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Title: Mass equidistribution of Hecke eigenforms on the Hilbert modular varieties
Abstract: We will discuss 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 $(2k, 2k, \cdots, 2k)$, one can associate a probability measure $d\mu_k$ on the Hilbert modular variety $\Gamma \backslash H^n$. We prove that $d\mu_k$ tends to the invariant measure on $\Gamma \backslash H^n$ 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.