On Zeta Functions of Orthogonal Groups of Single-class Positive Definite Quadratic Forms

Representations of Hecke–Shimura rings of integral single-class positive definite quadratic forms on relevant spaces of harmonic forms are considered, and the problem of simultaneous diagonalization of the corresponding Hecke operators is investigated. Explicit relations are deduced between zeta functions of the single-class quadratic forms in two and four variables corresponding to the harmonic eigenforms of genus 1 and 2, respectively, and zeta functions of the theta-series weighted by these eigenforms. Introduction By definition an (integral proper) automorph of a nonsingular integral quadratic form q(X) in m variables is an integral (m × m)-matrix D having positive determinant and satisfying the condition (0.1) q(DX) = μq(X) ( X = (x1, . . . , xm)); the number μ = μ(D) is the multiplier of the automorph. All automorphs form a semigroup A = A(q), the automorph semigroup of q, and the automorphs with multiplier μ = 1 form a subgroup E = E(q), called the group of (proper) units of q. The Hecke– Shimura ring or the automorph class ring of q over Z consists of all finite formal linear combinations with coefficients in Z of the symbols (EDE) = τ (D) corresponding in a one-to-one way to the double cosets EDE of A modulo E, also called the double cosets, (0.2) H = H(q) = { τ = ∑ α aατ (Dα) ∣∣∣ aα ∈ Z, Dα ∈ A}, where the product of two double cosets is defined by τ (D)τ (D′) = ∑ ED′′E⊂EDED′E c(D,D′;D′′)τ (D′′), and c(D,D′;D′′) is the number of pairs of representatives Di ∈ E\EDE and D′ j ∈ E\ED′E satisfying DiD j ∈ ED′′. Generally speaking, little can be said about the ring H, and it should be replaced by a more complicated construction of a matrix Hecke– Shimura ring, which we do not touch upon in this paper. However, in some cases, for example, if the form q belongs to a single-class genus, the ringH has many nice properties 2000 Mathematics Subject Classification. Primary 11F27, 11F46, 11F60, 14G10, 20C08.