Density results for automorphic forms on Hilbert modular groups II

BRUGGEMAN, R. W., MIATELLO R.J.,

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY

American Mathematical Society

Lugar: Providence; Año: 2010 vol. 362 p. 3841 - 3881

0002-9947

Abstract. 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 discrete subgroup of Hecke type  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 multieigenvaluesin various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see x1.2.4–1.2.13) 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  ([3], Theorem 2.1).bstract. 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 discrete subgroup of Hecke type  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 multieigenvaluesin various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips (see x1.2.4–1.2.13) 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  ([3], Theorem 2.1).