Journal Article

A local Weyl’s law, the angular distribution and multiplicity of cusp forms on product spaces

Tepper, David, Huntley, Jonathan

1992

Overview

Let Γ/H\\Gamma /\\mathcal {H} be a finite volume symmetric space with H\\mathcal {H} the product of half planes. Let Δi{\\Delta _i} be the Laplacian on the iith half plane, and assume that we have a cusp form ϕ\\phi, so we have Δiϕ=λiϕ{\\Delta _i}\\phi = {\\lambda _i}\\phi for i=1,2,…,ni = 1,2, \\ldots ,n. Let λ→=(λ1,…,λn)\\vec \\lambda = ({\\lambda _1}, \\ldots ,{\\lambda _n}) and let \\[ R=r12+⋯+rn2R = \\sqrt {r_1^2 + \\cdots + r_n^2} \\] with ri2+14=λir_i^2 + \\frac {1} {4} = {\\lambda _i}. Letting r→=(r1,…,rn)\\vec r = ({r_1}, \\ldots ,{r_n}), we let M(r→)M(\\vec r) denote the dimension of the space of cusp forms with eigenvalue λ→\\vec \\lambda. More generally, let M(r→,a)M(\\vec r,a) denote the number of independent eigenfunctions such that the r→\\vec r associated to an eigenfunction is inside the ball of radius aa, centered at r→\\vec r. We will define a function f(r→)f(\\vec r), which is generally equal to a linear sum of products of the ri{r_i}. We prove the following theorems. Theorem 1. \\[ M(r→)=O(f(r→)(log⁡R)n).M(\\vec r) = O\\left (\\frac {f(\\vec r)} {(\\log R)^n} \\right ). \\] Theorem 2. \\[ M(r→,A)=2nf(r→)+O(f(r→)log⁡R).M (\\vec {r}, A) = 2^n f(\\vec {r})+O\\left (\\frac {f(\\vec r)}{\\log R} \\right ). \\]

• Publisher: American Mathematical Society
• Subject: Algebra / Angular distribution / Coordinate systems / Eigenfunctions / Eigenvalues / Exact sciences and technology / Half planes / Mathematical cusps / Mathematical functions / Mathematical theorems / Mathematics / Number theory / Research article / Sciences and techniques of general use / Symmetry