The Borcherds-Zagier Isomorphism and a p-Adic Version of the Kohnen-Shimura Map

Let M be the space of even integer weight meromorphic modular forms on SL2(Z) with integer coefficients, leading coefficient equal to one, and whose zeros and poles are supported at cusps and imaginary quadratic irrationals. If r ≥ 0 is an integer, let M+r+1/2(Γ0(4)) be the space of modular forms of half-integral weight r + 1/2with respect to Γ0(4) which satisfy Kohnen’s plus-condition and whose poles are supported at the cusps of Γ0(4). (Recall that a modular form f(τ) satisfies Kohnen’s plus-condition if its q-expansion ∑ n≥0 c(n)q n has c(n) = 0 if (−1)n ≡ 2, 3mod 4.) In [3, Theorem 14.1], Borcherds establishes an isomorphism between the multiplicative group M and the additive group M+1/2(Γ0(4)). For example, an explicit construction of the Borcherds isomorphism is obtained by Zagier in [16], as we now describe. Using Zagier’s notation, if d ≡ 0, 3mod 4 is a positive integer, we denote by Qd the set of positive definite binary quadratic formsQ = [a, b, c] = aU + bUV + cV (a, b, c ∈ Z) with discriminant b − 4ac = −d with the usual action of the modular group Γ = PSL(2,Z) on Qd. To eachQ ∈ Qd we associate its unique root αQ in the upper half-plane and put wQ = 2 or 3 if Q is Γ-equivalent to [a, 0, a] or [a, a, a], and 1 otherwise. Let j = q+744+196884q+ · · · denote the modular invariant (q = exp(2πiτ) throughout). Consider the weight 0 modular form