Let X^d(N) be the quadratic twist of the modular curve X_0(N) through the
Atkin-Lehner involution w_N and a quadratic extension Q(\sqrt{d})/Q. The points
of X^d(N)(Q) are precisely the Q(\sqrt{d})-rational points of X_0(N) that are
fixed by \sigma composition w_N, where \sigma is the generator of
Gal(Q(\sqrt{d})/Q).Ellenberg asked the following question:

For which d and N does X^d(N) have rational points over every completion of
Q?

Given (N,d,p) we give necessary and sufficient conditions for the existence
of a Q_p-rational point on X^d(N), whenever p is not simultaneously ramified in
Q(\sqrt{d}) and Q(\sqrt{-N}), answering Ellenberg's question for all odd primes
p when (N,d)=1. The main theorem 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.