Jochnowitz Congruences at Residual Primes

We investigate relationships between the algebraic parts of L-values of weight two eigenforms f and g satisfying a congruence modulo a prime p, but whose signs in the functional equation are −1 and +1, respectively. By fixing an imaginary quadratic field satisfying certain hypotheses, we use the formula of Gross-Zagier and an explicit Waldspurger-type formula of Gross to give a certain congruence between Heegner points on GL2-type abelian varieties and toric periods on definite quaternion algebras. Such a relation may be viewed as a congruence between the algebraic parts of L′(f/K, 1) and L(g/K, 1), and are known as Jochnowitz congruences. This generalizes earlier work of Bertolini-Darmon and Vatsal to all level raising congruences for which such a sign change occurs.