On the dimension of the space of theta functions

Abstract

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

Series and integral representations of the Taylor coefficients of the Weierstrass sigma-function

“…In the present paper, we shall give new representations of the Taylor coefficients W r (Γ) either as series over Γ of Hermite-Gauss type, or as integrals over C of the σfunction. Before doing so, we fix the notations we shall use repeatedly in the sequel: Let {ω 1 , ω 2 } be an oriented arbitrary R-basis of C = R 2 and set Γ = Zω 1 + Zω 2 . The area of a fundamental cell Λ(Γ) of Γ is given by S = S Γ = (ω 1 ω 2 ), where (z) denotes the imaginary part of z ∈ C. We define ν ∈ R > :=]0, +∞[ and µ ∈ C by…”

“…C) can be viewed as the space of holomorphic sections of the holomorphic line bundle L = (C × C)/Γ. Its dimension is then given by the Pfaffian √ det E of the associated skew-symmetric form E (see [9,7,4,2,11]). In our case, E(z, w) := (ν/π) z, w .…”

Analytic and arithmetic properties of the $(Γ,χ)$-automorphic reproducing kernel function

“…So far, the focus of study in [6,7,5] has been the spectral properties of the so-called theta Bargmann-Fock Hilbert space O ν Γ,χ (C) associated to given full-rank oriented lattice Γ = Zω 1 + Zω 2 and pseudo-character χ (see also [2,10,11]). This space is defined to be the L 2functional space of holomorphic (Γ, χ)-theta functions f on C of magnitude ν > 0, provided that the functional equation f (z + γ) = χ(γ)e ν(z+ γ 2 )γ f (z) (1.1) holds for all z ∈ C and all γ ∈ Γ.…”

“…Notice for instance that ν ≥ π/S(Γ), otherwise O ν Γ,χ (C) is trivial. The proof of (1.5) can be handled using Riemann-Roch theorem which is well-known in the theory of abelian varieties (see [3,9,19,12,2,14,16]). It can also be done à la Selberg [17,8,2] by determining the trace of the integral operator associated to (Γ, χ)-automorphic kernel function K ν Γ,χ ([6]).…”

“…The proof of (1.5) can be handled using Riemann-Roch theorem which is well-known in the theory of abelian varieties (see [3,9,19,12,2,14,16]). It can also be done à la Selberg [17,8,2] by determining the trace of the integral operator associated to (Γ, χ)-automorphic kernel function K ν Γ,χ ([6]). In the present paper, we will concentrate on discussing some basic analytical and arithmetical properties of the reproducing kernel function K ν Γ,χ (z, w).…”