In this thesis we study the analogue of Arithmetic Quantum Unique
Ergodicity conjecture on the Hilbert modular variety. Let F
be a totally real number field with ring of integers 𝒪, and
let Γ = SL(2, 𝒪) be the Hilbert modular group. Given the
orthonormal basis of Hecke eigenforms in S2k(Γ),
the space of cusp forms of weight (2k, 2k,⋯, 2k),
one can associate a probability measure dμk
on the Hilbert modular variety Γ\ℍn. We
prove that dμk tends to the invariant measure on
Γ\ℍn weakly as k → ∞. This
shows that the analogue of Arithmetic Quantum Unique Ergodicity conjecture
is true on the average on Hilbert modular variety. Our result generalizes
Luo’s result [Lu] for the case F = ℚ.
Our approach is using Selberg trace formula, Bergman kernel, and
Shimizu’s dimension formula.