Heegner points and Jochnowitz congruences on Shimura curves

摘要

Given a rational elliptic curve E, a suitable imaginary quadratic field K anda quaternionic Hecke eigenform g of weight 2 obtained from E by level raisingsuch that the sign in the functional equation for L_K(E,s) (respectively,L_K(g,1)) is -1 (respectively, +1), we prove a ``Jochnowitz congruence''between the algebraic part of L'_K(E,1) (expressed in terms of Heegner pointson Shimura curves) and the algebraic part of L_K(g,1). This establishes arelation between Zhang's formula of Gross-Zagier type for central derivativesof L-series and his formula of Gross type for special values. Our resultsextend to the context of Shimura curves attached to division quaternionalgebras previous results of Bertolini and Darmon for Heegner points onclassical modular curves.