In the present paper, we provide a construction of the multiplicative Borcherds lift for unitary groups U(1, m), which takes weakly holomorphic elliptic modular forms as input functions and lifts them to automorphic forms having infinite product expansions and taking their zeros and poles along Heegner divisors. In order to transfer Borcherds' theory to unitary groups, we construct a suitable embedding of U(1, m) into O(2, 2m). We also derive a formula for the values taken by the Borcherds products at cusps of the symmetric domain of the unitary group. Further, as an application of the lifting, we obtain a modularity result for a generating series with Heegner divisors as coefficients, along the lines of Borcherds' generalization of the Gross-Zagier-Kohnen theorem.

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, a(n)a(n + r) (n ≥ 1) for a fixed positive integer r, concerning both non-vanishing and non-negativity.

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In [Borcherds products on unitary groups, preprint (2012); arXiv:1210.2542] the author constructed a multiplicative Borcherds lift for indefinite unitary groups U (1, n). In the present paper the case of U(1,1) is examined in greater detail. The lifting in this case takes weakly holomorphic elliptic modular forms of weight zero as inputs and lifts them to meromorphic modular forms for U(1,1), on the usual complex upper half-plane ℍ. In this setting, the Weyl-chambers can be described very explicitly and the associated Weyl-vectors can also be calculated. This is carried out in detail for weakly holomorphic modular forms with q-expansions of the form [Formula: see text], n > 0, and for a constant function, which together span the input space, [Formula: see text]. The general case for the lifting of an arbitrary [Formula: see text] comes by as a corollary. The lifted functions take their zeros and poles along Heegner-divisors, which consist of CM-points in ℍ. We find that their CM-order can to some extent be prescribed.

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