8 Rankin Triple Products and Quantum Chaos

In this dissertation we demonstrate the chaotic nature of certain special arithmetic quantum dynamical systems, using machinery from analytic number theory. We consider the quantized geodesic flow on finite-volume hyperbolic surfaces Γ\H, with Γ ⊂ SL 2 R consisting of the norm-1 units of an Eichler order in an indefinite quaternion algebra B over Q. Such Γ generalize the congruence subgroups of SL 2 Z and are co-compact whenever B is ramified. For Γ = SL 2 Z, we prove that high energy bound eigenstates obey the Random Wave conjecture of Berry/Hejhal for third moments. In fact we show that the third moment of a wave's amplitude distribution decays as E − 1 12 in the high energy limit. In the more general case of maximal orders, we reduce an optimal quantitative version of the Quantum Unique Ergodic-ity conjecture of Rudnick–Sarnak to the Lindelöf Hypothesis, itself a consequence of the Riemann Hypothesis, for particular families of automorphic L-functions. Furthermore , our analysis shows that any lowering of the exponent in the Phragmen–Lindelöf convexity bound implies QUE. In the moment problem as well, a decisive role is played by 'breaking convexity.' That is to say, the maximum non-trivial exponents precisely agree when translated between physical and arithmetical formulations for both of these problems. We accomplish this translation by proving identities expressing triple-correlation integrals of eigenforms in terms of central values of the corresponding Rankin triple-product L-functions. Very general forms of such identities were proved by Harris– Kudla, and certain (more precise) explicit classical versions of these were given by Gross–Kudla and Böcherer–Schulze-Pillot, for definite quaternion algebras. In using the Harris–Kudla method to prove our own classical identities, we have to solve two main problems. The first problem is to explicitly compute the adjoint of Shimizu's theta lift, which realizes the Jacquet–Langlands correspondence by transferring automorphic forms from GL 2 to GO(B), the latter being nearly the same as (B ×) 2. As is well known, theta liftings from metaplectic to orthogonal groups are generally more difficult to characterize than lifts in the opposite direction, which can be evaluated directly in terms of Whittaker functions. Since B × and hence GO(B) have 'multiplicity-one'— as Jacquet–Langlands proved with the trace-formula—we are able to determine the adjoint of Shimizu's lift by duality, from explicit knowledge of Shimizu's lift itself. It is, however, necessary to generalize Shimizu's original calculations, since he only considered averages of …