Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings

TL;DR

This paper develops a specialization method for characteristic ideals over certain normal deformation rings and applies it to Euler system bounds and Iwasawa main conjecture divisibilities in number theory.

Contribution

It introduces a new specialization technique for characteristic ideals in normal deformation rings and applies it to Euler systems and Iwasawa theory.

Findings

Proved Euler system bounds over Cohen-Macaulay normal domains.

Established divisibility results for the two-variable Iwasawa main conjecture.

Generalized previous results to broader classes of deformation rings.

Abstract

The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.