Abstract
E. Hecke initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference of multiplicities of certain conjugate representations of SL(Fp) on cusp forms of degree 1, level p, and weight ≥2 is given by the class number h(-p) of the field Q(√-p). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification of the Siegel moduli space of degree 2 to compute the values of characters of the representations of Sp4(Fp) on certain spaces of cusp forms of degree 2 and level p at parabolic elements of this group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp4(Fp) is a multiple of h(-p).
Keywords: cusp forms, Siegel space, symplectic group, Lefschetz formula
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