Bianchi modular symbols and p-adic L-functions

Abstract

In the present paper, we prove that the first homology group of Bianchi 3-fold is generated by the special Bianchi modular symbols. Also, we construct the integral valued p-adic L-function of p-ordinary Bianchi Hecke eigenforms by taking the Lefschetz-Poincaré Pairing of the cohomology class attached to Bianchi modular forms and Bianchi modular symbols on Bianchi 3-folds. By using the homology generation result, we prove that the μ-invariant of some isotopic components of the p-adic L-function of certain Bianchi Hecke eigenforms vanishes for a positive proportion of ordinary prime ideals. The main ingredient of the proof of the generation result is analyzing the fast convergent series expression of the integration of the Bianchi modular forms along the special modular symbols.