A ``class group'' obstruction for the equation \(Cy^d=F(x,z)\)
type: Item
instance of: scholarly article
title: A ``class group'' obstruction for the equation \(Cy^d=F(x,z)\)
zbMATH Open document ID: 1204.11063
author: Denis Simon
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-06-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10862
review text: In this paper, Diophantine equations of the form \[ Cy^d = F(x,z)\tag{1} \] are investigated, where \(F \in \mathbb Z[x,z]\) is an irreducible, primitive form of degree \(n \geq 2\), and \(C \neq 0\), \(d\geq2\) are integers. Let \(K = \mathbb Q (\theta)\) be the algebraic number field, given by \(F(\theta, 1)=0\). The author [C. R., Math., Acad. Sci. Paris 336, No. 1, 7--10 (2003; Zbl 1038.11073)] associates to the form \(F\) an order \(\mathbb Z_F\) within the ring of integers of \(K\), an ideal \(\mathfrak b\) of \(\mathbb Z_F\), and defines the class of \(F\) to be the class of \(\mathfrak b\) within the class group \(Cl(\mathbb Z_F)\). Each proper solution \((x_0, y_0, z_0)\) of (ref {1}) gives rise to an ideal \(\mathfrak D = \mathfrak b (x_0- \theta z_0)\) of \(\mathbb Z_F\) with very special properties, as described in Theorem 3. From this, necessary conditions for the existence of proper solutions of (ref {1}) are derived, in more detail, there must exist some ideal \(\mathfrak c\) of \(\mathbb Z_F\) with norm equal to \(C\) and divisible only by degree \(1\) prime ideals, and the classes of \(\mathfrak c\) and of \(F\) should only differ by a \(d\)-th power inside \(Cl(\mathbb Z_F)\). For \(n=2\) this reduces to previous results of Cassels and Darmon-Granville.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5572703
describes a project that uses: PARI/GP
describes a project that uses: Magma
describes a project that uses: ecdata
MaRDI profile type: Publication
OpenAlex ID: W2325070427
cites work: Notes on elliptic curves. I.
cites work: Q4177654
cites work: Q4341641
cites work: Number Theory
cites work: On the Equations zm = F (x, y ) and Axp + Byq = Czr
cites work: Decomposition of primes in non-maximal orders
cites work: Q5830705
cites work: Q4298518
cites work: Explicit 4-descents on an elliptic curve
cites work: Modular forms in Pari/GP
cites work: Q4061092
cites work: The invariant class of a binary form
DOI: 10.5802/JTNB.652
Recommended article: On the Diophantine equation \(Ax^{2m}-Bx^my^m+Cy^{2m}=Dz^2\)
Recommended article: On the Equations zm = F (x, y ) and Axp + Byq = Czr
Recommended article: Q3822247
Recommended article: Certain Diophantine Equations Linear In One Unknown
Recommended article: Q3820680
Recommended article: On the Diophantine equation $f(x)f(y)=f(z^2)$
Recommended article: On the diophantine equation f(x)f(y)=f(z)2
Recommended article: Q3684188
Recommended article: Q3794251
Recommended article: The diophantine equation \(X^2 - DY^2 = Z^m\).
description: scientific article; zbMATH DE number 5572703