Borcherds Products Associated with Certain Thompson Series

Let M ! 1/2 be the additive group consisting of nearly holomorphic modular forms of weight 1/2 for Γ0(4) whose Fourier coefficients are integers and satisfy the Kohnen’s “plus space” condition (i.e. n-th coefficients vanish unless n ≡ 0 or 1 modulo 4). We also let B be the multiplicative group consisting of meromorphic modular forms for some characters of SL2(Z) of integral weight with leading coefficient 1 whose coefficients are integers and all of whose zeros and poles are either cusps or imaginary quadratic irrationals. Borcherds [3] gave an isomorphism between M ! 1/2 and B by means of infinite products which we call modular products or Borcherds products. Let d denote a positive integer congruent to 0 or 3 modulo 4. We denote by Qd the set of positive definite binary quadratic forms Q = [a, b, c] = aX + bXY + cY 2 (a, b, c ∈ Z) of discriminant −d, with usual action of the modular group Γ = PSL2(Z). To each Q ∈ Qd, we associate its unique root αQ ∈ H (=upper half plane). We define the HurwitzKronecker class numberH(d) by H(d) = ∑