On the distribution of the of Frobenius elements on elliptic curves over function fields

Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $\phi:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the fiber $\mathcal{E}_P=\phi^{-1}(P)$ is smooth, the eigenvalues of the zeta-function of $\mathcal{E}_P$ over the residue field $\kappa_P$ of $P$ are of the form $q_P^{1/2}e^{i\theta_P},q_{P}e^{-i\theta_P}$, where $q_P=q^{\deg(P)}$ and $0\le\theta_P\le\pi$. The goal of this note is to determine given an integer $B\ge 1$, $\alpha,\beta\in[0,\pi]$ the number of $P\in C$ where the reduction of $E$ is good and such that $\deg(P)\le B$ and $\alpha\le\theta_P\le\beta$.Comment: 8 page

Rational points on curves over function fields

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

Finite quotients of the algebraic fundamental group of projective curves in positive characteristic

Let X be a smooth connected projective curve defined over an algebraically closed field k of characteristic p >0. Let G be a finite group whose order is divisible by p. Suppose that G has a normal p-Sylow subgroup. We give a necessary and sufficient condition for G to be a quotient of the algebraic fundamental group pi(1)(X) of X

PENSAMIENTO CRÍTICO EN LA INVESTIGACIÓN CIENTÍFICA Y ACADÉMICA COLECCIÓN CIENTÍFICA EDUCACIÓN, EMPRESA Y SOCIEDAD

PENSAMIENTO CRÍTICO EN LA INVESTIGACIÓN CIENTÍFICA Y ACADÉMICA COLECCIÓN CIENTÍFICA EDUCACIÓN, EMPRESA Y SOCIEDAD Primera Edición 2023 Vol. 21 Editorial EIDEC Sello Editorial EIDEC (978-958-53018) NIT 900583173-1 ISBN: 978-628-95884-1-5 Formato: Digital PDF (Portable Document Format) DOI: https://doi.org/10.34893/e1150-3660-8721-s Publicación: Colombia Fecha Publicación: 13/09/2023 Coordinación Editorial Escuela Internacional de Negocios y Desarrollo Empresarial de Colombia – EIDEC Centro de Investigación Científica, Empresarial y Tecnológica de Colombia – CEINCET Red de Investigación en Educación, Empresa y Sociedad – REDIEES Revisión y pares evaluadores Centro de Investigación Científica, Empresarial y Tecnológica de Colombia – CEINCET Red de Investigación en Educación, Empresa y Sociedad – REDIEE