Mass equidistribution of Hecke eigenforms on the Hilbert modular varieties.

摘要

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 G=SL2,O be the Hilbert modular group. Given the orthonormal basis of Hecke eigenforms in S2k(Gamma), the space of cusp forms of weight (2k, 2k, ···, 2k), one can associate a probability measure dmu k on the Hilbert modular variety G\Hn . We prove that dmuk tends to the invariant measure on G\Hn weakly as k → infinity. 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.