Quantum variance for Hecke eigenforms

We calculate the quantum variance for the modular surface. This variance, introduced by S. Zelditch, describes the fluctuations of a quantum observable. The resulting quadratic form is then compared with the classical variance. The expectation that these two coincide only becomes true after inserting certain subtle arithmetic factors, specifically the central values of corresponding L-functions. It is the off-diagonal terms in the analysis that are responsible for the rich arithmetic structure arising from the diagonalization of the quantum variance.