Rational points on the modular curves X0+(N)

Cycles on modular varieties and rational points on elliptic curves

This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over Q, the goal is to better understand the group E(Q̄) of algebra...

Rational Points on Atkin-lehner Twists of Modular Curves

These are the (more detailed) notes accompanying a talk that I am to give at the University of Pennsylvania on July 21, 2006. The topic is rational points on Atkin-Lehner twists of the modular curves X0(N). Apart from being an interesting Diophantine problem in its own right, there is an ulterior motive: Q-rational points correspond to “elliptic Q-curves” and thus to projective Galois represent...

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...