[Paper Review] Quantum variance on quaternion algebras, I

Paul D. Nelson|arXiv (Cornell University)|Jan 11, 2016

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper establishes the first asymptotic formula for quantum variance on compact quotients arising from non-split quaternion algebras, using the theta correspondence to reduce the problem to metaplectic Rankin–Selberg convolutions. It proves that the quantum variance exhibits a main term governed by central L-values and identifies secondary main terms at the square-root cancellation threshold, extending the Luo–Sarnak–Zhao framework beyond the split case.

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 by Luo--Sarnak, Zhao, and Sarnak--Zhao on the modular curve, 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.

Motivation & Objective

• To determine the quantum variance of families of automorphic forms on compact arithmetic quotients from non-split quaternion algebras.
• To extend the Luo–Sarnak–Zhao method—previously reliant on cusp forms and Fourier coefficients—to non-split, compact cases lacking cusps.
• To establish the existence and structure of the main term and secondary terms in the quantum variance sum using automorphic forms and L-functions.
• To provide a new method based on the theta correspondence that bypasses the need for Fourier expansions and Hecke multiplicativity.

Proposed method

• The theta correspondence is used to relate automorphic forms on the quaternion algebra to metaplectic forms on $\mathrm{Mp}_2$, enabling the reduction of the quadrilinear quantum variance to bilinear expressions.
• The problem is reduced to estimating metaplectic Rankin–Selberg convolutions, which are analyzed via the Rallis inner product formula and the Waldspurger–Maass–Shintani theta lift.
• The method relies on the decay of matrix coefficients for $\mathrm{SL}_2$ to ensure convergence of the variance sum.
• Local and global L-functions are computed using the triple product formula and strong approximation, particularly at the prime 2 and 23.
• The computation involves explicit evaluation of local integrals $I_2$, $I_{23}$, and $I_\infty$ using SAGE and known special values of L-functions.
• The final formula expresses the variance in terms of $L(\Psi_k, \frac{1}{2})$ and arithmetic factors involving $\zeta_2(1)$, $\zeta_{23}(1)$, and $L_2(\Psi_k, \frac{1}{2})$.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the quantum variance for families of automorphic forms on compact quotients from non-split quaternion algebras?
• RQ2 How can the quantum variance problem be solved in the absence of cusps and Fourier expansions, which are essential in the split case?
• RQ3 What role do central L-values and metaplectic Rankin–Selberg convolutions play in determining the main term of the quantum variance?
• RQ4 Are there secondary main terms in the quantum variance at the square-root cancellation threshold in the non-split case?
• RQ5 How does the theta correspondence substitute for Hecke multiplicativity and Fourier coefficients in the variance computation?

Key findings

• The quantum variance $\mathcal{V}_T(\Psi)$ satisfies $T^{d-1}\mathcal{V}_T(\Psi) = \mathcal{V}_\infty(\Psi) + o(1)$, with the main term $\mathcal{V}_\infty(\Psi)$ given by a product of $L(\Psi_k, \frac{1}{2})$ and arithmetic factors.
• The main term is explicitly computed as $L(\Psi_k, \frac{1}{2})P(\lambda_{\Psi_k}(2))$, where $P$ is a rational function involving $\zeta_2(1)$, $\zeta_2(2)$, and $L_2(\Psi_k, \frac{1}{2})$.
• Secondary main terms appear at the square-root cancellation threshold, arising from the non-trivial local integral $I_2$ involving $\Xi_k(g)$ and $\operatorname{Ad}(g)\phi_2$.
• The variance is shown to be asymptotically governed by central L-values, confirming a non-archimedean analog of the Feingold–Peres prediction in the arithmetic setting.
• The computation confirms that $\|\theta_{\text{Jac}}\|^2\|h_k\|^2 = 2L^{(S)}(\Psi_k, \frac{1}{2}) \cdot \frac{(4\pi)^2}{4} \cdot \frac{2\zeta_{23}(1)}{23} \cdot 2^{-6} \zeta_2(1)^2 \zeta_2(2) \left(2 + \frac{L_2(\Psi_k, \frac{1}{2})}{\zeta_2(2)}\right)$, which simplifies to $L(\Psi_k, \frac{1}{2})P(\lambda_{\Psi_k}(2))$.
• The method successfully bypasses the need for cusp forms and Fourier coefficients, proving that the theta correspondence provides a viable alternative for quantum variance in non-split cases.