Related papers: Kloosterman sums and Hecke polynomials in characte…

We prove upper bounds for sums of Kloosterman sums against general arithmetic weight functions. In particular, we obtain power cancellation in sums of Kloosterman sums over arithmetic progressions, which is of square-root strength in any…

We define L-functions for meromorphic modular forms that are regular at cusps, and use them to: (i) find new relationships between Hurwitz class numbers and traces of singular moduli, (ii) establish predictions from the physics of…

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by D\k{a}browski and Reeder. This is achieved by representing the sums in an explicit…

In this paper, we construct a binary linear code connected with the Kloosterman sum for $GL(2,q)$. Here $q$ is a power of two. Then we obtain a recursive formula generating the power moments 2-dimensional Kloosterman sum, equivalently that…

A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of q elements, where q is a power of a prime p > 3. His proof required deep algebro-geometric techniques, and he…

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)}…

In this paper, we investigate the average behavior of the $n^{th}$ normalized Fourier coefficients of the $j^{th}$ ($j \geq 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,sym^{j}f)$), attached to a primitive holomorphic…

We establish existence theorems for the image of the normalized character map of the $p$-adic Heisenberg algebra $S$ taking values in the algebra of Serre $p$-adic modular forms $M_p$. In particular, we describe the construction of an…

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\bar{\rho}_{f}$ is irreducible. Under some additional hypotheses we prove that…

Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of)…

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…

We consider the logarithm of the central value $\log L(1/2)$ in the orthogonal family ${L(s,f)}_{f \in H_k}$ where $H_k$ is the set of weight $k$ Hecke-eigen cusp form for $SL_2(\mathbb{Z})$, and in the symplectic family…

In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.

Let $j\geq 2$ be a given integer. Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma=SL(2,\mathbb{Z})$. Denote by $\lambda_{\text{sym}^{j}f}(n)$ the $n$th normalized…

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are…

In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in $\mathbb{Z}/m\mathbb{Z}$ for arbitrary integers $m$. These results have been motivated by a wide variety of…

Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet L-functions in the context of the calculation of moments and connections with Random Matrix Theory. According…

For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \mathbb{F}_q[x]$ with respect to finite characters $\chi$ when $p$ is larger than an explicit…

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on $\ell$-adic properties…

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-(2n,2^r)$. Then we obtain four infinite…