STATISTICS OF PRIME DIVISORS IN FUNCTION FIELDS

2009 ◽  
Vol 05 (01) ◽  
pp. 141-152 ◽  
Author(s):  
ROBERT C. RHOADES
Keyword(s):  
Simple Proof ◽  
Function Field ◽  
Function Fields ◽  
Random Polynomial ◽  
Prime Divisors ◽  
Number Of Prime Divisors ◽  
Field Setting ◽  
The Way

We show that the prime divisors of a random polynomial in 𝔽q[t] are typically "Poisson distributed". This result is analogous to the result in ℤ of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdös–Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {ω(f)}f∈𝔽q[t] than was previously known, where ω(f) is the number of prime divisors of f.

Large values of Dirichlet L-functions over function fields

2020 ◽  
Vol 16 (05) ◽  
pp. 1081-1109
Author(s):  
Dragan Đokić ◽  
Nikola Lelas ◽  
Ilija Vrećica
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Function Field ◽  
Function Fields ◽  
Resonance Method ◽  
Dirichlet Characters ◽  
Field Setting

In this paper, we investigate the existence of large values of [Formula: see text], where [Formula: see text] varies over non-principal characters associated to prime polynomials [Formula: see text] over finite field [Formula: see text], as [Formula: see text], and [Formula: see text]. When [Formula: see text], we provide a lower bound for the number of such characters. To do this, we adapt the resonance method to the function field setting. We also investigate this problem for [Formula: see text], where now [Formula: see text] varies over even, non-principal, Dirichlet characters associated to prime polynomials [Formula: see text] over [Formula: see text], as [Formula: see text]. In addition to resonance method, in this case, we use an adaptation of Gál-type sums estimate.

Introduction

2019 ◽  
pp. 1-61
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Finite Field ◽  
Function Field ◽  
Algebraic Curve ◽  
Mass Formula ◽  
Essential Feature ◽  
Function Fields ◽  
Rational Functions ◽  
Algebraic Stack ◽  
Field Setting

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field F q. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with Fq-valued points of an algebraic stack BunG(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of BunG(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals FINITE GROUPS AS GALOIS GROUPS OF FUNCTION FIELDS WITH INFINITE FIELD OF CONSTANTS

2010 ◽  
Vol 88 (3) ◽  
pp. 301-312
Author(s):  
C. ÁLVAREZ-GARCÍA ◽  
G. VILLA-SALVADOR
Keyword(s):  
Finite Groups ◽  
Function Field ◽  
Function Fields ◽  
Galois Groups ◽  
Infinite Field ◽  
Separable Extension ◽  
Separable Extensions

AbstractLetE/kbe a function field over an infinite field of constants. Assume thatE/k(x) is a separable extension of degree greater than one such that there exists a place of degree one ofk(x) ramified inE. LetK/kbe a function field. We prove that there exist infinitely many nonisomorphic separable extensionsL/Ksuch that [L:K]=[E:k(x)] andAutkL=AutKL≅Autk(x)E.

scholarly journals On the number of prime divisors of the order of elliptic curves modulo p

2005 ◽  
Vol 117 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Jörn Steuding ◽  
Annegret Weng
Keyword(s):  
Elliptic Curves ◽  
Prime Divisors ◽  
Modulo P ◽  
Number Of Prime Divisors

scholarly journals The distribution of the number of prime divisors of sums a + b

1988 ◽  
Vol 29 (1) ◽  
pp. 94-99 ◽  
Author(s):  
P.D.T.A Elliott ◽  
A Sárközy
Keyword(s):  
Prime Divisors ◽  
Number Of Prime Divisors

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

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