scholarly journals Computing Singular Elements Modulo Squares

2021 ◽  
Vol 179 (3) ◽  
pp. 227-238
Author(s):  
Przemysław Koprowski
Keyword(s):  
Function Field ◽  
Global Function ◽  
Global Function Field

The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtwängler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and S-singular elements (modulo squares) in a global function field.

Zsigmondy theorem for arithmetic dynamics induced by a drinfeld module

2019 ◽  
Vol 15 (06) ◽  
pp. 1111-1125
Author(s):  
Zhengjun Zhao ◽  
Qingzhong Ji
Keyword(s):  
Prime Ideal ◽  
Function Field ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Arithmetic Dynamics ◽  
Global Function ◽  
Ideal Formula ◽  
Global Function Field ◽  
The Ideal

Let [Formula: see text] be a Drinfeld [Formula: see text]-module defined over a global function field [Formula: see text] Let [Formula: see text] be a non-torsion point of [Formula: see text] with infinite [Formula: see text]-orbit. For each [Formula: see text] write the ideal [Formula: see text] as a quotient of relatively prime integral ideals. We establish an analogue of the classical Zsigmondy theorem for the ideal sequence [Formula: see text] i.e. for all but finitely many [Formula: see text] there exists a prime ideal [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]

scholarly journals Wild sets in global function fields

2020 ◽  
Vol 70 (2) ◽  
pp. 259-272
Author(s):  
Alfred Czogała ◽  
Przemysław Koprowski ◽  
Beata Rothkegel
Keyword(s):  
Function Field ◽  
Function Fields ◽  
The Self ◽  
Global Function ◽  
Global Function Fields ◽  
Global Function Field ◽  
Set Of Points

Abstract Given a self-equivalence of a global function field, its wild set is the set of points where the self-equivalence fails to preserve parity of valuation. In this paper we describe structure of finite wild sets.

scholarly journals On the Computation of Integral Closures of Cyclic Extensions of Function Fields

2007 ◽  
Vol 10 ◽  
pp. 141-160
Author(s):  
Robert Fraatz
Keyword(s):  
Function Field ◽  
Strong Approximation ◽  
Approximation Theorem ◽  
Function Fields ◽  
Proper Subset ◽  
Global Function ◽  
Global Function Field ◽  
Integral Closures ◽  
Strong Approximation Theorem

AbstractLet S be a non-empty proper subset of the set of places of a global function field F and E a cyclic Kummer or Artin–Schreier–Witt extension of F. We present a method of efficiently computing the ring of elements of E which are integral at all places of S. As an important tool, we include an algorithmic version of the strong approximation theorem. We conclude with several examples.

scholarly journals Zero-cycles and rational points on some surfaces over a global function field

2012 ◽  
Vol 155 (1) ◽  
pp. 63-70
Author(s):  
J.-L. Colliot-Thélène ◽  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Function Field ◽  
Rational Points ◽  
Global Function ◽  
Global Function Field
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