Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions

Abstract

We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg.