Descent via (3, 3)-isogeny on Jacobians of Genus 2 Curves
نویسنده
چکیده
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.
منابع مشابه
Descent 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 کاملIsogenies and the Discrete Logarithm Problem on Jacobians of Genus 3 Hyperelliptic Curves
We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic genus 3 curves to Jacobians of non-hyperelliptic genus 3 curves, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to (Z/2Z) for any hyperelliptic genus 3 curve. These algorithms provide a rational isogeny...
متن کاملSolving Diophantine Problems on Curves via Descent on the Jacobian
The theory of Jacobians of curves has largely been developed in a vacuum, with little computational counterpart to the abstract theory. A recent development has been the explicit construction of Jacobians & formal groups, and workable methods of descent [6],[7] to find the rank. We suggest that the following plan will provide a powerful tool for finding the set of Q-rational points C(Q) on a cu...
متن کاملFamilies of Explicitly Isogenous Jacobians of Variable-separated Curves
We construct six infinite series of families of pairs of curves (X, Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny JX → JY splitting multiplication by 2, 3, or 4. The families are derived from Cassou–Noguès and Couveignes’ explicit classification of pairs (f, g) of polynomials such that f(x1)− g(x2) is reducible.
متن کاملOn a Theorem of Coleman
A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 Introduction The following classical result of Chabauty [3] is a curiosity of the literature in that there has been a 50 year period during which applications have been ...
متن کامل