A generalization of a theorem of Hecke for SL_2(F_p) to fundamental discriminants

Abstract

Let p > 3 be an odd prime, p ≡ 3 mod 4 and let π+, π− be the pair of cuspidal representations of SL_2(F_p). It is well known by Hecke that the difference m_π+ −m_π− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(√−p). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q(√−D) and prove that an alternating sum of multiplicities of certain irreducibles of SL_2(Z/DZ) is an explicit multiple, up to a sign and a power of 2, of either the class number h(−D) or of the sums h(−D)+h(−D/2), h(−D)+2h(−D/2); the last two possibilities occur in some of the cases when D ≡ 0 mod 8. The proof uses the holomorphic Lefschetz number.

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