Ribet bimodules and the specialization of Heegner

Authors: Santiago Molina

Publish Date: 2011/10/06

Volume: 189, Issue: 1, Pages: 1-38

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Abstract

For a given order R in an imaginary quadratic field K we study the specialization of the set CMR of Heegner points on the Shimura curve X = X 0D N at primes p DN As we show if p does not divide the conductor of R a point P ∈ CMR 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 0D N and K Ribet’s theory of bimodules we give a construction of a correspondence Φ between CMR 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 PicR and of an AtkinLehner 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

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