Rational points on Jacobians of hyperelliptic curves
نویسنده
چکیده
We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.
منابع مشابه
The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point
We prove that when all hyperelliptic curves of genus n ≥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. It follows that (the limsup of) the average rank of the Mordell-Weil group of their Jacobians is at most 3/2. The method of Chabauty can then be used to obtain an effective bound on the number of rational p...
متن کاملRational Points on Hyperelliptic Curves: Recent Developments
We give an overview over recent results concerning rational points on hyperelliptic curves. One result says that ‘most’ hyperelliptic curves of high genus have very few rational points. Another result gives a bound on the number of rational points in terms of the genus and the Mordell-Weil rank, provided the latter is sufficiently small. The first result relies on work by Bhargava and Gross on ...
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 کاملHanoi lectures on the arithmetic of hyperelliptic curves
Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...
متن کاملFields of definition of torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields
This paper deals with fields of definition of the l-torsion points on the Jacobians of genus 2 hyperelliptic curves over finite fields in order to speed Gaudry and Schost’s point counting algorithm for genus 2 hyperelliptic curves up. A result in this paper shows that the extension degrees of the fields of difinition of the l-torsion points can be in O(l) instead of O(l). The effects of the res...
متن کامل