Joint Gaussian value distribution conjecture for orthogonal Hecke–Maass cusp forms

Let $J\geq2$, and let $f_j$, $1\leq j\leq J$, be pairwise orthogonal real-valued Hecke–Maass cusp forms for $\operatorname{SL}_2(\mathbb{Z})$, with spectral parameters $t_{f_j}$. Assume

$$\frac{1}{\operatorname{vol}(\Gamma\backslash\mathbb{H})}\int_{\Gamma\backslash\mathbb{H}}|f_j(z)|^2\,\mathrm{d}\mu z=1,\qquad 1\leq j\leq J.$$

Let $a_j$ be positive integers. Joint Gaussian value distribution conjecture. The powers $\{f_j^{a_j}\}_{j=1}^J$ are statistically independent: for every $\psi\in\mathcal{C}_c^\infty(\Gamma\backslash\mathbb{H})$,

$$\int_{\Gamma\backslash\mathbb{H}}\psi(z)\prod_{j=1}^J f_j(z)^{a_j}\,\mathrm{d}\mu z=\left(\prod_{j=1}^Jc_{a_j}\right)\int_{\Gamma\backslash\mathbb{H}}\psi(z)\,\mathrm{d}\mu z+o(1)$$

as $\min(t_{f_1},\ldots,t_{f_J})\to\infty$, where $c_a=(2a-1)!!$ for even $a$ and $c_a=0$ for odd $a$. This predicts that distinct orthogonal Hecke–Maass cusp forms behave like independent random waves in the semiclassical limit. The paper proves two conditional joint-moment results supporting the conjecture, but the full assertion remains open.