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

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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 $SL_2(\Bbb F_q)$
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Abstract:
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.

Mathematics Subject Classification (1991): Primary 11F41; Secondary 10D21, 12A50

Mathematical Reviews Number: MR1633870

Received: 1997-12-24

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The Spaces of Hilbert Cusp Forms for Totally Real Cubic Fields and Representations of SL2(Fq)SL_2(\Bbb F_q)

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