DENSITY RESULTS FOR AUTOMORPHIC FORMS ON HILBERT MODULAR GROUPS II

摘要

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for SL 2 over a totally real number field F, with a discrete subgroup of Hecke type Γ_0 (I) for a non-zero ideal I in the ring of integers of F. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see §1.2.4-1.2.13) and products of prescribed small intervals for all but one of the infinite places of F. The main tool in the derivation is a sum formula of Kuznetsov type (Sum formula for SL2 over a totally real number field, Theorem 2.1).