REPRESENTATIONS OF FINITE GROUPS AND HILBERT MODULAR FORMS

ABSTRACT

Hecke studied the representation of in the space of elliptic modular cusp forms for the principal congruence subgroup with level odd prime . He showed that the difference of the multiplicities of certain two irreducible representations in this representation is the class number of . McQuillan extended Hecke's result to the case where the level of the principal congruence subgroup is , where are distinct odd primes. In the case of Hilbert modular forms for real quadratic fields, Meyer and Sczech gave a result which is analogous to that of Hecke. In this paper, we generalize their result to the case where the level of the principal congruence subgroup is , where are distinct prime ideals lying over odd primes. We get this result with the use of the holomorphic Lefschetz formula. Our result can be thought of a generalization of that of McQuillan to the real quadratic field case. We also give a generalization of a result of Eichler [3].

ACKNOWLEDGMENTS

I express my sincere gratitude to Professor Takayuki Oda for his interest and constant encouragement. My sincere gratitude also goes to Professor T. Shoji, who read the first draft of the present paper and gave some comments. In particular, he simplified the proof of Proposition 4.3. My sincere gratitude also goes to Professor R. Tsushima for his helpful advice. I was partially supported by Grant-in-Aid for Scientific Research, Ministry of Education, Japanese Government, No. 10740020.