Related papers: The second Dirichlet coefficient starts out negati…

It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along…

We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $\ell$. Among the applications of this we show that there are $\gg \sqrt{X}/\log \log…

We prove an extension of Hoermander's classical result on hypoelliptic second order equations, where the coefficients of the related vector fields are globally Lipschitz and satisfy the classical Hoermander condition on a dense set while…

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 vanishing of Fourier coefficients of holomorphic $\eta$-quotients. We investigate examples of two different types: the first one involves integral weight CM newforms, while the second one involves half-integral…

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the…

The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.

For a second-order elliptic equation of nondivergence form in the plane, we investigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a…

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

The Zygmund functions form an intermediate class between Lipschitz and H\"older functions; their second order divided differences are uniformly bounded. It is well known that for $d \geq 1$ the graph of any Lipschitz function $f:\R^d…

Let $f$ be a normalized newform of weight 2 on $\Gamma_0(N)$ whose coefficients lie in $\mathbb{Q}$ and let $\chi_M$ be a primitive quadratic Dirichlet character with conductor $M$. In this paper, under mild assumptions on $M$, we give a…

In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincar\'e series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which…

We consider second-order divergence form uniformly parabolic and elliptic PDEs with bounded and $VMO_{x}$ leading coefficients and possibly linearly growing lower-order coefficients. We look for solutions which are summable to the $p$th…

A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a…

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.

Let $f$ be a normalized primitive Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,\mathbb{Z})$. For integers $j \geq 2$, let $\lambda_{sym^j f}(m)$ denote the $m$th Fourier coefficient of the $j$th…

This work is a survey of relations between Drinfeld modules and higher dimensional fields of positive characteristic. The main new result stated is the expression of vanishing orders of certain modular forms through partial zeta values.

We study one-parameter families of elliptic curves over $\mathbb{Q}(T)$, which are of the form $y^2=x^3+A(T)x+B(T)$, with non-constant $j$-invariant. We define the $r$\textsuperscript{th} moment of an elliptic curve to be $A_{r,E}(p) :=…

Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on…

The concept of a modular value of an observable of a pre- and post-selected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an…