Functional equation for the Selmer group of nearly ordinary Hida deformation of Hilbert modular forms

TL;DR

This paper proves a duality and functional equation for Selmer groups related to nearly ordinary Hilbert modular forms and Hida deformations, providing evidence for the Iwasawa main conjecture in this setting.

Contribution

It establishes a duality result and a functional equation for the characteristic ideal of Selmer groups associated to nearly ordinary Hilbert modular forms and Hida deformations.

Findings

Proves a duality for Selmer groups over cyclotomic extensions.

Establishes a functional equation for the characteristic ideal of Selmer groups.

Provides evidence supporting the Iwasawa main conjecture for Hida families.

Abstract

We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the `big' Selmer group associated to an ordinary Hida family of elliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginary quadratic field.