scholarly journals On the splitting rate of rate of a tower of Artin-Schreier type

2020 ◽  
Vol 44 (173) ◽  
pp. 1167-1173
Author(s):  
Horacio Navarro
Keyword(s):  
Finite Field ◽  
Asymptotic Behaviour ◽  
Function Fields ◽  
Rational Places ◽  
Tower Of Function Fields

In this note we study the asymptotic behaviour of the number of rational places in a tower of function fields of Artin-Schreier type over a finite field with 2s elements, where s > 0 is an odd integer.

scholarly journals ON INVARIANTS OF TOWERS OF FUNCTION FIELDS OVER FINITE FIELDS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250190 ◽  
Author(s):  
FLORIAN HESS ◽  
HENNING STICHTENOTH ◽  
SEHER TUTDERE
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Function Fields ◽  
Finite Extension ◽  
Towers Of Function Fields ◽  
Asymptotic Number ◽  
Tower Of Function Fields

In this paper, we consider a tower of function fields [Formula: see text] over a finite field 𝔽q and a finite extension E/F0 such that the sequence [Formula: see text] is a tower over the field 𝔽q. Then we study invariants of [Formula: see text], that is, the asymptotic number of the places of degree r in [Formula: see text], for any r ≥ 1, if those of [Formula: see text] are known. We first give a method for constructing towers of function fields over any finite field 𝔽q with finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽q attaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with qr a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.

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.

scholarly journals Number fields and function fields: coalescences, contrasts and emerging applications

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140315 ◽  
Author(s):  
J. P. Keating ◽  
Z. Rudnick ◽  
T. D. Wooley
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Analytic Number Theory ◽  
Number Fields ◽  
Past Half Century ◽  
Recent Developments ◽  
Substantial Progress ◽  
And Function ◽  
Past Half ◽  
Field Setting

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained.

scholarly journals Weierstrass Semigroups in an Asymptotically Good Tower of Function Fields

1998 ◽  
Vol 4 (4) ◽  
pp. 381-392 ◽  
Author(s):  
Ruud Pellikaan ◽  
Henning Stichtenoth ◽  
Fernando Torres
Keyword(s):  
Function Fields ◽  
Weierstrass Semigroups ◽  
Tower Of Function Fields
Sign in / Sign up

Export Citation Format

Share Document