Tome (2007)

Sur les courbes hyperelliptiques cyclotomiques et les équations

x^{p} - y^{p} = c z ²

Wilfrid Ivorra

GDML_Books, (2007), p.

Résumé

Let p be a prime number ≥ 11 and c be a square-free integer ≥ 3. In this paper, we study the diophantine equation $x^{p} - y^{p} = c z ²$ in the case where p belongs to 11,13,17. More precisely, we prove that for those primes, there is no integer solution (x,y,z) to this equation satisfying gcd(x,y,z) = 1 and xyz ≠ 0 if the integer c has the following property: if ℓ is a prime number dividing c then ℓ ≢ 1 mod p. To obtain this result, we consider the hyperelliptic curves $C_{p} : y ² = Φ_{p} (x)$ and $D_{p} : p y ² = Φ_{p} (x)$ , where $Φ_{p}$ is the pth cyclotomic polynomial, and we determine the sets $C_{p} (ℚ)$ and $D_{p} (ℚ)$ . Using the elliptic Chabauty method, we prove that $C_{p} (ℚ) = (- 1, - 1), (- 1, 1), (0, - 1), (0, 1)$ and $D_{p} (ℚ) = (1, - 1), (1, 1)$ for p ∈ 11,13,17.

Zbl 1116.11018

EUDML-ID : urn:eudml:doc:286030

@book{bwmeta1.element.bwnjournal-article-doi-10_4064-dm444-0-1,
     author = {Wilfrid Ivorra},
     title = {Sur les courbes hyperelliptiques cyclotomiques et les \'equations $x^{p} - y^{p} = cz$^2$$
            },
     series = {GDML\_Books},
     year = {2007},
     zbl = {1116.11018},
     language = {fra},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-dm444-0-1}
}

Wilfrid Ivorra. Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$
            . GDML_Books (2007),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-dm444-0-1/