On uniform bounds for rational points on rational curves of arbitrary degree

On Uniform Bounds for Rational Points on Non-rational Curves

We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ > 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O (H2/d+ ). We also show that one can take δ = 1/450 in the case d = 3.

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...

Rational Points on Curves

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rat...

Uniform Bounds for the Number of Rational Points on Hyperelliptic Curves of Small Mordell-weil Rank

We show that there is a bound depending only on g, r and [K : Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell-Weil rank r of its Jacobian is at most g − 3. If K = Q, an explicit bound is 8rg + 33(g − 1) + 1. The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian log...