ﻻ يوجد ملخص باللغة العربية
Given a positive integer $Q$, denote by $mathcal{C}_Q$ the multiplicative cyclic group of order $Q$. Let $n$ be a divisor of $Q$ and $r$ a divisor of $Q/n$. Guided by the well-known formula of Vinogradov for the indicator function of the set of primitive elements of a finite field $mathbb{F}_q$, we derive an expression for the indicator function for the set of $(r,n)$-free elements of $mathcal{C}_Q$, i.e., the subset of the subgroup $mathcal{C}_{Q/n}$ comprising elements that are $r$-free in $mathcal{C}_{Q/n}$, i.e., are not $p$-th powers in $mathcal{C}_{Q/n}$ for any prime $p$ dividing $r$. We deduce a general lower bound for the the number of elements $theta in mathbb{F}_q$ for which $f(theta)$ is $(r,n)$-free and $F(theta)$ is $(R,N)$-free, where $f, F inmathbb{F}_q[x]$ and $n,N$ are divisors of $q-1$ with $rmid (q-1)/n$, $Rmid (q-1)/N$. As an application, we consider the existence of $mathbb{F}_q$-primitive points (i.e., points whose coordinates are primitive elements) on curves like $y^n=f(x)$. In particular, elliptic curves $y^2=f(x)$, where $f$ is a square-free cubic, are studied. We find, for example, all the odd prime powers $q$ for which the elliptic curves $y^2=x^3 pm x$ contain an $mathbb{F}_q$-primitive point.
Let $p=2n+1$ be an odd prime, and let $zeta_{p^2-1}$ be a primitive $(p^2-1)$-th root of unity in the algebraic closure $overline{mathbb{Q}_p}$ of $mathbb{Q}_p$. We let $ginmathbb{Z}_p[zeta_{p^2-1}]$ be a primitive root modulo $pmathbb{Z}_p[zeta_{p^2
Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun a
Let $mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) in mathbb{F}_q[x]$ with $textrm{deg}(f_i(x)) leq m_i$, for $i = 1, 2$, and such that the rational function $f
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $mathbb{Z}$ that has an $mathbb{R}$-point and a $mathbb{Z}_p$-point for every prime $p$ but no $mathbb{Z}$-point. This is best possible: we also prove that any stacky cu