The Quantum Variance of The Modular Surface

We discuss the quantum variance, which is introduced by Zelditch and
describes the fluctuations of a quantum observable, on the phase space
of modular surface. We asymptotically evaluate the quantum variance
and show that it is equal to the classical variance of the geodesic
flow on the phase space after inserting the correction factor of
certain L-function’s central value on each irreducible subspace. It is
also very close to the arithmetic variance studied by Luo, Rudnick and
Sarnak. This talk is based on a joint work with Peter Sarnak.