From Binary Hermitian Forms to parabolic cocycles of Euclidean Bianchi groups

TL;DR

This paper introduces a new family of functions based on binary Hermitian forms over Euclidean imaginary quadratic fields, constructing parabolic cocycles for Bianchi groups and linking their averages to special L-values, enabling efficient computations.

Contribution

It presents a novel class of functions derived from binary Hermitian forms, leading to the construction of nontrivial parabolic cocycles and new formulas for L-value computations.

Findings

Constructed nontrivial cocycles for Euclidean Bianchi groups.

Linked average values of functions to special L-values.

Provided efficient computational formulas for L-values.

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 $d_K$. 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 $L(\chi_{d_K},s)$. Using the properties of these functions we give new and computationally efficient formulas for computing some special values of $L(\chi_{d_K},s)$.