On the conductor of an elliptic curve with a rational point of order 2

Abstract

Let C be an elliptic curve (an abelian variety of dimension one) defined over the field Q of rational numbers. A minimal Weierstrass model for C at all primes p in the sense of Néron [3] is given by a plane cubic equation of the formwhere aj belongs to the ring Z of integers of Q, the zero of C being the point of infinity.

Mod-2 dihedral Galois representations of prime conductor

“…The bulk of the paper is devoted to making these counts (Theorems 2 and 12) using class field theory plus the theory of modular Galois representations. As a byproduct, we recover some nonexistence results of Setzer [34], Hadano [14], and Kida [20] for elliptic curves of conductor N or 2N with N prime, derived using a totally different approach: a diophantine analysis of discriminants of Weierstrass equations due to Ogg [26].…”

On elliptic curves whose conductor is a product of two prime powers

“…Theorem 3.5. Let E/Q be an elliptic curve such that E(Q) [6] = {0}. Assume moreover that N E = p α q β where p = q are primes, and α, β > 0.…”

“…x → x t 2 (s − t) 6 , y → y t 3 (s − t) 9 to obtain the following integral Weierstrass equation y 2 + t(s − t) 3 + s(2s − t)(3s 2 − 3st + t 2 ) xy − ts(s − t) 5 (2s − t)(3s 2 − 3st + t 2 )(2s 2 − 2st + t 2 )y = x 3 − s(s − t) 2 (2s − t)(3s 2 − 3st + t 2 )(2s 2 − 2st + t 2 )x 2 where ∆ E = s 12 t 2 (s − t) 12 (2s − t) 6 (3s 2 − 3st + t 2 ) 4 (2s 2 − 2st + t 2 ) 3 (6s 2 − 6st + t 2 )…”

“…In particular, we prove that Szpiro's conjecture holds for these elliptic curves.problem under the condition that E has a rational torsion point. For example,in [6] the elliptic curves with conductor p m , p is prime, and 2-torsion points were listed.It was shown in [5] that all elliptic curves with conductor 2 m p n where p ≡ 3 or 5 mod 8, p = 3, that have a rational point of order 2, are effectively determined under the truth of the conjecture of Ankeny-Artin-Chowla.It is worth mentioning that the complete list of elliptic curves with a prime conductor has already been produced. The following theorem gives this list explicitly.Theorem 1.1 (Theorem 5.3.2, [10]).…”

Courbes elliptiques ayant bonne réduction en dehors de 3

“…D'autre part, on connait toutes les courbes elliptiques deίinies sur Q qui ont bonne reduction en dehors de 3, cf. [1], appendice. Ces courbes se partagent en trois classes de Q-isogenie; l'une de ces classes a pour conducteur 3 3, les deux autres ont pour conducteur 3 5 ; en particulier, le facteur local de la fonction L en 3 est 1 pour chacune des trois classes.…”