Ribet Bimodules and the Specialization of Heegner points
Rights accessOpen Access
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 = X0(D,N) at primes p | DN. As we show, if p does not divide the conductor of R, a point P in CM(R) specializes to a singular point (resp. a connected component) of the special fiber Xp of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X0(D,N) and K. Ribet’s theory of bimodules, we give a construction of a correspondence between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebras 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 Xp. We also illustrate our results with an explicit example.
CitationMolina, S. Ribet Bimodules and the Specialization of Heegner points. "Israel journal of mathematics", 2012, núm. 189, p. 1-38.