A generalization of a theorem of Hecke for [formula omitted] to fundamental discriminants

Abstract

Let p>3 be an odd prime, p≡3mod4 and let π+,π− be the pair of cuspidal representations of SL2(Fp). It is well known by Hecke that the difference mπ+−mπ− 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 SL2(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≡0mod8. The proof uses the holomorphic Lefschetz number.