Functional equation of the $p$-adic $L$-function of Bianchi modular forms

TL;DR

This paper derives the functional equation for the $p$-adic $L$-function of Bianchi modular forms over imaginary quadratic fields and extends it to $ ext{Sigma}$-smooth base-change forms using $p$-adic families.

Contribution

It establishes the functional equation for the $p$-adic $L$-function of small slope Bianchi modular forms and extends the result to a broader class via $p$-adic families.

Findings

Functional equation for small slope Bianchi forms' $p$-adic $L$-function.

Extension of the functional equation to $ ext{Sigma}$-smooth base-change forms.

Use of $p$-adic families to generalize the results.

Abstract

Let $K$ be an imaginary quadratic field with class number 1, in this paper we obtain the functional equation of the $p$-adic $L$-function of small slope $p$-stabilised Bianchi modular forms. Then, using $p$-adic families of Bianchi modular forms, we extend our result to $\Sigma$-smooth base-change Bianchi modular forms.