Ribet bimodules and the specialization of Heegner points

Abstract

For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X 0(D, N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a irreducible component) of the special fiber $\tilde X$ of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X 0(D, N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence Φ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebra that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points in $\tilde X$ . We also illustrate our results with an explicit example.