Related papers: Heegner divisors, $L$-functions and harmonic weak …

We introduce the $L$-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic…

We show the existence of "Zagier duality" between vector valued harmonic weak Maass forms and vector valued weakly holomorphic modular forms of integral weight. This duality phenomenon arises naturally in the context of harmonic weak Maass…

Moduli spaces of stable coherent sheaves on a surface are of much interest for both mathematics and physics. Yoshioka computed generating functions of Poincare polynomials of such moduli spaces if the surface is the projective plane P2 and…

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in…

We show that certain space of vector valued harmonic weak Maass forms of half integral weight is isomorphic to a space of scalar valued ones whose Fourier coefficients are supported on suitable progressions. This kind of result for…

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish,…

We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a…

The primary goal of this paper is to construct the basis of the space of weight 3/2 mock modular forms which is an extension of the Borcherd-Zagier basis of weight 3/2 weakly holomorphic modular forms. The shadows of the members of this…

We construct a natural basis for the space of weak harmonic Maass forms of weight 5/2 on the full modular group. The non-holomorphic part of the first element of this basis encodes the values of the ordinary partition function p(n). We…

In this paper, we use theta integrals to give a different construction of mock Maass forms studied by Sander Zwegers. With this method, we construct new real-analytic modular forms, whose Fourier coefficients are logarithms of algebraic…

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the "central $L$-value" of the modular $j$-invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this…

In 1975, Zagier introduced the highly influential hyperbolic Poincar\'e series $f_{k,D}$. We connect the divisor modular form of $f_{k,D}$ to a new weak Maass form $\omega_{k+1,D}$. Furthermore, we show that the generating function of…

Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these…

In this article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein…

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating their generating functions as well…

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight $2$ newforms under the $\xi$-operator. Our work uses a new non-analytic completion of the Kleinian $\zeta$-function from the…

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…

In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle…

We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface…

In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from $q$-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular…