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

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.