I saw a nice talk by Matt Young recently (joint work with Sheng-Chi Liu and Riad Masri) on the following problem.
For a fundamental discriminant of an imaginary quadratic field , one has points in with complex multiplication by the ring of integers of . Choose a prime which splits in . One obtains a set of points in , given explicitly as follows:
for in the class group and one of the two primes above in . The complex points can be thought of as being tiled by copies of the fundamental domain in the upper half plane.
Problem: How large does have to be to guarantee that every one of the copies of contains one of the CM points by ?
This is the question that Young and his collaborators answer. Namely, one gets an upper bound of the shape (with some explicit , possibly 20), the point being that this is a polynomial bound. Note that this proof is not effective, since it trivially gives a lower bound on the order of the class group which is a power bound in the discriminant, and no such effective bounds are known.
I idly wondered during the talk about the following "mod-" version of this problem. To be concrete, suppose that (the general case will be similar). We now suppose that is chosen so that is inert in . Then all the points in are supersingular, which means that they all reduce to the same curve with -invariant . Now, as above, choose a prime which splits in . The pre-image of in consists of exactly points.
Problem: How large does have to be to ensure that these points all come from the reduction of one of the CM points by as above?
Since is supersingular, we know that is an order in the quaternion algebra ramified at and . In fact, it is equal to the integral Hamilton quaternions . If and are lifts of , then there is naturally a degree preserving injection:
The degree on the LHS is the degree of an isogeny, and it is the canonical norm on the RHS.
In particular, if and , then one obtains a natural map:
preserving norms. The norm map on is . The image of the natural isogeny is simply , whose image has norm . Hence the problem becomes:
Problem: If one considers all the -maps:
do the images of cover the elements of of norm ?
Given a field in which is inert, it wasn’t obvious how to explicitly write down the maps , but this problem does start to look similar in flavour to the original one. Moreover, to make things even more similar, in the original formulation over one can replace modular curves by definite quaternion algebras ramified at (say) and , and then the Archimidean problem now also becomes a question of a class group surjecting onto a finite set of supersingular points. In fact, this Archimedean analogue may well be *equivalent* to the version I just described! Young told me that his collaborators had mentioned working with various quotients coming from quaternion algebras as considered by Gross, which I took to mean the finite quotients coming from definite quaternion algebras as above. Hence, with any luck, they will provide an answer this problem.