Author
Liu, Sheng-Chi

Year
2009

Advisor
Luo, Wenzhi

Abstract

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 $\mathcal{O}$, and let $\Gamma = SL(2, \mathcal{O})$ be the Hilbert modular group. Given the orthonormal basis of Hecke eigenforms in $S_{2k}(\Gamma)$, the space of cusp forms of weight $(2k, 2k, …., 2k)$, one can associate a probability measure $d\mu_k$ on the Hilbert modular variety $\Gamma \backslash \mathbb{H}^n$. We prove that $d\mu_k$ tends to the invariant measure on $\Gamma \backslash \mathbb{H}^n$ weakly as $k \to \infty$. 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 $F= \mathbb{Q}$.

Our approach is using Selberg trace formula, Bergman kernel, and Shimizu's dimension formula.

Thesis
thesis.pdf