The Spaces of Hilbert Cusp Forms for Totally Real Cubic Fields and Representations of SL2(Fq)

Abstract. Let S2m(Γ(p)) be the space of Hilbert modular cusp forms for the principal congruence subgroup with level p of SL2(OK) (here OK is the ring of integers ofK, and p is a prime ideal ofOK). Then we have the action of SL2(Fq) on S2m(Γ(p)), where q = Np. When q is a power of an odd prime, for each SL2(Fq) we have two irreducible characters which have conjugate values mutually. In the case where K is the field of rationals, M. Eichler gives a formula for the difference of multiplicites of these characters in the trace of the representation of SL2(Fq) on S2m(Γ(p)). In the case where K is a real quadratic field, H. Saito gives a formula analogous to that of Eichler for the difference. The purpose of this paper is to give a formula analogous to that of Eichler in the case where K is a totally real cubic field. 1