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A “class group” obstruction for the equation $$Cy^d=F(x,z)$$. (English) Zbl 1204.11063
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.
##### MSC:
 11D41 Higher degree equations; Fermat’s equation 11R29 Class numbers, class groups, discriminants
##### Software:
ecdata; PARI/GP; Magma
Full Text:
##### References:
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