A “class group” obstruction for the equation C y d = F ( x , z )

Denis Simon[1]

  • [1] LMNO - UMR 6139 Université de Caen – France Campus II – Boulevard Mal Juin BP 5186 – 14032 Caen Cedex

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 3, page 811-828
  • ISSN: 1246-7405

Abstract

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In this paper, we study equations of the form C y d = F ( x , z ) , where F ∈ ℤ [ x , z ] is a binary form, homogeneous of degree n , which is supposed to be primitive and irreducible, and d is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.

How to cite

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Simon, Denis. "A “class group” obstruction for the equation $Cy^d=F(x,z)$." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 811-828. <http://eudml.org/doc/10862>.

@article{Simon2008,
abstract = {In this paper, we study equations of the form $Cy^d = F(x,z)$, where $F\in \mathbb\{Z\}[x,z]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.},
affiliation = {LMNO - UMR 6139 Université de Caen – France Campus II – Boulevard Mal Juin BP 5186 – 14032 Caen Cedex},
author = {Simon, Denis},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {811-828},
publisher = {Université Bordeaux 1},
title = {A “class group” obstruction for the equation $Cy^d=F(x,z)$},
url = {http://eudml.org/doc/10862},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Simon, Denis
TI - A “class group” obstruction for the equation $Cy^d=F(x,z)$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 811
EP - 828
AB - In this paper, we study equations of the form $Cy^d = F(x,z)$, where $F\in \mathbb{Z}[x,z]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result can be used to prove that these equations have no proper solution. Numerous examples are given to illustrate this result. In a second part, we make a link between this condition and the properties of the different in the considered number field.
LA - eng
UR - http://eudml.org/doc/10862
ER -

References

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  11. MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/magma/ 
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  13. pari/gp, The Pari group (K. Belabas, H. Cohen,...). http://pari.math.u-bordeaux.fr/ 
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  15. D. Simon, La classe invariante d’une forme binaire. Comptes Rendus Mathématiques 336, Issue 1 , (2003) 7–10. Zbl1038.11073MR1968893

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