Related papers: Non-repetition of second coefficients of Hecke pol…

Let $T_m(N, k, \chi)$ be the $m$-th Hecke operator of level $N$, weight $k \ge 2$, and nebentypus $\chi$, where $N$ is coprime to $m$. We first show that for any given $m \ge 1$, the second coefficient of the characteristic polynomial of…

For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $\chi$ be a Dirichlet character modulo $N$ with $\chi(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,\chi)$ denote the restriction of the $m$-th Hecke operator to the space…

In this paper, we define the multiplicative Hecke operators $\mathcal{T}(n)$ for any positive integer on the integral weight meromorphic modular forms for $\Gamma_{0}(N)$. We then show that they have properties similar to those of additive…

Let $f_1,...,f_d$ be an orthogonal basis for the space of cusp forms of even weight $2k$ on $\Gamma_0(N)$. Let $L(f_i,s)$ and $L(f_i,\chi,s)$ denote the $L$-function of $f_i$ and its twist by a Dirichlet character $\chi$, respectively. In…

We study the recurrence coefficients of the monic polynomials $P_n(z)$ orthogonal with respect to the deformed (also called semi-classical) Freud weight \begin{equation*} w_{\alpha}(x;s,N)=|x|^{\alpha}{\rm…

We study the Hankel determinant and orthogonal polynomials with respect to the two-parameter weight function $$ w(x)=w(x;t_1, t_2):=\exp(-x^6-t_2 x^4-t_1 x^2),\qquad x\in\mathbb{R}, $$ with $t_1,\; t_2 \in \mathbb{R}$. This problem arises…

Let T^{N,chi}_{p,k}(x) be the characteristic polynomial of the Hecke operator T_p acting on the space of cusp forms S_k(N,chi). We describe the factorization of T^{N,chi}_{p,k}(x) mod l as k varies, and we explicitly calculate those…

We prove that the trace of the Hecke operator $T_2$ acting on the vector space of cusp forms of level one takes no repeated values, except for 0, which only occurs when the space is trivial.

We shall introduce and study certain truncated sums of Hecke eigenvalues of $GL_2$-automorphic forms along quadratic polynomials. A power saving estimate is established and new applications to moments of critical $L$-values associated to…

Inspired by Borcherds' questions, Guerzhoy constructed a new type of Hecke operators $\mathcal{T}(p)$, called the multiplicative Hecke operators, which acts on the space of meromorphic modular forms on the full modular group ${\rm SL}(\Z)$.…

Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k for the full modular group. We show that if there exists n>1 such that T_{n,k}(X) is irreducible and has the full…

In this paper, we study the polynomial representation of the double affine Hecke algebra of type $(C^\vee_n, C_n)$ for specialized parameters. Inductively and combinatorially, we give a linear basis of the representation in terms of linear…

This paper treats the problem of determining conditions for the Fourier coefficients of a Maass-Hecke newform at cusps other than infinity to be multiplicative. To be precise, the Fourier coefficients are defined using a choice of matrix in…

We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition…

In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of…

Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if…

The affine Hecke algebra of type $A$ has two parameters $\left( q,t\right) $ and acts on polynomials in $N$ variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys-Murphy…

In arXiv:1603.03910 [math.NT] we introduced some $C_{n}$ in $Z/2[t]$ defined by a linear recurrence and showed that each $C_{n}$, $n\equiv 0 \bmod{4}$, is a sum of $C_{k}$, $k<n$. Combining this with results from arXiv:1508.07523 [math.NT]…

We define Hecke operators U_m that sift out every m-th Taylor series coefficient of a rational function in one variable, defined over the reals. We prove several structure theorems concerning the eigenfunctions of these Hecke operators,…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…