Finding rational points on bielliptic genus 2 curves

Finding Rational Points on Bielliptic Genus 2 Curves

We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X + f2X + f1X + f0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family ...

Rational Points on Curves of Genus 2: Experiments and Speculations

We present results of computations providing statistics on rational points on (small) curves of genus 2 and use them to present several conjectures. Some of these conjectures are based on heuristic considerations, others are based on our experimental results.

Covering Techniques and Rational Points on Some Genus 5 Curves

We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curve defined over a number field. This method is used to solve some arithmetic problems that remained open.

Rational points on curves

Rational points on curves

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...