Rational solutions of certain Diophantine equations involving norms

Maciej Ulas

Acta Arithmetica, Tome 166 (2014), p. 47-56 / Harvested from The Polish Digital Mathematics Library

Résumé

We present some results concerning the unirationality of the algebraic variety $_{f}$ given by the equation $N_{K/k} (X₁ + αX₂ + α²X₃) = f(t)$, where k is a number field, K=k(α), α is a root of an irreducible polynomial h(x) = x³ + ax + b ∈ k[x] and f ∈ k[t]. We are mainly interested in the case of pure cubic extensions, i.e. a = 0 and b ∈ k∖k³. We prove that if deg f = 4 and $_{f}$ contains a k-rational point (x₀,y₀,z₀,t₀) with f(t₀)≠0, then $_{f}$ is k-unirational. A similar result is proved for a broad family of quintic polynomials f satisfying some mild conditions (for example this family contains all irreducible polynomials). Moreover, the unirationality of $_{f}$ (with a non-trivial k-rational point) is proved for any polynomial f of degree 6 with f not equivalent to a polynomial h satisfying h(t) = h(ζ₃t), where ζ₃ is the primitive third root of unity. We are able to prove the same result for an extension of degree 3 generated by a root of the polynomial h(x) = x³ +ax + b ∈ k[x], provided that f(t) = t⁶ + a₄t⁴ + a₁t + a₀ ∈ k[t] with a₁a₄ ≠ 0.

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Publié le : 2014-01-01

Zbl 06345771

EUDML-ID : urn:eudml:doc:279029