, classical results (see Lang [8, pp. 89-90]) show that W = ker(E) = ker(H) is a subspace of ℂ^{n}. Let Λ_{0} denote the image of Λ in ℂ^{n}/W, (L_{0}, M_{0}) the induced type on ℂ^{n}/W, and E_{0} and H_{0} the induced forms on ℂ^{n}/W. It is easy to show that Λ_{0} is a lattice, E_{0} is nondegenerate, and H_{0} is positive definite. Again, picking a ℤ-basis for Λ_{0}, \sqrt{det E_{0}} is an integer, called the reduced Pfaffian of E. It is not hard to see that dim T_{L,M} = dim T_{L_{0},M_{0}}, so Theorem 1.1 implies the following stronger result.

Theorem 1.2. Let Λ be a lattice in ℂ^{n}, and let (L, M) be a type of theta functions, and E and H as in (1.2). Let W denote ker(E), Λ_{0} the image of Λ in ℂ^{n}/W, H_{0} the reduced form of H, and P f_{RED} the reduced Pfaffian of E. Then,

dim T_{L,M} = det(H_{0})Vol((ℂ/W)/Λ_{0}) = P f_{RED}.

The proof of Theorem 1.1 follows immediately from the first observation in Section 5 and Theorems 5.2 and 5.3.

2. Properties of The Reproducing Kernel

A reproducing kernel in a finite-dimensional Hilbert space H ⊂ Ł^{2}(X, μ) is function Φ : X × X → ℂ satisfying the following three properties:

1. For fixed w, f(z) = Φ(z, w) ∈ H,

2. For every f ∈ H, f(z) = ∫_{X} Φ(z, w)f(w) dμ(w),

3. Φ(z, w) = \overline{Φ(w, z)}.

If H is a finite-dimensional vector space, then integrating the reproducing kernel for H “along the diagonal” yields the dimension of H.

Theorem 2.1. Let H ⊂ L^{2}(X, μ) be a finite-dimensional Hilbert space. Then, there exists an unique reproducing kernel Φ and

dim H = ∫_{X} Φ(z, z) dμ(z).

Proof. Let \{ϕ_{1}, …, ϕ_{n}\} be an orthonormal basis for H. Then Φ(z, w) = ∑_{i=1}^{n} ϕ_{i}(z)\overline{ϕ_{i}(w)} is the unique reproducing kernel for H and integrating along the diagonal clearly gives n.

3. Properties of Theta Functions

In this section we collect several results from Lang [8, pp. 83-89]. Recall that a theta function on ℂ^{n} with respect to a lattice Λ is a meromorphic function, not identically zero, satisfying (1.1). The theta multiplier, J, satisfies the cocycle identity

J(z, λ + ν) = J(z, λ)J(z + λ, ν). (3.1)

Theorem 3.1. Let θ be a theta function of (L, M) type, and define K(z) = M(z)− \frac{1}{2}L(z, z). Then θ satisfies

θ(z + λ) = θ(z)J(z, λ) = θ(z) exp [2πi(L(z, λ) + \frac{1}{2}L(λ, λ) + K(λ))] (3.2)

where L is defined on ℂ^{n} × ℂ^{n} and is ℂ-linear in the first component and ℝ-linear in the second component, and K is defined on ℂ^{n} and is ℝ-linear.

Let θ be a theta function as in Theorem 3.1, and define E and H as in (1.2).

Theorem 3.2. E is alternating and real-valued for all z, w ∈ ℂ^{n} and integervalued on all z, w ∈ Λ. Moreover, the form q(z, w) = E(iz, w) is symmetric. (In other words, E is a Riemann form. The form H is Hermitian and positive, and E(z, w) = Im H(z, w).

A theta function θ is a normalized theta function if it satisfies (3.2) and H(z, w) = 2iL(z, w) and K(z) ∈ ℝ for all z ∈ ℂ^{n}. In this case relation (3.2) can be written in terms of H and K as

θ(z + λ) = θ(z) exp [πH(z, λ) + \frac{π}{2} H(λ, λ) + 2πiK(λ)]. (3.3)

and the theta multiplierJ becomes

J(z, λ) = exp [πH(z, λ) + \frac{π}{2} H(λ, λ) + 2πiK(λ)]. (3.4)

Normalized entire theta function satisfy the following growth condition.

Theorem 3.3. If θ is a normalized entire theta function with associated Hermitian form H, then there exists a constant C such that |θ(z)|^{2} e^{−πH(z,z)} < C.

For normalized entire theta functions we will denote the type by (H, K) instead of (L, M) and the associated vector space as N_{H,K}. Finally, we have the following isomorphism theorem.

Theorem 3.4. Let θ(z) be an entire theta function of (L, M) type. Then there exists a unique normalized entire theta function θ_{N}(z) of (H, K) type and a trivial theta function θ_{T} (z) such that θ_{N}(z) = θ(z)θ_{T} (z), up to multiplication by a constant. Moreover, dim T_{L,M} = dim N_{H,K}.

4. Auxiliary Space

Let H be an n × n positive definite Hermitian form. Then, H(z, w) = w^{∗}Hz, where the second H is a Hermitian matrix. By the spectral theorem, there exists a unitary matrix U and a diagonal matrix D such that

U^{∗}HU = D = [\begin{matrix}Δ_{1} & 0 & … & 0 \\ 0 & Δ_{2} & … & 0 \\ ⋮ & ⋮ & ... & 0 \\ 0 & 0 & … & Δ_{n} \end{matrix}]

and all the Δ_{i} are real and positive. Moreover, Δ = ∏_{i=1}^{n} Δ_{i} = det(H).

Define the measure μ as dμ(z) = Δe^{−πH(z,z)} dz (where dz is Lebesgue measure), and let H_{H} be the space of all analytic functions f : ℂ^{n} → ℂ for which

∫_{ℂ^{n}} |f(z)|^{2} dμ(z) = ∫_{ℂ^{n}} |f(z)|^{2}Δe^{−πH(z,z)} dz < ∞.

It is clear that H_{H} is a subset of L^{2}(μ), and is therefore a pre-Hilbert space with the inner product

(f_{1}, f_{2}) = ∫_{ℂ^{n}} f_{1}(z)\overline{f_{2}(z)} dμ(z).

Following Igusa [7, pp. 31-33], we will show that it is in fact a Hilbert space.

Theorem 4.1. Let

ϕ_{(p_{1},... ,p_{n})}(z) = ∏_{i=1}^{n} \sqrt{\frac{(πΔ_{i})^{p_{i}}}{p_{i}!}} (U^{−1}z)_{i}^{p_{i}}

where (p_{1}, …, p_{n}) range over all n-tuples of non-negative integers. Then, {ϕ_{(p_{1},... ,p_{n})}} is an orthonormal set in H_{H}.

Proof. Making the substitution w = U^{−1}z, the theorem reduces to the one-dimensional case where it can be easily checked.

Lemma 4.2. Let {ϕ_{(p_{1},... ,p_{n})}} be as in Theorem 4.1. Then,

∑_{(p_{1},... ,p_{n})} ϕ_{(p_{1},... ,p_{n})}(z)\overline{ϕ_{(p_{1},... ,p_{n})}(w)} = e^{πH(z,w)}.

Proof. Since (p_{1}, …, p_{n}) range over all n-tuples of non-negative integers, the summation becomes

∑_{(p_{1},... ,p_{n})} ϕ_{(p_{1},... ,p_{n})}(z) \overline{ϕ_{(p_{1},... ,p_{n})}(w)} = ∑_{(p_{1},... ,p_{n})} ∏_{i=1}^{n} \frac{(πΔ_{i})^{p_{i}}}{p_{i}!} (U^{−1}z)_{i}^{p_{i}} \overline{(U^{−1}w)_{i}^{p_{i}}}

= ∏_{i=1}^{n} (∑_{j=0}^{∞} \frac{(πΔ_{i})^{j}}{j!} (U^{−1}z_{i})^{j}\overline{(U^{−1}w_{i})^{j}})

= ∏_{i=1}^{n} e^{πΔ_{i}(U^{−1}z)_{i}\overline{(U^{−1}w)_{i}}}.

Since H is Hermitian, ∑_{i=1}^{n}(w^{∗}U)_{i}Δ_{i}(U^{∗}z)_{i} = w^{∗}UDU^{∗} = w^{∗}H^{∗}z = w^{∗}Hz, so the proof is complete.

The following lemma bounds the value of |f(z)| in terms of ∥f∥; this estimate is helpful in relating uniform convergence on compact sets to L^{2}(μ) convergence.

Lemma 4.3. For any f ∈ H_{H} and any z ∈ ℂ^{n}, |f(z)| ≤ e^{\frac{π}{2} H(z,z)}∥f∥.

Proof. Since f(Uz) is analytic, upon making the substitution z → U^{−1}z f has a power series expansion which may be written as

f(z) = ∑_{(p_{1},... ,p_{n})} c_{(p_{1},... ,p_{n})}ϕ_{(p_{1},... ,p_{n})}(z), (4.1)

for some suitable constants c_{(p_{1},... ,p_{n})} and orthonormal functions ϕ_{(p_{1},... ,p_{n})} as in Theorem 4.1. Since f ∈ H_{H},

∑_{(p_{1},... ,p_{n})} |c_{(p_{1},... ,p_{n})}|^{2} = ∥f∥^{2} < ∞. (4.2)

By the Cauchy-Schwarz Inequality and Lemma 4.2,

|f(z)| = | ∑_{(p_{1},... ,p_{n})} c_{(p_{1},... ,p_{n})}ϕ_{(p_{1},... ,p_{n})}(z)|

≤ \sqrt{∑_{(p_{1},... ,p_{n})} |c_{(p_{1},... ,p_{n})}|^{2}}\sqrt{∑_{(p_{1},... ,p_{n})} |ϕ_{(p_{1},... ,p_{n})}(z)|^{2}} = e^{\frac{π}{2} H(z,z)}∥f∥.

Theorem 4.4. H_{H} is a Hilbert space.

Proof. The only property we must show is completeness. By definition H_{H} ⊂ L^{2}(μ) with the ∥ · ∥ norm being the L^{2}(μ) norm. So, suppose that {f_{i}} is a Cauchy sequence in H_{H}; we must show that its limit is also in H_{H}. Since H_{H} ⊂ L^{2}(μ), {f_{i}} is Cauchy in L^{2}(μ), there exists a function f ∈ L^{2}(μ) such that f_{i} → f in the ∥ · ∥ norm.

On the other hand, since \{f_{i}\} is Cauchy and since we can bound e^{\frac{π}{2} H(z,z)} on any compact set B, Lemma 4.3 implies that the sequence {f_{i}} is uniformly Cauchy on compact sets. This means that f_{i} converges to some analytic function g pointwise and uniformly on compact sets.

Thus if we can show that f = g, we will have that f_{i} → f in the ∥ · ∥ norm, with f analytic and ∥f∥ < ∞. In other words, we will have that f ∈ H_{H}.

Suppose on the contrary f ≠ g. If B ⊂ ℂ^{n} is compact, define ∥ · ∥_{B} by ∥h∥_{B}^{2} = ∫_{B} |h(z)|^{2} dμ(z). It is easy to see that ∥ · ∥_{B} is a norm and that ∥h∥_{B} ≤ ∥h∥ for any function h. Assuming f ≠ g, ∥f −g∥_{B} > δ for some compact set B and δ > 0. Since f_{i} → g uniformly on the compact set B, for l sufficiently large, ∥g −f_{l}∥_{B} < δ/2. On the other hand, since f_{i} → f in the ∥ · ∥ norm, for k sufficiently large, ∥f − f_{k}∥_{B} < δ/2. Taking m = max(k, l), the triangle inequality yields ∥f − g∥_{B} < δ, which is a contradiction.

The Hilbert space H_{H} is the Fock space ([1], [11], [7]). We have shown that the set defined in Theorem 4.1 is orthonormal. The next theorem shows that it is also complete. Having found a complete, orthonormal set, we will be ready to construct a reproducing kernel for H_{H}.

Theorem 4.5. The orthonormal system {ϕ_{(p_{1},... ,p_{n})}} defined in Theorem 4.1 is complete.

Proof. We must show that for any f ∈ H_{H}, the partial sums

∑_{p_{1},... ,p_{n}<N}(f, ϕ_{(p_{1},... ,p_{n})})ϕ_{(p_{1},... ,p_{n})}

converge to f in the ∥ · ∥ norm.

Since f is analytic, it has a power series expansion (4.1). Then, since {ϕ_{(p_{1},... ,p_{n})}} form an orthonormal set, we have (f, ϕ_{(p_{1},... ,p_{n})}) = c_{(p_{1},... ,p_{n})}. By Bessel’s Equality and (4.2),

∥f − ∑_{p_{1},... ,p_{n}<N}(f, ϕ_{(p_{1},... ,p_{n})})ϕ_{(p_{1},... ,p_{n})}∥^{2} = ∥f∥^{2} − ∑_{p_{1},... ,p_{n}<N} |(f, ϕ_{(p_{1},... ,p_{n})})|^{2}

= ∑_{(p_{1},... ,p_{n})} |c_{(p_{1},... ,p_{n})}|^{2} − ∑_{p_{1},... ,p_{n}<N} |c_{(p_{1},... ,p_{n})}|^{2}.

As N → ∞, the expression on the right converges to 0, completing the proof.

Theorem 4.6. Let Φ(z, w) = e^{πH(z,w)}. Then Φ is the reproducing kernel for H_{H}.

Proof. In light of Lemma 4.2 and Theorem 4.5, it is clear that Φ satisfies all three properties of being a reproducing kernel for H_{H}.

5. Proof of Theorem 1.1

In light of Theorem 3.4 it is sufficient to prove Theorem 1.1 only in the case of normalized entire theta functions, i.e., for the space N_{H,K} where H is positive definite. Given this fact, the proof follows immediately from Theorem 5.2 and 5.3. The first of these theorems is proved by constructing a reproducing kernel for N_{H,K}; the second follows by direct calculation.

Throughout this section, let H and K be fixed. Define the inner product on N_{H,K} by

(θ_{1}, θ_{2}) = ∫_{ℂ^{n}/Λ} θ_{1}(w)\overline{θ_{2}(w)} dμ(w)

with Δ = det(H) and dμ(z) = Δ e^{−πH(z,z)} dz as in the previous section. It is easily checked that the inner product is well-defined and N_{H,K} is a finite-dimensional Hilbert space.

Letting t be a positive real number, let H_{tH} refer to the space H_{H} defined in the previous section with the Hermitian form H replaced by t times H. As det(tH) = t^{n} det(H) = t^{n}Δ, the corresponding measure dμ_{t}(z) is t^{n}Δe^{−πtH(z,z)} dz and the reproducing kernel, given in Theorem 4.6, is Φ_{t}(z, w) = e^{πtH(z,w)}.

For t > 0, λ ∈ Λ, and J as in (3.4), we define

Φ_{t}^{θ,λ}(z, w) = Φ_{t}(z, w + λ)J(w, λ)e^{−πtH(w+λ,w+λ)+πtH(w,w)},
Φ_{t}^{θ}(z, w) = ∑_{λ∈Λ} Φ_{t}^{θ,λ}(z, w),

and

Φ^{θ,λ}(z, w) = Φ_{1}^{θ,λ}(z, w) and Φ^{θ}(z, w) = Φ_{1}^{θ}(z, w). (5.1)

Using the fact that H is linear in the first-component and conjugate-linear in the second, we may simplify Φ^{θ,λ}(z, w) to obtain:

Φ^{θ,λ}(z, w) = exp [πH(z, w + λ) − \frac{π}{2} H(λ, λ) − πH(λ, w) + 2πiK(λ)] . (5.2)

Theorem 5.1. Φ^{θ} as in (5.1) is the reproducing kernel for N_{H,K}.

Proof. We must verify the three properties of the reproducing kernel given in Section 2. We first check property (2), which follows from the following stronger statement: for t ≥ 1 and θ ∈ N_{H,K}

θ(z) = ∫_{ℂ^{n}/Λ} Φ_{t}^{θ}(z, w)θ(w) dμ(w). (5.3)

Showing that the integrand is periodic is well-defined is a routine calculation whose proof we suppress. Since θ is an entire theta function, according to Theorem 3.3, there exists a constant C such that |θ(z)|^{2}e^{−πH(z,z)} < C for all z ∈ ℂ^{n}. Thus,

∥θ∥_{t} = ∫_{ℂ^{n}} |θ(z)|^{2} dμ_{t}(z) < t^{n}Δ ∫_{ℂ^{n}} Ce^{(1−t)πH(z,z)} dz.

Suppose first that t > 1. Since H is positive definite, the integral above converges, and so θ(z) ∈ H_{tH}. Using the reproducing kernel for H_{tH} (Theorem 4.6) and the theta multiplier (3.3),

θ(z) = ∫_{ℂ^{n}} Φ_{t}(z, w)θ(w)dμ_{t}(w) = ∫_{ℂ^{n}/Λ} ∑_{λ∈Λ} Φ^{θ,λ}(z, w + λ)θ(w + λ) dμ(w + λ)

which gives (5.3) when t > 1.

Now consider the case when t = 1. We have

Φ^{θ}(z, w)θ(w)e^{−πH(w,w)} = ∑_{λ∈Λ} Φ(z, w + λ)J(w, λ)e^{−πH(w+λ,w+λ)+πH(w,w)}θ(w)e^{−πH(w,w)}
= e^{−πH(z,w)}θ(w)e^{−πH(w,w)} ∑_{λ∈Λ} e^{− \frac{π}{2} H(λ,λ)+terms linear in λ}.

By assumption H is positive definite, and so this summation is uniformly convergent on compact sets. Of course ℂ^{n}/Λ is compact and therefore, as t → 1 from above,

∫_{ℂ^{n}/Λ} Φ_{t}^{θ}(z, w)θ(w) dμ_{t}(w) → ∫_{ℂ^{n}/Λ} Φ^{θ}(z, w)θ(w) dμ(w),

establishing (5.3) when t = 1.

Next we check the property (3), i.e., that

Φ^{θ}(z, w) = \overline{Φ^{θ}(w, z)}. (5.4)

Using the representation of Φ^{θ,λ} as in (5.2) together with the facts that H is Hermitian and K is linear and real-valued, one easily checks that Φ^{θ,λ}(z, w) = \overline{Φ^{θ,−λ}(w, z)}, and adding up over all λ ∈ Λ, we get the desired result.

Finally, we check that for any fixed w ∈ ℂ^{n}, the function f(z) = Φ^{θ}(z, w) is a normalized entire theta function of (H, K) type, i.e., f ∈ N_{H,K}. In other words, we must show that Φ^{θ}(z + ν, w) = Φ^{θ}(z, w)J(z, ν). Using identities (3.1) and (3.3) together with the facts that H is Hermitian and K is linear and real-valued,

Φ^{θ,λ}(z, w + ν) = Φ^{θ,λ+ν}(z, w)\overline{J(z, ν)}.

Adding up over all λ ∈ Λ and using (5.4), we obtain the desire result.

Thus, Φ^{θ} satisfies all three properties of being a reproducing kernel, completing the proof.

Theorem 5.2. dim N_{H,K} = det(H)Vol(ℂ^{n}/Λ).

Proof. Expanding Φ^{θ} as in (5.1), J as in (3.3) and letting S_{λ}(z) = e^{π(H(z,λ)−H(λ,z))}, Theorem 2.1 shows that

dim N_{H,K} = ∫_{ℂ^{n}/Λ} Φ^{θ}(z, z) dμ(z) = Δ ∑_{λ∈Λ} e^{−πH(λ,λ)+2πiK(λ)} ∫_{ℂ^{n}/Λ} S_{λ}(z) dz.

Since we are considering only normalized theta functions, H(z, w) = 2iL(z, w), and so S_{λ}(z) = e^{2πi(L(z,λ)−L(λ,z))}. Clearly S_{λ}(z + w) = S_{λ}(z)S_{λ}(w). Moreover, by Theorem 3.2, for ν ∈ Λ, L(ν, λ) − L(λ, ν) = E(ν, λ) is an integer, and so S_{λ}(ν) = 1 and S_{λ}(z + ν) = S_{λ}(z). Thus, S_{λ} is a group character on ℂ^{n}/Λ, which means that the integral is zero unless S_{λ} is identically 1. Therefore, the only surviving term in the summation is the one with λ = 0, which gives

dim N_{H,K} = Δ ∫_{ℂ^{n}/Λ} 1 dz = ΔVol(ℂ^{n}/Λ) = det(H)Vol(ℂ^{n}/Λ).

Theorem 5.3. If H is associated to a nondegenerate Riemann from E with Pfaffian P f, then dim N_{H,K} = P f.

Proof. For z, w ∈ ℂ^{n}, let us write z = x + iy and w = u + iv where u, v, x, y are all real. By the Spectral Theorem, H = U^{∗}DU for some unitary matrix U and positive real diagonal matrix D. Write U = A + iB with A and B real. Then, considering ℂ^{n} as a complex vector space (not as an ℝ-module over a ℤ-basis for Λ), we have (by Theorem 3.2)

E(z, w) = Im(H(z, w)) = Im((u + iv)^{∗}(A + iB)^{∗}D(A + iB)(x + iy))
= [\begin{matrix}u \\ v \end{matrix}]^{T} [\begin{matrix}A & −B \\ B & A \end{matrix}]^{T} [\begin{matrix}0 & −D \\ D & 0 \end{matrix}] [\begin{matrix}A & −B \\ B & A \end{matrix}] [\begin{matrix}x \\ y \end{matrix}] .

Suppose that Λ is spanned over ℤ by {τ_{1}, …, τ_{2n}}, let R + iS be the n × 2n matrix whose j-th column is τ_{j}. Thus, in this basis we have

E(a, b) = b^{T} [\begin{matrix}R \\ S \end{matrix}]^{T} [\begin{matrix}A & −B \\ B & A \end{matrix}]^{T} [\begin{matrix}0 & D \\ −D & 0 \end{matrix}] [\begin{matrix}A & −B \\ B & A \end{matrix}] [\begin{matrix}R \\ S \end{matrix}] a.

It is clear that

|det [\begin{matrix}R \\ S \end{matrix}]| = Vol(ℂ^{n}/Λ).

On the other hand, since U = A + iB → det [\begin{matrix}A & −B \\ B & A \end{matrix}] is a continuous homomor-

phism of the connected compact group of n × n unitary matrices into ℝ^{×}, so its image is constant; that is,

det [\begin{matrix}A & −B \\ B & A \end{matrix}] = 1.

Thus, by Theorem 5.2,

P f = \sqrt{det(E)} = det(D)Vol(ℂ^{n}/Λ) = det(H)Vol(ℂ^{n}/Λ).

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Department of Mathematics, Stanford University, Stanford, CA 94305

E-mail address: bump@math.stanford.edu

Department of Mathematics, Stanford University, Stanford, CA 94305

E-mail address: apekker@stanfordalumni.org

Current address: 1841 Palisades Drive, Santa Rosa, CA 95403