Abstract
In the present paper, we provide a construction of the multiplicative Borcherds lift for unitary groups \(\mathrm U (1,m)\), 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 \(\mathrm U (1,m)\) into \(\mathrm O (2,2m)\). 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.
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Acknowledgments
Much of the present paper is a short version of the author’s thesis [10] completed at the TU Darmstadt. I am indebted to my thesis adviser, J. Bruinier for his insight and many helpful suggestions. I would also like to thank the anonymous referee, whose comments have helped to considerably improve this paper.