Displaying similar documents to “On rational torsion points of central ℚ -curves”

Siegel’s theorem and the Shafarevich conjecture

Aaron Levin (2012)

Journal de Théorie des Nombres de Bordeaux

Similarity:

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k , one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S . We show here that an extension of this result to an effective Shafarevich conjecture for of hyperelliptic curves of genus g would imply an effective version of Siegel’s theorem for integral points...

On a theorem of Mestre and Schoof

John E. Cremona, Andrew V. Sutherland (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229 . We extend this result to all finite fields with q > 49 , and all prime fields with q > 29 .

Computing the cardinality of CM elliptic curves using torsion points

François Morain (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let ℰ / ℚ ¯ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field 𝕂 . The field of definition of ℰ is the ring class field Ω of the order. If the prime p splits completely in Ω , then we can reduce ℰ modulo one the factors of p and get a curve E defined over 𝔽 p . The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose,...

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

Similarity:

Let K be a number field, and let A / K be an abelian variety. Let c denote the product of the Tamagawa numbers of A / K , and let A ( K ) tors denote the finite torsion subgroup of A ( K ) . The quotient c / | A ( K ) tors | is a factor appearing in the leading term of the L -function of A / K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over ℚ or quadratic extensions K / ℚ , and for abelian surfaces A / ℚ . The smallest possible...

Twists of Hessian Elliptic Curves and Cubic Fields

Katsuya Miyake (2009)

Annales mathématiques Blaise Pascal

Similarity:

In this paper we investigate Hesse’s elliptic curves H μ : U 3 + V 3 + W 3 = 3 μ U V W , μ ∈ Q - { 1 } , and construct their twists, H μ , t over quadratic fields, and H ˜ ( μ , t ) , μ , t ∈ Q over the Galois closures of cubic fields. We also show that H μ is a twist of H ˜ ( μ , t ) over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, R ( t ; X ) : = X 3 + t X + t , t ∈ Q - { 0 , - 27 / 4 } , to parametrize all of quadratic fields and cubic ones. It should be noted that H ˜ ( μ , t ) is a twist of H μ as algebraic curves because it may not always have any...