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Courbes elliptiques sur ℚ, ayant un point d’ordre 2 rationnel sur ℚ, de conducteur 2 N p

Wilfrid Ivorra — 2004

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...

Sur les courbes hyperelliptiques cyclotomiques et les équations x p - y p = c z ²

Wilfrid Ivorra — 2007

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,...

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