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

Eigenvalues of Hecke operators on Hilbert modular groups

Abstract. Let F be a totally real field, let I be a nonzero ideal of the ring of integers OF Q of F, let 0(I) be the congruence subgroup of Hecke type of G = dj =1 SL2(R) embedded diagonally in G, and let be a character of 0(I) of the form ac b d = (d), where d 7! (d) is a character of OF modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring HI , generated by the Hecke operators T(p2), p 2 P (see x3.2) acting on (; )- automorphic forms on G. Given the cuspidal space L2;cusp 0(I)nG; , we let V$run through an orthogonal system of irreducible G-invariant subspaces so that each V$ is invariant under HI . For each 1 j d, let $= ($; j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V$, and for each p 2 P, we take$;p 0 so that 2 $;p N(p) is the eigenvalue on V$ of the Hecke operator T(p2) For each family of expanding boxes t 7! t , as in (3) in Rd, and fixed an interval Jp in [0;1), for each p 2 P, we consider the counting function N( t; (Jp)p2P) := X $;$2 t : $;p2Jp ;8p2P jcr($)j2 : Here cr($) denotes the normalized Fourier coecient of order r at 1 for the elements of V$, with r 2 O0 F r pO0 F for every p 2 P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the t , the asymptotic distribution of the function N( t; (Jp)p2P), as t ! 1. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues $are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe$; j 2 j for all infinite places j , l and$;p 2 Jp for all finite places p in P (for fixed intervals j and Jp) and then allow j$;lj to grow to 1, then there are infinitely many such $, and their positive density is as described in Theorem 1.1.Fil: Bruggeman, Roelof W.. Utrecht University; Países BajosFil: Miatello, Roberto Jorge. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentin