The Barnes-Hurwitz zeta cocycle on [formula omitted].

We introduce a 1-cocycle Z of the group G = PGL 2 (Q) with values in a module D of distributions (in the sense of Stevens and Hu-Solomon). This cocycle is essentially constructed from the Barnes' double zeta function and it has the advantage of defining a family of maps that depend meromorphically on the usual parameter s ∈ C. In particular, this permits the extension of the cocycle property to any Taylor coefficient of such zeta function at s = 0. Furthermore, we show that the class of Z in the first cohomology group H 1 (G , D) is nonzero, and we use basic facts about the arithmetic of real quadratic fields to prove the vanishing of H 0 (G , D) , the group of G -invariant elements in D. [ABSTRACT FROM AUTHOR]