Bianchi Modular Forms and the Rationality of Periods

Abstract

Using an explicit Eichler-Shimura-Harder isomorphism, we establish the analogue of Manin's rationality theorem for Bianchi periods and hence special values of $L$-functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.