Related papers: On $L$-functions of certain exponential sums

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…

Hooley proved that if $f\in \Bbb Z [X]$ is irreducible of degree $\ge 2$, then the fractions $\{ r/n\}$, $0<r<n$ with $f(r)\equiv 0\pmod n$, are uniformly distributed in $(0,1)$. In this paper we study such problems for reducible…

Zeros of Bessel functions $J_\alpha$ play an important role in physics. They are a motivation for studying zeros of exponential polynomials defined over $\overline{\mathbb{Q}}$, and more generally of $E$-functions. In this paper we…

Given an entire function $f$ of finite order $\rho$, let $L(z,f)=\sum_{j=0}^{m}b_{j}(z)f^{(k_{j})}(z+c_{j})$ be a linear delay-differential polynomial of $f$ with small coefficients in the sense of $O(r^{\lambda+\varepsilon})+S(r,f)$,…

We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z),…

We relate the $p$-adic heights of generalized Heegner cycles to the derivative of a $p$-adic $L$-function attached to a pair $(f, \chi)$, where $f$ is an ordinary weight $2r$ newform and $\chi$ is an unramified imaginary quadratic Hecke…

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$, and let $\mathbb{F}_q[t][u]$ be the…

For a newform for Gamma_0(N) of even weight k, we prove that its attached p-adic L-function is not identically zero on the group Z_p of the p-adic units. If p >3, we prove that the order of vanishing at any p-adic integer is finite.

In this paper, for the generalized Fibonacci sequence $\left\{W_n\left(a,b,p,q\right)\right\}$, by using elementary methods and techniques, we give the asymptotic estimation values of…

In this document will be proved a formula to compute the $p$-adic valuation of a hyperfactorial. We call a hyperfactorial the result of multiplying a given number of consecutive integers from 1 to the given number,each raised to its own…

Let $A$ be a finite subset of $\ffield$, the field of Laurent series in $1/t$ over a finite field $\mathbb{F}_q$. We show that for any $\epsilon>0$ there exists a constant $C$ dependent only on $\epsilon$ and $q$ such that…

In this note we define anticyclotomic p-adic measures attached to a finite set of places S above p, a modular elliptic curve E over a general number field F and a quadratic extension K/F. We study the exceptional zero phenomenon that arises…

Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic…

This article studies the finite--slope analogue of Loeffler's conjectural framework for Rankin--Selberg $p$-adic $L$-functions in universal deformation families. Starting from residual representations $\bar\rho_1,\bar\rho_2$ of tame…

Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this…

The classical $n$-variable Kloosterman sums over finite fields are well understood by Deligne's theorem from complex point of view and by Sperber's theorem from $p$-adic point of view. In this paper, we study the complex and $p$-adic…

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $\Gamma_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the…

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer…

Functionals with values in Non-Archimedean field of Laurent series applied to the definition of generalized solution (in the form of soliton and shock wave) of the Hopf equation and equations of elasticity theory. Calculation method for the…

We construct a $p$-adic Rankin-Selberg $L$-function associated to the product of two families of modular forms, where the first is an ordinary (Hida) family, and the second an arbitrary universal-deformation family (without any ordinarity…