LIMITS OF TRACES OF SINGULAR MODULI

Abstract

Let f and g be weakly holomorphic modular functions on Gamma(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 infinity under the action of Gamma(0)(4N). In this paper, we prove that when z goes radially to r, the limit Q((H) over cap (f))(r) of the sum H(f)(z) = Sigma(d>0) Tr-d(f)e(2 pi idz) 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 Q((H) over cap (f))(r) = Q((H) over cap (g))(r) for all r equivalent to i infinity under the action of Gamma(0)(4N), then Tr-d(f) = Tr-d(g) for all integers d.