## The Spaces of Hilbert Cusp Forms for Totally Real Cubic Fields and Representations of $SL_2(\Bbb F_q)$

J. Math. Sci. Univ. Tokyo
Vol. 5 (1998), No. 2, Page 367--399.

Hamahata, Yoshinori
The Spaces of Hilbert Cusp Forms for Totally Real Cubic Fields and Representations of $SL_2(\Bbb F_q)$
Let $S_{2m}(Î(\frak p))$ be the space of Hilbert modular cusp forms for the principal congruence subgroup with level $\frak p$ of $SL_2(O_K)$ (here $O_K$ is the ring of integers of $K$, and $\frak p$ is a prime ideal of $O_K$). Then we have the action of $SL_2(\Bbb F_q)$ on $S_{2m}(Î(\frak p))$, where $q=N\frak p$. When $q$ is a power of an odd prime, for each $SL_2(\Bbb F_q)$ 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 $SL_2(\Bbb F_q)$ on $S_{2m}(Î(\frak p))$. 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.