Automorphic Products on Unitary Groups

The dissertation provides a construction for Borcherds products on unitary groups of signature (1,q). The starting point for this is the multiplicative lifting due to R. E. Borcherds. He employs the singular theta-correspondence to construct a lifting, which takes as inputs weakly holomorphic vector valued modular forms, transforming under the Weil-representation of SL(2,Z) for a quadratic lattice, and lifts these to meromorphic automorphic forms for an arithmetic subgroup of O(2,n). The resulting functions have expansions as infinite products and take their zeros and poles along Heegner divisors. In order to transfer this result to unitary groups, we construct an embedding between the symmetric domain of the unitary group and that of an orthogonal group, respectively. This embedding is compatible with the complex structures of either symmetric domain and a suitable choice of cusps. The main result is the construction of Borcherds products, on unitary groups of signature (1,q). In this setting we prove a result which is analogous to that of Borcherds. As in the case of orthogonal groups, the infinite products thus constructed have their zeros and poles on Heegner divisors. Here, the role of the quadratic lattice is taken by a hermitian lattice, which we assume to have as multiplier system the ring of integers of an imaginary quadratic number field. Further, we study the behavior of these automorphic products on the boundary of the symmetric domain. It turns out that the values taken on the boundary points can be interpreted as CM-values of generalized eta-products. In the finial chapter, we construct examples for the unitary group SU(1,1) and unimodular lattices, which in this case are simply hyperbolic planes over the rings of integers of imaginary quadratic number fields. In this case, the resulting products can be viewed as meromorphic elliptic modular forms on the (classical) complex upper half-plane.

Druckausg.: München, Verl. Dr. Hut, 2011, ISBN 978-3-86853-842-7 [Darmstadt, TU, Diss., 2011]

Druckausg.: München, Verl. Dr. Hut, 2011, ISBN 978-3-86853-842-7 [Darmstadt, TU, Diss., 2011]