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

TL;DR

This paper constructs the $p$-adic $L$-function for Bianchi modular forms, proves homology groups are generated by special symbols, and derives results on $$-invariants and non-vanishing of $L$-values.

Contribution

It introduces a new construction of $p$-adic $L$-functions for Bianchi forms and establishes their homological generation by modular symbols.

Findings

The $p$-adic $L$-function is explicitly constructed.

Homology groups are generated by Bianchi modular symbols.

The $$-invariant vanishes for certain forms and primes.

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 $\mu$-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.