Equidistribution of Hecke eigenforms on the Hilbert modular varieties

Abstract

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 2 k ( Γ ) , one can associate a probability measure d μ k on the Hilbert modular variety Γ \\ H n . We prove that d μ k tends to the invariant measure on Γ \\ H n weakly as k → ∞ . This generalizes Luo's result [W. Luo, Equidistribution of Hecke eigenforms on the modular surface, Proc. Amer. Math. Soc. 131 (2003) 21–27] for the case F = Q .