Heegner cycles and congruences between anticyclotomic p-adic L-functions over CM-extensions

Abstract

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 λʳ. 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 λʳ (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 λʳ, thus extending recent work of Kriz and Li [17] concerning elliptic modular forms.