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 $L({\chi }_{d_K},s)$.

中文翻译：

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从二元 Hermitian 形式到 Euclidean Bianchi 群的抛物线 cocycles

我们研究了一系列以非常简单的方式定义的函数，它们是具有判别式的欧几里得虚二次域K的整数环中的系数的二元厄密形式的幂和$d_{ķ}$. 使用这些函数，我们构造了一个属于欧几里得 Bianchi 群上抛物线余环空间的非平凡余环。我们还表明，这些函数的平均值与$\mathrm{大号}({\chi }_{d_{ķ}},s)$. 利用这些函数的性质，我们给出了计算一些特殊值的新的和计算效率高的公式$\mathrm{大号}({\chi }_{d_{ķ}},s)$.