A new paper on the subject by my Ph.D. student, Ekin Ozman, is up on the arXiv today.

The twists in question are isomorphic to X_0(N) over a quadratic field K = Q(sqrt(d)), but not over Q; the twist is via the Atkin-Lehner involution w_N, which is to say that the rational points on such a twist are in bijection with the points P of X_0(N)(K) satisfying

where is the generator of Gal(K/Q). Call this twist X^d(N).

Why do we care about these twists? For one thing, X^d(N) parametrizes certain classes of *Q-curves*, elliptic curves over Qbar which are isogenous to all their Galois conjugates. These guys turn out to be more flexible than elliptic curves over Q for constructing “Frey curves” attached to Diophantine equations, but have all the same handsome modularity properties as elliptic curves over Q, allowing the usual Mazur-Frey-Serre-Ribet-Wiles style argument to go through.

More generally, though, the X^d(N) form a very natural class of modular curves whose arithmetic hasn’t been much thought about. For instance: is there an analogue of Mazur’s theorem? That is, do these curves have rational points? Here one immediately encounters a divergence from the untwisted case — X^d(N) might not even have local points! The cusps, which provide Q_p-points on X_0(N) for every N and p, are not rational points on X^d(N); without them, there’s no reason for X_0(N)(Q_p) to be nonempty. In a 2004 survey paper about Q-curves, I asked (Problem A in the linked paper) for which d and N the curve X^d(N) had local points everywhere; this certainly has to be settled before any investigation of the global points can start. Pete Clark (see the appendix to this paper), Jordi Quer, and Josep Gonzalez got partial results on the problem; now Ekin has almost entirely solved it, getting an exact criterion for the existence of local points whenever K and Q(sqrt(-N)) have no common primes of ramification.

I was never able to see a simple way to do this — and it turns out that’s because the answer, as Ekin works it out, is actually pretty complicated! So I won’t state her theorem here; I’ll just say that it’s competely explicit and it allows you to compute just about whatever you want. For example: the number of squarefree d < X such that X^d(17) has local points everywhere is on order of . (She could compute the constant, too, if for some terrifying reason you needed it…)

In the end, a lot of the X^d(N) have local points everywhere; and because they are all covers of the quotient X_0(N)/w_N, which has finitely many rational points once N is big enough, many of them don’t have any global rational points. In other words, you have a healthy population of curves violating the Hasse Principle. (This observation is due to Clark.) Are these failures of the Hasse principle always due, as some people expect, to the Brauer-Manin obstruction? In the last section of the paper, Ekin works out one case in full — namely, X^{17}(23) — and shows that it is indeed so.