The Barnes-Hurwitz zeta cocycle on PGL2(Q)

We introduce a 1-co cycle 3 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 is an element of 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 3 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. (c) 2022 Elsevier Inc. All rights reserved.