Related papers: Zagier duality for harmonic weak Maass forms of in…
A modular grid is a pair of sequences $(f_m)_m$ and $(g_n)_n$ of weakly holomorphic modular forms such that for almost all $m$ and $n$, the coefficient of $q^n$ in $f_m$ is the negative of the coefficient of $q^m$ in $g_n$. Zagier proved…
We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients…
Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more…
In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which…
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…
While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating…
Second-order automorphic forms are similar to the usual automorphic forms but have a weaker automorphy condition. We answer a question of Zagier and find the dimensions of spaces of holomorphic, even weight, second-order forms. We also…
In this paper, we study the parallel cases of Zagier's and Folsom-Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their $p$-adic properties under the action of Hecke…
Let $q:=e^{2 \pi iz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $\Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th…
Duke and the second author defined a family of linear maps from spaces of weakly holomorphic modular forms of negative integral weight and level 1 into spaces of weakly holomorphic modular forms of half integral weight and level 4 and…
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…
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$,…
Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where…
Recently, Bruinier and Ono proved that the coefficients of certain weight -1/2 harmonic weak Maa{\ss} forms are given as "traces" of singular moduli for harmonic weak Maa{\ss} forms. Here, we prove that similar results hold for the…
In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke…
This note is a compilation of related research on modular relations for multiple zeta values. Roughly speaking, modular relations are (homogeneous) linear relations of multiple zeta values of fixed weight whose coefficients are `originated'…
Zagier proved that the generating series for the traces of singular moduli is a \textit{weakly holomorphic} modular form of weight 3/2 on $\Gamma_0(4)$. Bruinier and Funke extended the results of Zagier to modular curves of arbitrary genus.…
Recently, Mertens, Ono, and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a…
Let $M_k^\sharp(N)$ be the space of weight $k$, level $N$ weakly holomorphic modular forms with poles only at the cusp at $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of…
After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group…