Anticyclotomic main conjecture for modular forms and integral Perrin-Riou twists

Abstract

We prove a one-sided divisibility relation for the anticyclotomic Iwasawa main conjecture for modular forms in terms of the $p$-adic $L$-function constructed by Bertolini-Darmon-Prasanna and Brakočević. The divisibility relation is known by Castella if $p$ is ordinary and by Castella-Wan for the elliptic curve case. Here we prove the higher weight non-ordinary case with a treatment that works uniformly for both ordinary and non-ordinary cases. In the proof, we establish a theory of integral Perrin-Riou twist. It enables us not only to twist systems of generalized Heegner cycles (which are not norm-compatible) by any continuous $p$-adic anticyclotomic characters but also to investigate the denominators of resulting systems explicitly.