Let E be an elliptic curve with CM by the imaginary quadratic order OD of negative discriminant D. Given p a prime, if p is inert or ramified in the quadratic field generated by √D then E has supersingular reduction at a(ny) fixed place above p. By a variant of Duke’s equidistribution theorem, as D grows along such discriminants, the proportion of CM elliptic curves with CM by OD whose reduction at such place is a given supersingular curve converge to a natural (non-zero) limit. A further step is to fix several (distinct) primes p1, . . . ,ps and to look for the proportion of CM curves whose reduction above each of these primes is prescribed. In this talk, we will explain how a powerful result of Einsiedler and Lindenstrauss classifying joinings of rank 2 actions on products of locally homogeneous spaces implies that as D grows along adequate subsequences of negative discriminants, this proportion converge to the product of the limits for each individual pi (a sort of asymptotic Chinese Reminder Theorem for reductions of CM elliptic curves if you wish). This is joint work with M. Aka, M. Luethi and A.Wieser. If time permits, we will also describe a further refinement — obtained with the additional collaboration of R. Menares — of these equidistribution results for the formal groups attached to these curves.
This video is part of the Number Theory Web Seminar series.
