Luo–Sarnak variance formula for Hecke eigenfunctions

Let $X=\mathbf H^2/SL(2,\mathbb Z)$, and let $\{\varphi_k\}$ be the orthonormal basis of Hecke eigenfunctions on $X$. For $f\in C_0^\infty(X)$, there exists a quadratic form $B(f)$ such that

$$\frac{1}{N(\lambda)}\sum_{\lambda_j\leq\lambda}\left|\int_X f|\varphi_j|^2\,d\operatorname{vol}-\frac{1}{\operatorname{Vol}(X)}\int_X f\,d\operatorname{Vol}\right|^2=\frac{B(f,f)}{\lambda}+o\left(\frac{1}{\lambda}\right).$$

Luo–Sarnak variance formula. The displayed asymptotic holds for the Hecke eigenfunction basis. The result is rigorous for the stated arithmetic surface and holomorphic Hecke eigenforms; the analogous assertion for smooth Maass–Hecke eigenfunctions is described as expected, while the broader variance conjecture remains open.