Density results for automorphic forms on Hilbert modular groups

We give density results for automorphic representations of Hilbert modular groups. In particular, we show that there are infinitely many automorphic representations that have a prescribed discrete series factor at some (but not all) real places.Comment: 35 pages, LaTe

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 \SL_2 over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ 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 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.Comment: Accepted for publication by the Transactions of the American Mathematical Societ