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Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

Luis Santiago Palacios

Abstract

Let $K$ be an imaginary quadratic field. In this article, we construct $p$-adic $L$-functions of non-cuspidal Bianchi modular forms by introducing the notions of $C$-cuspidality and partial Bianchi modular symbols. When $p$ splits in $K$, we focus on $p$-adic $L$-functions of non-cuspidal base change Bianchi modular forms, showing that they factor as products of two Katz $p$-adic $L$-functions.

Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

Abstract

Let be an imaginary quadratic field. In this article, we construct -adic -functions of non-cuspidal Bianchi modular forms by introducing the notions of -cuspidality and partial Bianchi modular symbols. When splits in , we focus on -adic -functions of non-cuspidal base change Bianchi modular forms, showing that they factor as products of two Katz -adic -functions.
Paper Structure (21 sections, 23 theorems, 95 equations)

This paper contains 21 sections, 23 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Lg-cuspidal Bianchi modular forms and partial symbols
  3. Non-cuspidal base change and Katz Lg-adic Lg-functions
  4. Lg-cuspidal Bianchi modular forms
  5. Notations
  6. Bianchi modular forms
  7. Fourier expansion and cuspidal conditions
  8. Lg-function of Lg-cuspidal Bianchi modular forms
  9. Partial Bianchi modular symbols
  10. Partial modular symbols
  11. Partial modular symbols and Lg-cuspidal forms
  12. Lg-adic Lg-function of Lg-cuspidal Bianchi modular forms
  13. Locally analytic distributions
  14. Overconvergent partial Bianchi modular symbols
  15. Admissible distributions
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists a locally analytic distribution $L_p^{\iota}(\mathcal{F},-)$ on the ray class group $\mathrm{Cl}_K(p^{\infty})$, such that for any Hecke character $\psi$ of $K$ of conductor $\mathfrak{f}|(p^\infty)$ and infinity type $(q,r)$ satisfying $0\leqslant q \leqslant k, 0\leqslant r \leqslant where $\psi_{p-\mathrm{fin}}$ is the $p$-adic avatar of $\psi$, $-D_K$ is the discriminant of $K$,

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 69 more