Joint distribution of Hecke eigenforms on $\mathbb{H}^3$
Didier Lesesvre, Luca Marchesini, Nicole Raulf
TL;DR
The paper investigates the joint value distribution of Hecke-Maaß cusp forms on the hyperbolic 3-space $\\mathbb{H}^3$, extending Gaussian-moment heuristics from quantum chaos to automorphic forms on a 3-dimensional locally symmetric space. It develops a spectral-framework combining Parseval, the Watson-Ichino triple-product formula, and Rankin-Selberg unfolding to relate inner products like $\\langle \\psi, f^2 g\\rangle$ to central $L$-values with controlled archimedean factors, and then derives sharp bounds via subconvexity-type estimates. The main result asserts a smoothed joint equidistribution: for all $\\psi \\in C_c^{\\infty}(\\Gamma\\backslash\\mathbb{H}^3)$, as $t_f\\to\\infty$, $$\\langle \\psi, f^2 g\\rangle = \langle \\psi, g\\rangle \\mathbf{1}_{t_f > t_g - t_g^{\\varepsilon}} + O_{\\psi}((t_f t_g^{1/2})^{-1+\\varepsilon}),$$ under a Generalized Lindelöf Hypothesis (or, with stronger subconvex bounds, to an $o(1)$-type error). This establishes an effective decoupling of orthogonal Hecke-Maaß cusp forms in dimension three and provides a robust blueprint for extensions to other number fields and level aspects, reinforcing the Gaussian-molec ular moment picture in higher rank settings.
Abstract
We prove a joint value equidistribution statement for Hecke-Maaß cusp forms on the hyperbolic three-space $\mathbb{H}^3$. This supports the conjectural statistical independence of orthogonal cusp forms.