Rank Generating Functions as Weakly Holomorphic Modular Forms

Introduction and statement of results. Recent works have illustrated that the Fourier coefficients of harmonic weak Maass forms of weight 1/2 contain a wealth of number-theoretic and combinatorial information. After these works, it is known that many enigmatic q-series (the “mock theta functions” of Ramanujan, and certain rank-generating functions from the theory of partitions, for example) arise naturally as the “holomorphic parts” of such forms. See, for example, Bringmann and Ono [5, 6], Bringmann, Ono, and Rhoades [7], Zwegers [19], Bringmann and Lovejoy [4], Lovejoy and Osburn [12], or see the survey paper [13] for an overview. As another striking example, Bruinier and Ono [9] show that the coefficients of the holomorphic parts of weight 1/2 Maass forms determine the fields of definition of certain Heegner divisors in the Jacobians of modular curves, which in turn determine the vanishing or non-vanishing of derivatives of modular L-function