Limits of traces of singular moduli

Abstract

Let f and g be weakly holomorphic modular functions on Γ 0 (N ) with the trivial character. For an integer d, let Tr d (f ) denote the modular trace of f of index d. Let r be a rational number equivalent to i∞ under the action of Γ 0 (4N ). In this paper, we prove that, when z goes radially to r, the limit QĤ (f ) (r) of the sum H(f )(z) = d>0 Tr d (f )e 2πidz is a special value of a regularized twisted L-function defined by Tr d (f ) for d ≤ 0. It is proved that the regularized L-function is meromorphic on C and satisfies a certain functional equation. Finally, under the assumption that N is square free, we prove that if QĤ (f ) (r) = QĤ (g) (r) for all r equivalent to i∞ under the action of Γ 0 (4N ), then Tr d (f ) = Tr d (g) for all integers d.