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

Luis Santiago Palacios

TL;DR

The article establishes a functional equation for the $p$-adic $L$-function of small slope Bianchi modular forms and extends it to $\Sigma$-smooth base-change cases via $p$-adic families. It builds on overconvergent modular symbols to define $L_p(\mathcal{F},-)$ and relates $p$-stabilised L-functions to their un-stabilised counterparts through explicit Euler factors. A key advance is handling both the small slope and critical slope regimes, with the latter using a three-variable $p$-adic $L$-function to obtain a slope-insensitive functional equation for base-change forms. The results enhance our understanding of $p$-adic $L$-functions in the Bianchi setting and supply a robust framework for interpolating and relating special values across p-adic families, with implications for arithmetic geometry over imaginary quadratic fields.

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 $Σ$-smooth base-change Bianchi modular forms.

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

TL;DR

The article establishes a functional equation for the -adic -function of small slope Bianchi modular forms and extends it to -smooth base-change cases via -adic families. It builds on overconvergent modular symbols to define and relates -stabilised L-functions to their un-stabilised counterparts through explicit Euler factors. A key advance is handling both the small slope and critical slope regimes, with the latter using a three-variable -adic -function to obtain a slope-insensitive functional equation for base-change forms. The results enhance our understanding of -adic -functions in the Bianchi setting and supply a robust framework for interpolating and relating special values across p-adic families, with implications for arithmetic geometry over imaginary quadratic fields.

Abstract

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

Paper Structure

This paper contains 19 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Bianchi modular forms
  3. Notation
  4. Background
  5. Fourier-Whittaker expansions
  6. Hecke operators
  7. Twisted series of Bianchi modular forms
  8. $L$-function
  9. Definition of the $L$-function
  10. Functional equation of the complex $L$-function
  11. $p$-stabilisations of Bianchi modular forms
  12. $L$-function of a $p$-stabilisation
  13. Modular symbols
  14. Overconvergent modular symbols
  15. $p$-adic $L$ function
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{F}\in S_{(k,k)}(\Gamma_0(\mathfrak{n}))$ be a Bianchi newform and $\psi$ be a Hecke character of $K$ of conductor $\mathfrak{f}$ with $(\mathfrak{n},\mathfrak{f})=1$ and infinity type $0 \leqslant (q,r) \leqslant (k,k)$, we have where $\mathfrak{n}=(\nu)$, and $\epsilon(\mathfrak{n})=\pm1$ is the eigenvalue of $\mathcal{F}$ for the Fricke involution $W_\mathfrak{n}$.

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 47 more