Let E be a CM-field, and suppose that f,g are two primitive Hilbert cusp forms over E⁺ of weight 2 satisfying a congruence modulo λ^r. Under appropriate hypotheses, we show that the complex L-values of f and g twisted by a ring class character over E, and divided by the motivic periods, also satisfy a congruence relation mod λ^r (after removing some Euler factors). We treat both the even and odd cases for the sign in the functional equation -- this generalizes classical work of Vatsal [23] on congruences between elliptic modular forms twisted by Dirichlet characters. In the odd case, we also show that the p-adic logarithms of Heegner points attached to f and g satisfy a congruence relation modulo λ^r, thus extending recent work of Kriz and Li [17] concerning elliptic modular forms.

The authors thank Antonio Cauchi, Daniel Disegni and Ming-Lun Hsieh for patiently answering their questions during the preparation of this article. The bulk of the work was undertaken during a two week visit of the first named author to Université Laval in March 2019, and he thanks the Mathematics Department at Laval warmly for their hospitality. The second named author's research is supported through a NSERC Discovery Grants Program (no. 05710).