Asian Journal of Mathematics

Volume 21 (2017)

Number 3

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

Pages: 397 – 428



Somnath Jha (Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India)

Dipramit Majumdar (Department of Mathematics, IIT Madras, Chennai, India)


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 an ‘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 a modest 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 $\mathbb{Z}^2_p$ extension of an imaginary quadratic field.


Selmer group, Iwasawa theory of Hida family, Hilbert modular forms

2010 Mathematics Subject Classification

11F33, 11F80, 11R23

Received 24 April 2015

Published 5 July 2017