restricted access Euler systems and Jochnowitz congruences

This article relates the Gross-Zagier formula with a simpler formula of Gross for special values of L-series, via the theory of congruences between modular forms. Given two modular forms f and g (of different levels) which are congruent but whose functional equations have sign -1 and 1 respectively, and an imaginary quadratic field K satisfying certain auxiliary conditions, the main result gives a congruence between the algebraic part of L'(f/K, 1) (expressed in terms of Heegner points) and the algebraic part of the special value L(g/K, 1). Congruences of this type were anticipated by Jochnowitz, and for this reason are referred to as "Jochnowitz congruences."