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 representations. We will see that this leads to a realization of infinitely many new groups PSL2(Fp) as Galois groups conditional on the Birch Swinnerton-Dyer conjecture, and to a “natural” infinite sequence of curves violating the Hasse principle. These last two results, which are taken from a 2005 note [Cl1] and a very recent preprint [Cl2] of mine, might sound deep and/or impressive, but the proofs are easy to the point of raising the question of why they were not done before. In response, I would have to say that this circle of objects and ideas – so close to the numbertheoretic mainstream (what I was taught in grad school were the three mainstays of number theory – Diophantine geometry, Galois theory, and automorphic forms – are all clearly present and up to their usual tricks) – seems profoundly underexplored. I think there are many interesting and tractable problems here, and I will try to justify this impression (to the extent that I am able; I have almost nothing intelligent to say about the automorphic side of things) as much as to showcase any result of mine.