Hida duality and the Iwasawa main conjecture

TL;DR

This paper refines Hida's duality theorem for ordinary Lambda-adic modular forms, providing a simpler proof of a key isomorphism related to the Iwasawa main conjecture and extending Ohta's results by removing restrictive hypotheses.

Contribution

It offers a necessary condition for integral duality in Lambda-adic modular forms and simplifies the proof of the Iwasawa main conjecture over Q by extending Ohta's approach.

Findings

Refined Hida duality theorem with integral conditions.

Proved isomorphism between Hecke algebra modulo Eisenstein ideal and Iwasawa algebra.

Extended Ohta's proof of the Iwasawa main conjecture, removing restrictive hypotheses.

Abstract

The central result of this paper is a refinement of Hida's duality theorem between ordinary Lambda-adic modular forms and the universal ordinary Hecke algebra. Specifically, we give a necessary condition for this duality to be integral with respect to particular submodules of the space ordinary Lambda-adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt p-adic L-function. The motivation behind these results stems from Ohta's proof of the Iwasawa main conjecture over Q. Specifically, the most general application of this argument, which employs results on congruence modules and requires one to make some restrictive hypotheses. Using our results we are able to extend Ohta's argument and remove these hypotheses.