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

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.