Related papers: Harmonic weak Maass forms and periods II

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite…

In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.

We give an explicit construct of a harmonic weak Maass form $F_{\Theta}$ that is a "lift" of $\Theta^3$, where $\Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $\Theta^3$ are related to class numbers of…

In this paper, we consider the Fourier coefficients of meromorphic Jacobi forms of negative index. This extends recent work of Creutzig and the first two authors for the special case of Kac-Wakimoto characters which occur naturally in Lie…

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…

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted…

In this paper, we provide an explicit construction of weight $0$ meromorphic modular forms. Following work of Petersson, we build these via Poincar\'e series. There are two main aspects of our investigation which differ from his approach.…

We study the sum of additively twisted Fourier coefficients of a symmetric-square lift of a Maass form invariant under the full modular group. Our bounds are uniform in terms of the spectral parameter of the Maass form, as well as in terms…

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $\mathbb{Z}+\frac{1}{2}$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental…

We discuss polyharmonic Maass forms of even integer weight on PSL(2, Z)\H, which are a generalization of classical Maass forms. We explain the role of real-analytic Eisenstein series E_k(z, s) and the differential operator…

We generalize the notions of locally and polar harmonic Maass forms to general orthogonal groups of signature $(2, n)$ with singularities along real analytic and algebraic cycles. We prove a current equation for locally harmonic Maass forms…

This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a…

We prove a spectral summation formula for the product of four Fourier coefficients of half-integral weight cusp forms in Kohnen's subspace. The other side of the formula involves certain generalized class numbers of pairs of quadratic forms…

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has…

In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincar\'e series. This relates to Rademacher's "expansion of zero" principle, which was recently…

Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper,…

This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression…

We investigate non-correlation of Fourier coefficients of Maass forms against a class of real oscillatory functions, in analogy to known results with Frobenius trace functions. We also establish an equidistribution result for twisted…

In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit…