BORCHERDS PRODUCTS FOR U(1,1)

作者：Hofmann Eric^*

来源：International Journal of Number Theory, 2013, 9(7): 1801-1820.

DOI：10.1142/S1793042113500589

• Borcherds product
• unitary modular form
• Heegner-divisor
• CM point
• unitary modular variety

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 q(-n) + O(q), n > 0, and for a constant function, which together span the input space, M-0(!) (Gamma). The general case for the lifting of an arbitrary f is an element of M-0(!) (Gamma) 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.

• 出版日期2013-11