Asian Journal of Mathematics

Volume 21 (2017)

Number 3

Anticyclotomic Iwasawa invariants and congruences of modular forms

Pages: 499 – 530



Chan-Ho Kim (School of Mathematics, Korea Institute for Advanced Study, Seoul, Korea)


The main purpose of this article is to examine how congruences between Hecke eigensystems of modular forms affect the Iwasawa invariants of their anticyclotomic $p$-adic $L$-functions. We apply Greenberg–Vatsal and Emerton–Pollack–Weston’s ideas on the variation of Iwasawa invariants under congruences to the anticyclotomic setting. As an application, we establish infinitely many new examples of the anticyclotomic main conjecture for modular forms, which are not treated by Skinner–Urban’s work. An explicit example is given.


Iwasawa theory, $p$-adic $L$-functions, congruences, modular forms, quaternion algebras

2010 Mathematics Subject Classification

Primary 11R23. Secondary 11F33.

Received 2 August 2015

Published 5 July 2017