For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ 2−k (resp. D k−1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are… (More)
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as " generating functions " for central values and derivatives of quadratic twists of weight 2 modular… (More)
In this paper we study the distribution of the coefficients a(n) of half integral weight modular forms modulo odd integers M. As a consequence we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in [O-S]. Moreover, we find a simple criterion… (More)
Generalizing work of Gross–Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to quadratic spaces over an arbitrary totally real base field. As a special case,… (More)
Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan's mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass… (More)
Recently, the authors  constructed generalized Borcherds products where modular forms are given as infinite products arising from weight 1/2 harmonic Maass forms. Here we illustrate the utility of these results in the special case of Ramanujan's mock theta function ω(q). We obtain identities and congruences modulo 512 involving the coefficients of ω(q).