Eigenvalues of Hecke operators on Hilbert modular groups

Abstract

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.