Iwasawa Theory of Hilbert modular forms for anticyclotomic extension

TL;DR

This paper advances Iwasawa theory for Hilbert modular forms by removing the need for Ihara's lemma, thereby strengthening the main conjecture results in the anticyclotomic extension context.

Contribution

It eliminates the Ihara's lemma condition in the proof of the Iwasawa main conjecture for Hilbert modular forms, broadening the applicability of previous results.

Findings

Removed the Ihara's lemma condition from key proofs.

Extended the validity of the Iwasawa main conjecture.

Strengthened the theoretical framework for Hilbert modular forms.

Abstract

Following Bertolini and Darmon's method, with "Ihara's lemma" among other conditions Longo and Wang proved one divisibility of Iwasawa main conjecture for Hilbert modular forms of weight $2$ and general low parallel weight respectively. In this paper, we remove the "Ihara's lemma" condition in their results.