Unit computation in purely cubic function fields of unit rank 1

1998 ◽  
pp. 592-606
Author(s):  
Renate Scheidler ◽  
Andreas Stein
Keyword(s):  
Function Fields ◽  
Cubic Function Fields ◽  
Cubic Function

scholarly journals Tabulation of cubic function fields via polynomial binary cubic forms

2012 ◽  
Vol 81 (280) ◽  
pp. 2335-2359 ◽  
Author(s):  
Pieter Rozenhart ◽  
Michael Jacobson ◽  
Renate Scheidler
Keyword(s):  
Function Fields ◽  
Cubic Forms ◽  
Cubic Function Fields ◽  
Cubic Function

An Explicit Treatment of Cubic Function Fields with Applications

2010 ◽  
Vol 62 (4) ◽  
pp. 787-807 ◽  
Author(s):  
E. Landquist ◽  
P. Rozenhart ◽  
R. Scheidler ◽  
J. Webster ◽  
Q. Wu
Keyword(s):  
Function Fields ◽  
Standard Form ◽  
Irreducible Polynomial ◽  
Zeta Functions ◽  
Cube Root ◽  
Integral Bases ◽  
Root Extraction ◽  
Necessary And Sufficient Criteria ◽  
Cubic Function Fields ◽  
Cubic Function

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

scholarly journals The arithmetic of certain cubic function fields

2003 ◽  
Vol 73 (245) ◽  
pp. 387-413 ◽  
Author(s):  
Mark L. Bauer
Keyword(s):  
Function Fields ◽  
Cubic Function Fields ◽  
Cubic Function

EFFECTIVE DETERMINATION OF PRIME DECOMPOSITIONS OF CUBIC FUNCTION FIELDS

2010 ◽  
Vol 06 (02) ◽  
pp. 437-448
Author(s):  
YUSHENG ZHAO ◽  
WEI LI ◽  
XIANKE ZHANG
Keyword(s):  
Function Fields ◽  
Explicit Methods ◽  
Prime Decomposition ◽  
Cubic Function Fields ◽  
Cubic Function

In this paper, we determine completely the prime decomposition of cubic function fields by effective and explicit methods.

scholarly journals Cubic function fields with prescribed ramification

2021 ◽  
pp. 1-35
Author(s):  
Valentijn Karemaker ◽  
Sophie Marques ◽  
Jeroen Sijsling
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Explicit Procedure ◽  
Cubic Function Fields ◽  
Cubic Function ◽  
Elliptic And Hyperelliptic Curves

This paper describes cubic function fields [Formula: see text] with prescribed ramification, where [Formula: see text] is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure [Formula: see text] of [Formula: see text] is of genus zero, and a description of the twists of [Formula: see text] up to isomorphism over [Formula: see text]. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on [Formula: see text]. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

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