Regulators of rank one quadratic twists

Christophe Delaunay; Xavier-François Roblot

Displaying similar documents to “Regulators of rank one quadratic twists”

The rank of hyperelliptic Jacobians in families of quadratic twists

Sebastian Petersen (2006)

Journal de Théorie des Nombres de Bordeaux

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The variation of the rank of elliptic curves over $ℚ$ in families of quadratic twists has been extensively studied by Gouvêa, Mazur, Stewart, Top, Rubin and Silverberg. It is known, for example, that any elliptic curve over $ℚ$ admits infinitely many quadratic twists of rank $\geq 1$ . Most elliptic curves have even infinitely many twists of rank $\geq 2$ and examples of elliptic curves with infinitely many twists of rank $\geq 4$ are known. There are also certain density results. This paper studies the variation...

Non-trivial $Ш$ in the Jacobian of an infinite family of curves of genus 2

Anna Arnth-Jensen, E. Victor Flynn (2009)

Journal de Théorie des Nombres de Bordeaux

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We give an infinite family of curves of genus 2 whose Jacobians have non-trivial members of the Tate-Shafarevich group for descent via Richelot isogeny. We prove this by performing a descent via Richelot isogeny and a complete 2-descent on the isogenous Jacobian. We also give an explicit model of an associated family of surfaces which violate the Hasse principle.

On rational torsion points of central $ℚ$ -curves

Fumio Sairaiji, Takuya Yamauchi (2008)

Journal de Théorie des Nombres de Bordeaux

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Let $E$ be a central $ℚ$ -curve over a polyquadratic field $k$ . In this article we give an upper bound for prime divisors of the order of the $k$ -rational torsion subgroup $E_{t o r s} (k)$ (see Theorems 1.1 and 1.2). The notion of central $ℚ$ -curves is a generalization of that of elliptic curves over $ℚ$ . Our result is a generalization of Theorem 2 of Mazur [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].

On elliptic curves and random matrix theory

Mark Watkins (2008)

Journal de Théorie des Nombres de Bordeaux

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Rubinstein has produced a substantial amount of data about the even parity quadratic twists of various elliptic curves, and compared the results to predictions from random matrix theory. We use the method of Heegner points to obtain a comparable (yet smaller) amount of data for the case of odd parity. We again see that at least one of the principal predictions of random matrix theory is well-evidenced by the data.

On a theorem of Mestre and Schoof

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

Journal de Théorie des Nombres de Bordeaux

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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 $q > 29$ .

Displaying similar documents to “Regulators of rank one quadratic twists”

The rank of hyperelliptic Jacobians in families of quadratic twists

Non-trivial Ш in the Jacobian of an infinite family of curves of genus 2

On rational torsion points of central ℚ -curves

On elliptic curves and random matrix theory

On a theorem of Mestre and Schoof

Non-trivial $Ш$ in the Jacobian of an infinite family of curves of genus 2

On rational torsion points of central $ℚ$ -curves