Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bianchi modular symbols and $p$-adic $L$-functions

Jaesung Kwon

Abstract

In the present paper, we constructed the $p$-adic $L$-function of Bianchi modular form. Also we proved that the first homology groups are generated by the special Bianchi modular symbols. As a corollary, the $μ$-invariant of some isotopic component of the $p$-adic $L$-function vanishes for Bianchi newforms and a positive proportion of ordinary primes. Also we obtain the residual non-vanishing result of the integral $L$-values.

Bianchi modular symbols and $p$-adic $L$-functions

Abstract

In the present paper, we constructed the -adic -function of Bianchi modular form. Also we proved that the first homology groups are generated by the special Bianchi modular symbols. As a corollary, the -invariant of some isotopic component of the -adic -function vanishes for Bianchi newforms and a positive proportion of ordinary primes. Also we obtain the residual non-vanishing result of the integral -values.

Paper Structure

This paper contains 30 sections, 23 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Generation of homology groups by special modular symbols
  3. Integral $L$-values and residual non-vanishing
  4. Vanishing of $\mu$-invariant
  5. Skectch of the proof of Theorem A
  6. Lattice estimation
  7. $\ell$-adic Galois representations and ordinary primes
  8. Technical Difficulties
  9. Notations
  10. Cusp forms on GL(2)
  11. Approximate functional equation
  12. Integral representation of special $L$-values
  13. Functional equations
  14. Twisted cusp forms
  15. Modular symbols and Eichler-Shimura isomorphism
  16. ...and 15 more sections

Key Result

Proposition 2.2

Let $\mathscr{F}$ be the set of fractional ideals of $F$. For each $f\in S_{k}(\mathfrak{N},\chi)$, there exists a function $a_f:\mathscr{F}\rightarrow\mathbb{C}$ satisfying the following properties:

Theorems & Definitions (52)

  • Definition 2.1: Hida hida1994critical
  • Proposition 2.2: Hida hida1994critical
  • Proposition 3.1
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Remark 4.3
  • Proposition 4.4
  • proof
  • ...and 42 more