New York Journal of Mathematics
Volume 26 (2020), 496-525


Daniel Delbourgo and Antonio Lei

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

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Published: May 22, 2020.
Keywords: Iwasawa theory, p-adic L-functions, Hilbert modular forms.
Subject: Primary 11F33; Secondary 11F41, 11G40, 11R23.

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.


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).

Author information

Daniel Delbourgo:
Department of Mathematics and Statistics
University of Waikato
Gate 8, Hillcrest Road
Hamilton 3240, New Zealand


Antonio Lei:
Département de mathématiques et de statistique
Université Laval
1045 avenue de la Médecine
Québec QC G1V 0A6, Canada