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We establish a duality result proving the `functional equation' of thecharacteristic ideal of the Selmer group associated to a nearly ordinaryHilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of atotally real number field. Further, we use this result to establish a dualityor algebraic `functional equation' for the `big' Selmer groups associated tothe corresponding nearly ordinary Hida deformation. The multivariablecyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbertmodular forms is not established yet and this can be thought of as an evidenceto the validity of this Iwasawa main conjecture. We also prove a functionalequation for the `big' Selmer group associated to an ordinary Hida family ofelliptic modular forms over the $\mathbb{Z}_{p}^{2}$ extension of an imaginaryquadratic field.

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