Rational points on twists of X0(63)

Independence of Rational Points on Twists of a given Curve

In this paper, we study bounds for the number of rational points on twists C ′ of a fixed curve C over a number field K, under the condition that the group of K-rational points on the Jacobian J ′ of C ′ has rank smaller than the genus of C ′. The main result is that with some explicitly given finitely many possible exceptions, we have a bound of the form 2r + c, where r is the rank of J ′(K) a...

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...

A Uniform Bound for Rational Points on Twists of a given Curve

THEOREM 1. Let K/Q be a number field, and let C/K be a smooth projective curve of genus at least 2. For each^class ^ e i / 1 (Gal (AT/AT), Aut(C)), let C% be the twist of C by x (cf. [11, X §3]) and le\Jx = J ac (Q be the Jacobian variety of Cx. There is a constant y = y{C/K) such that #C,( /Q^r7 a n k * ( K ) forallxeH{G2\{K/K),A\xi{C)). We illustrate this general theorem by applying it to a p...

Rational and Integral Points on Quadratic Twists of a given Hyperelliptic Curve

We show that the abc-conjecture implies that few quadratic twists of a given hyperelliptic curve have any non-trivial rational or integral points; and indicate how these considerations dovetail with other predictions.

Distribution of Rational Points: A Survey