Regularity and symbolic defect of points on rational normal curves

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...

Rational Curves and Points on K3 Surfaces

— We study the distribution of algebraic points on K3 surfaces.

Finite Descent and Rational Points on Curves

In this paper, we discuss the information obtainable from descent via torsors under finite k-group schemes on the location of the k-rational points within the adelic points on a smooth projective k-variety X , where k is a number field. When X is a curve of genus ≥ 2, we conjecture that the information coming from “finite abelian descent” cuts out precisely the rational points; we provide theor...