We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and other does not. begin with torsors Jacobians curves Q F_q(t), conclude pair hyperkaehler 4-folds Q. The latter is independently interesting as new example transcendental Brauer-Manin obstruction to Hasse principle.

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...

1. Uniformity and correlation 1 1.

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...