Rankin Triple Products and Quantum Chaos

TL;DR

This paper establishes explicit formulas linking triple product L-values of automorphic forms to period integrals, and uses these to prove Quantum Unique Ergodicity and decay of third moments under certain hypotheses.

Contribution

It provides explicit Harris-Kudla type formulas for triple products on hyperbolic quotients and proves QUE assuming subconvexity, advancing understanding of quantum chaos.

Findings

Proved explicit formulas relating L-values to period integrals.

Established QUE under subconvexity assumptions.

Demonstrated decay of third moments in high energy limit.

Abstract

We prove explicit Harris-Kudla type formulas for triples of Maass forms, holomorphic forms, and combinations thereof, on the hyperbolic plane modulo congruence groups and co-compact lattices arising from Eichler orders of quaternion algebras. These formulas relate the central value of the corresponding Rankin triple product L-function to a squared trilinear period integral. Assuming subconvexity estimates for these L-values, we prove Quantum Unique Ergodicity on such quotients; the relevant Lindelof hypotheses imply a quantitative form of QUE, with an optimal rate. In connection with the Berry/Hejhal Random Wave conjecture, we prove decay of third moments in the high energy limit, making use of a subconvexity result of Iwaniec/Ivic/Jutila and Kim-Shahidi's result on cuspidality of the symmetric cube.