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

TL;DR

This paper develops a broad algebraic functional equation framework for big Galois representations over multiple $ ext{Z}_p$-extensions, unifying various Iwasawa-theoretic cases and generalizing prior results.

Contribution

It introduces a general approach to algebraic functional equations applicable to a wide range of Galois representations in Iwasawa theory, including complex cases like Hida families and Rankin-Selberg deformations.

Findings

Establishes algebraic functional equations in Selmer group and complex settings.

Applies to triple products of Hida families and Rankin-Selberg deformations.

Generalizes many previously known algebraic functional equations.

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.