Quantum variance on quaternion algebras, I

Abstract

We determine the quantum variance of a sequence of families of automorphic forms on a compact quotient arising from a non-split quaternion algebra. Our results compare to those obtained on SL 2 (Z)\ SL 2 (R) in work of Luo, Sarnak and Zhao, whose method required a cusp. Our method uses the theta correspondence to reduce the problem to the estimation of metaplectic Rankin-Selberg convolutions. We apply it here to the first non-split case.8 This is the special case |λ(n)| 2 = d|n λ(n 2 /d 2 ) of the Hecke multiplicatively exploited by Luo-Sarnak-Zhao. 9 The analysis of triple product averages mentioned above exploits the related identity L(ϕ × ϕ × Ψ, 1/2) = L(sym 2 ϕ × Ψ, 1/2)L(Ψ, 1/2).

Small‐scale equidistribution of Hecke eigenforms at infinity

“…where 𝑊 𝑖 (𝑦) = 𝑉( 1 4𝜋𝑦 )𝑦 −2𝑖 for 𝑦 ∈ ℝ + , and the implied constant is of the form (20). Since 𝜃 < 1∕2, we can choose 𝑁 large enough such that the dominating error term in 𝑘 is 𝑘 −1−𝜃+𝜀 .…”

Quantum variance for holomorphic Hecke cusp forms on the vertical geodesic

“…Closed geodesics on the modular surface were investigated my Luo, Rudnick and Sarnak in [LRS09]. In the compact setting of quaternion algebras Nelson [Nel16], [Nel17], [Nel19] evaluated the quantum variance using the theta correspondence. Further papers indicating the active field of research are for example given by work of Huang [Hua21] for the variance of Eisenstein series, Huang-Lester [HL20] for the variance of dihedral Maass cusp forms and the work of Nordentoft, Petridis and Risager [NPR21] on small scale equidistribution at infinity.…”

Quantum variance for dihedral Maass forms

“…Recently, Sarnak-Zhao [52] obtained the asymptotic formula of quantum variance for several phase space observables, that is for Hecke-Maass cusp forms on PSL 2 (Z)\ PSL 2 (R). Additionally, in the setting of quaternion algebras Nelson [38,39,40] has computed an asymptotic formula for the quantum variance. In each of the previously mentioned works the leading order constant of the quantum variance is given by classical variance of the geodesic flow V (ψ) along with an additional arithmetic factor, which is related to the central value of an L-function.…”

“…In contrast to prior work on the quantum covariance (such as [52] and Nelson [38,39,40]) our proof of Theorem 1.3 uses the aforementioned Watson-Ichino formula together with estimates for moments of central values of Rankin-Selberg L-functions. This approach is more closely related to work of the first named author on the quantum variance of the Eisenstein series.…”