From binary Hermitian forms to parabolic cocycles of Euclidean Bianchi groups

Abstract

We study a family of functions defined in a very simple way as sums of powers of binary Hermitian forms with coefficients in the ring of integers of an Euclidean imaginary quadratic field K with discriminant dK. Using these functions we construct a nontrivial cocycle belonging to the space of parabolic cocycles on Euclidean Bianchi groups. We also show that the average value of these functions is related to the special values of L(χdK,s). Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L(χdK,s).