Kloosterman Sums and Traces of Singular Moduli

where q = e. Let d ≡ 0, 3 (mod 4) be a positive integer, so that −d is a negative discriminant. Denote by Qd the set of positive definite integral binary quadratic forms Q(x, y) = ax + bxy + cy = [a, b, c] with discriminant −d = b − 4ac, including imprimitive forms (if such exist). We let αQ be the unique complex number in the upper half plane H which is a root of Q(x, 1) = 0. Values of j at the points αQ are known as singular moduli. Singular moduli are algebraic integers which play prominent roles in number theory. For example, Hilbert class fields of imaginary quadratic fields are generated by singular moduli, and isomorphism classes of elliptic curves with complex multiplication are distinguished by singular moduli. Because of the modularity of j, the singular modulus j(αQ) depends only on the equivalence class of Q under the action of Γ = PSL2(Z). We define ωQ ∈ {1, 2, 3} as