Rankin Triple Products and Quantum Chaos

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.

58 pages. A reformatted version, with minor revisions, of my Ph.D. thesis (Princeton 2002, adviser: Peter Sarnak). For the original, see 0810.0425v1.

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