منابع مشابه
Descent via Isogeny in Dimension 2
A technique of descent via 4-isogeny is developed on the Jacobian of a curve of genus 2 of the form: Y 2 = q1(X)q2(X)q3(X), where each qi(X) is a quadratic defined over Q. The technique offers a realistic prospect of calculating rank tables of Mordell-Weil groups in higher dimension. A selection of worked examples is included as illustration.
متن کاملDescent via (3, 3)-isogeny on Jacobians of Genus 2 Curves
We give parametrisation of curves C of genus 2 with a maximal isotropic (Z/3) in J [3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3, 3)-isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 3-part of the Tate-Shafarevich group.
متن کاملDescent via (5, 5)-isogeny on Jacobians of Genus 2 Curves
We describe a family of curves C of genus 2 with a maximal isotropic (Z/5) in J [5], where J is the Jacobian variety of C, and develop the theory required to perform descent via (5, 5)isogeny. We apply this to several examples, where it can shown that non-reducible Jacobians have nontrivial 5-part of the Tate-Shafarevich group.
متن کاملExplicit isogeny descent on elliptic curves
In this note, we consider an `-isogeny descent on a pair of elliptic curves over Q. We assume that ` > 3 is a prime. The main result expresses the relevant Selmer groups as kernels of simple explicit maps between finitedimensional F`-vector spaces defined in terms of the splitting fields of the kernels of the two isogenies. We give examples of proving the `-part of the Birch and Swinnerton-Dyer...
متن کاملDescent via isogeny on elliptic curves with large rational torsion subgroups
We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1994
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-66-1-23-43