Traces of singular values and Borcherds products

Abstract

Let $p$ be a prime for which the congruence group ${\Gamma }_0(p{)}^{\ast }$ is of genus zero, and $j_p^{\ast }$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on ${\Gamma }_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^{\ast }$ and extend Zagier's trace formula for singular values of $j_p^{\ast }$. Further we investigate the twisted traces of sigular values of $j_p^{\ast }$ and construct Borcherds products related to them.

21 pages