On the Hecke Equivariance of the Borcherds Isomorphism

LetM be the multiplicative group of integral weight meromorphic modular forms for some character of SL2(Z) with integer coefficients, leading coefficient 1, and whose zeros and poles are supported at cusps and imaginary quadratic irrationals. Denote by Mk,h ⊂ M the subset which consists of modular forms of weight k and for which order of the pole at infinity is h. We have M = ∪k∈Z,h∈ 1 12 ZMk,h and Mk1,h1Mk2,h2 ⊂ Mk1+k2,h1+h2 . Let M + 1 2 (Γ0(4)) be the space of modular forms of weight 1/2 with 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) of weight 1/2 satisfies Kohnen’s plus-condition if its q-expansion ∑ n≫−∞ c(n)q n has c(n) = 0 for n ≡ 2, 3 mod 4. Throughout, we let q = e with Iτ > 0.) In [1, Theorem 14.1], Borcherds establishes an isomorphism between the multiplicative group M and the additive group M1 2 (Γ0(4)). Denote this isomorphism by B : M → M + 1 2 (Γ0(4)).