Density Results for Automorphic Forms on Hilbert Modular Groups Ii

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for SL2 over a totally real number field F , with a discrete subgroup of Hecke type Γ0(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).