Displaying similar documents to “On elliptic curves and random matrix theory”

Regulators of rank one quadratic twists

Christophe Delaunay, Xavier-François Roblot (2008)

Journal de Théorie des Nombres de Bordeaux

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We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of a rank one quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions. ...

Elliptic curves associated with simplest quartic fields

Sylvain Duquesne (2007)

Journal de Théorie des Nombres de Bordeaux

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We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much...

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

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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...