Abstract
Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.
Highlights
Let D be a negative integer with D ≡, so that D is an imaginary quadratic discriminant
The Z-lattice OD = [τD, ] becomes a quadratic order of discriminant D = dK · t in the imaginary quadratic eld K = Q(τD) where dK is a fundamental discriminant of K and a positive integer t is the conductor of OD
Let QD be the set of all positive de nite integral binary quadratic forms of discriminant D, namely, QD = {ax + bxy + cy ∈ Z[x, y] | a >, b − ac = D}
Summary
Let D be a negative integer with D ≡ , (mod ) so that D is an imaginary quadratic discriminant. The Z-lattice OD = [τD , ] becomes a quadratic order of discriminant D = dK · t in the imaginary quadratic eld K = Q(τD) where dK is a fundamental discriminant of K and a positive integer t is the conductor of OD. Let QD be the set of all positive de nite integral binary quadratic forms of discriminant D, namely, QD = {ax + bxy + cy ∈ Z[x, y] | a > , b − ac = D}. The modular group Γ( ) = SL (Z)/{±I } acts on the set QD from the right by the rule γ = γ γ : Q(x, y) = ax + bxy + cy → Qγ(x, y) = Q(γ x + γ y, γ x + γ y),.
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