Counting rational points near planar curves

Theory L . Caporaso COUNTING RATIONAL POINTS ON ALGEBRAIC CURVES

We describe recent developments on the problem of finding examples of algebraic curves of genus at least 2 having the largest possible number of rational points. This question is related to the Conjectures of Lang on the distribution of rational points on the varieties of general type.

Counting Rational Points on Curves over Finite Fields (Extended Abstract)

We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F E Fp[x, y, 21, which is rational over the ground field IFp. More precisely, we show that if we are given a projective plane curve C of degree n, and if C has only ordinary multiple points, then one can compute the number of IFp-rational points on C in randomized time (logp)" where A = ...

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...