The Shintani cocycle. II: Partial \(\zeta\)-functions, cohomologous cocycles and \(p\)-adic interpolation
In a previous paper, the author [Compos. Math. 112, 333--364 (1998; Zbl 0920.11026)] gave an algebraic definition of a 1-cocycle on \(\text{PGL}_2(\mathbb Q)\) whose values are expressible in terms of (ratios of) two-variable formal power series. This Shintani cocycle is similar to cocycles constructed analytically by \textit{G. Stevens} [Proc. Int. Number Theory Conf., Univ. Laval 1987, de Gruyter, Berlin, 887--927 (1989; Zbl 0684.10028)] and \textit{R. Sczech} [Comment. Math. Helvetici 67, 363--382 (1992; Zbl 0776.11021)]. The present article explores the relations between the Shintani cocycle and values at non-positive integers of partial ray class zeta functions of real quadratic fields. Let \(K\) be a real quadratic field, \(\mathfrak c\) a ray class for \(K\), and \(m\) a non-negative integer. The author defines a linear operator \(V_K^{(m)}\) whose input is a pair \((\Phi, \mathfrak c)\), where \(\Phi\) lies in a certain group of cocycles on \(\text{GL}_2(\mathbb Z)\). Its output is a real number that is zero when \(\Phi\) is a coboundary and is \(\zeta_K(-m, \mathfrak c)\) (with slight modifications when \(m=0\)) when \(\Phi\) is the restriction to \(\text{GL}_2(\mathbb Z)\) of the \(2m\)th homogeneous part of the Shintani cocycle. This shows that the Shintani cocycle is not a coboundary. The author makes two cohomologous deformations of the cocycle, one of which is closely related to those of Stevens and Sczech, and the other of which has good \(p\)-adic interpolation properties. This gives a new method of constructing \(p\)-adic analogues of the functions \(\zeta_K(s,\mathfrak c)\), which were first constructed by \textit{J. Coates} and \textit{W. Sinnott} [Invent. Math. 25, 253--279 (1974; Zbl 0305.12008)].
scientific article; zbMATH DE number 1274813
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