scholarly journals Generic polynomials for cyclic function field extensions over certain finite fields

2017 ◽  
Vol 4 (2) ◽  
pp. 585-602 ◽  
Author(s):  
Sophie Marques
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Field Extensions ◽  
Generic Polynomials ◽  
Cyclic Function

scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

scholarly journals Function Field Sieve Method for Discrete Logarithms over Finite Fields

1999 ◽  
Vol 151 (1-2) ◽  
pp. 5-16 ◽  
Author(s):  
Leonard M. Adleman ◽  
Ming-Deh A. Huang
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Sieve Method ◽  
Discrete Logarithms

scholarly journals On the multiplicative order of elements in Wiedemann's towers of finite fields

2015 ◽  
Vol 7 (2) ◽  
pp. 220-225
Author(s):  
R. Popovych
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Field Extensions ◽  
Fermat Numbers ◽  
Multiplicative Order ◽  
Order Of Elements

We consider recursive binary finite field extensions $E_{i+1} =E_{i} (x_{i+1} )$, $i\ge -1$, defined by D. Wiedemann. The main object of the paper is to give some proper divisors of the Fermat numbers $N_{i} $ that are not equal to the multiplicative order $O(x_{i} )$.

On the Local Behavior of Specializations of Function Field Extensions

2018 ◽  
Vol 2019 (9) ◽  
pp. 2951-2980 ◽  
Author(s):  
Joachim König ◽  
François Legrand ◽  
Danny Neftin
Keyword(s):  
Function Field ◽  
Local Behavior ◽  
Field Extensions

scholarly journals THE FOURTH MOMENT OF DIRICHLET L-FUNCTIONS FOR THE RATIONAL FUNCTION FIELD

2014 ◽  
Vol 10 (01) ◽  
pp. 183-218 ◽  
Author(s):  
NATTALIE TAMAM
Keyword(s):  
Lower Bound ◽  
Asymptotic Formula ◽  
Rational Function ◽  
Finite Fields ◽  
Polynomial Ring ◽  
Function Field ◽  
Fourth Moment ◽  
Rational Function Field ◽  
Central Value

We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We obtain an asymptotic formula of the fourth moment for the central value of these Dirichlet L-functions. In addition, we find a lower bound for the 2k th moment of these L-functions. These results agree up to constants with the polynomial ring analog of the Keating and Snaith Conjecture for the asymptotic of leading terms.

scholarly journals Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

2020 ◽  
Vol 156 (4) ◽  
pp. 733-743
Author(s):  
John R. Doyle ◽  
Bjorn Poonen
Keyword(s):  
Finite Fields ◽  
Function Field ◽  
Uniform Boundedness ◽  
Number Fields ◽  
Periodic Points ◽  
Irreducible Components

Fix $d\geqslant 2$ and a field $k$ such that $\operatorname{char}k\nmid d$. Assume that $k$ contains the $d$th roots of $1$. Then the irreducible components of the curves over $k$ parameterizing preperiodic points of polynomials of the form $z^{d}+c$ are geometrically irreducible and have gonality tending to $\infty$. This implies the function field analogue of the strong uniform boundedness conjecture for preperiodic points of $z^{d}+c$. It also has consequences over number fields: it implies strong uniform boundedness for preperiodic points of bounded eventual period, which in turn reduces the full conjecture for preperiodic points to the conjecture for periodic points. Our proofs involve a novel argument specific to finite fields, in addition to more standard tools such as the Castelnuovo–Severi inequality.

The Sato-Tate distribution in thin families of elliptic curves over high degree extensions of finite fields

2019 ◽  
Vol 15 (03) ◽  
pp. 469-477
Author(s):  
Igor E. Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Prime Fields ◽  
High Degree

Over the last two decades, there has been a wave of activity establishing the Sato-Tate kind of distribution in various families of elliptic curves over prime fields. Typically the goal here is to prove this for families which are as thin as possible. We consider a function field analogue of this question, that is, for high degree extensions of a finite field where new effects allow us to study families, which are much thinner that those typically investigated over prime fields.

scholarly journals Elementary properties of power series fields over finite fields

2001 ◽  
Vol 66 (2) ◽  
pp. 771-791 ◽  
Author(s):  
Franz-Viktor Kuhlmann
Keyword(s):  
Power Series ◽  
Finite Fields ◽  
Open Problem ◽  
Special Interest ◽  
Function Field ◽  
Positive Characteristic ◽  
Axiom System ◽  
Elementary Property ◽  
Ground Field ◽  
Local Uniformization

AbstractIn spite of the analogies between ℚp and which became evident through the work of Ax and Kochen, an adaptation of the complete recursive axiom system given by them for ℚp, to the case of does not render a complete axiom system. We show the independence of elementary properties which express the action of additive polynomials as maps on . We formulate an elementary property expressing this action and show that it holds for all maximal valued fields. We also derive an example of a rather simple immediate valued function field over a henselian defectless ground field which is not a henselian rational function field. This example is of special interest in connection with the open problem of local uniformization in positive characteristic.

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