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Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$

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Ivorra W.

Instytut Matematyczny Polskiej Akademii Nauk

EN

Let p be a prime number ≥ 11 and c be a square-free integer ≥ 3. In this paper, we study the diophantine equation $x^{p} - y^{p} = cz²$ 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 $C_{p}: y² = Φ_{p}(x)$ and $D_{p}: py² = Φ_{p}(x)$, 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}.

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Courbes elliptiques sur ℚ, ayant un point d'ordre 2 rationnel sur ℚ, de conducteur $2^{N}p$

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Ivorra W.

Instytut Matematyczny Polskiej Akademii Nauk

EN

Let p be a prime number ≥ 29 and N be a positive integer. In this paper, we are interested in the problem of the determination, up to ℚ-isomorphism, of all the elliptic curves over ℚ whose conductor is $2^{N}p$, with at least one rational point of order 2 over ℚ. This problem was studied in 1974 by B. Setzer in case N = 0. Consequently, in this work we are concerned with the case N ≥ 1. The results presented here are analogous to those obtained by B. Setzer and allow one in practice to find a complete list of such curves.

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Sur les équations $x^p + 2^{β}y^p = z^2$ et $x^p + 2^{β}y^p = 2z^2$

84%

Ivorra W.

Acta Arithmetica

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2003

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tom 108

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nr 4

327-338

Ograniczanie wyników

1 Acta Arithmetica

3 Ivorra W.

1 2007

1 2004

1 2003

Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$

Courbes elliptiques sur ℚ, ayant un point d'ordre 2 rationnel sur ℚ, de conducteur $2^{N}p$

Sur les équations $x^p + 2^{β}y^p = z^2$ et $x^p + 2^{β}y^p = 2z^2$