Displaying similar documents to “The rank of hyperelliptic Jacobians in families of quadratic twists”

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. ...

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.

Siegel’s theorem and the Shafarevich conjecture

Aaron Levin (2012)

Journal de Théorie des Nombres de Bordeaux

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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 the maximal unramified pro-2-extension over the cyclotomic ℤ 2 -extension of an imaginary quadratic field

Yasushi Mizusawa (2010)

Journal de Théorie des Nombres de Bordeaux

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For the cyclotomic ℤ 2 -extension k ∞ of an imaginary quadratic field k , we consider the Galois group G ( k ∞ ) of the maximal unramified pro- 2 -extension over k ∞ . In this paper, we give some families of k for which G ( k ∞ ) is a metabelian pro- 2 -group with the explicit presentation, and determine the case that G ( k ∞ ) becomes a nonabelian metacyclic pro- 2 -group. We also calculate Iwasawa theoretically the Galois groups of 2 -class field towers of certain cyclotomic 2 -extensions.

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 [].