Points on Quadratic Twists of the Classical Modular Curve

In this talk, we will give results on be the quadratic twist of the
modular curve X_0(N) through the Atkin-Lehner involution w_N and a
quadratic extension K/Q. Given (N,d,p) we give necessary and
sufficient conditions for the existence of a Q_p-rational point on the
twisted curve when (N,d)=1. The main result yields a population of
curves which have local points everywhere but no points over Q; in
several cases we show that this obstruction to the Hasse Principle is
explained by the Brauer-Manin obstruction. If time permits, we will
mention some generalizations of this question to Shimura curves.