In this paper we investigate a result of Ueno on the modularity of generating series associated to the zeta functions of binary Hermitian forms previously studied by Elstrodt et al. We improve his result by showing that the generating series are Eisenstein series. As a consequence we obtain an explicit formula for the special values of zeta functions associated with binary Hermitian forms.
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(χ d K , s). Using the properties of these functions we give new and computationally efficient formulas for computing some special values of L(χ d K , s).
Contents1. Introduction 1 2. Definition and Elementary Properties of H k,∆ 5 3. Continuity of H k,∆ 14 4. The average value of H k,∆ and Cohen-Zagier type formulas 18 5. Cocycle property of H k,∆ 21 References 34
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