Related papers: Multiplicative Hecke operators and their applicati…
This is basically a summary of [Mu]. The focus of the paper is the explicit computation of Hecke operators for period functions. In particular we compute the matrix representations of the 2nd Hecke operator on period functions for the full…
The Fourier coefficients of a Maass form $\phi$ for SL$(n,\mathbb Z)$ are complex numbers $A_\phi(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form…
Physicists showed that the generating function of orbifold elliptic genera of symmetric orbifolds can be written as an infinite product. We show that there exists a geometric factorization on space level behind this infinite product formula…
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…
The aim of this paper is to show how differential characters of Abelian varieties can be used to construct differential modular forms of weight 0 and order 2 which are eigenvectors of Hecke operators. These differential modular forms have…
We study the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups. We provide representations as polynomials in the false Eisenstein series with coefficients in the space of Drinfeld modular forms (the…
In the article, we consider a question concerning the estimation of summatory function of the Fourier coefficients of Hecke eigenforms indexed by a sparse set of integers. In particular, we provide an estimate for the following sum;…
We investigate non-vanishing properties of $L(f,s)$ on the real line, when $f$ is a Hecke eigenform of half-integral weight $k+{1\over 2}$ on $\Gamma_0(4).$
Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight…
Let P be a simple lattice polytope. We define an action of the Hecke operators on E (P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E (P). We also describe how the Brion-Vergne formula transforms under the…
In this paper we want to define the Kohnen plus space for Hilbert modular forms with a odd square-free level and a quadratic character by a representation-theoretic way. We will show that in the classical case the one we defined is the same…
In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form.…
We present an explicit set of matrices giving the action of the Hecke operators $T(p)$, $T_j(p^2)$ on Siegel modular forms.
We present a new proof of the "arithmetic" large sieve inequality, starting from the corresponding "harmonic" inequality, which is based on an amplification idea. We show that this also adapts to give some new sieve inequality for modular…
For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular…
In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in…
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…
In this note, we explicitly construct mock modular forms with integral Fourier coefficients by evaluating regularized Petersson inner products involving their shadows, which are unary theta functions of weights 1/2 and 3/2 . In addition, we…
We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak…
For each prime $p$, we determine the distribution of the $p^{th}$ Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As $p\to\infty$, this distribution tends to the Sato--Tate distribution.