Mass equidistribution of Hecke eigenforms on the Hilbert modular varieties

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 S_2k(Γ), the space of cusp forms of weight (2k, 2k,⋯, 2k), one can associate a probability measure dμ_k on the Hilbert modular variety Γ\ℍⁿ. We prove that dμ_k 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.