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Algebraic properties of Shintani’s generating functions: Dedekind sums and cocycles on \(\text{PGL}_2(\mathbb{Q})\). (English) Zbl 0920.11026
From the author’s introduction: This paper seeks to make a new contribution to a circle of ideas linking special values of zeta-functions with generalised Dedekind sums. The special values in question are those taken at non-positive integers by the partial zeta-function \(\zeta_K(s,{\mathfrak a})\) associated to a ray-class \({\mathfrak a}\) of a real quadratic field \(K\). Siegel was the first to obtain explicit general formulae for these values in [C. L. Siegel, Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1968, 7-38 (1968; Zbl 0273.12002) and ibid. 1969, 87-102 (1969; Zbl 0186.08804)]. His formulae involve sums of products of values of Bernoulli polynomials, sums which can be recognised as generalisations of the sums ‘\(s (h,k)\)’ of Dedekind appearing in the transformation law for the logarithm of the \(\eta\)-function Siegel’s formulae show in particular that these special values are rational, a result extended by Klingen and Siegel to an arbitrary totally-real numher field \(F\) of degree \(d\) over \(\mathbb{Q}\). In his paper [T. Shintani, J. Fac. Sci. Univ. Tokyo, Sect. IA 23, 393-417 (1976; Zbl 0349.12007)], Shintani gave a new proof of this fact by means of some remarkable explicit formulae for \(\zeta_F(-k,{\mathfrak a})\), \(k=0,1,2,\dots\). His method is entirely different from Siegel’s. It involves the construction of cones in \(\mathbb{R}^d\) to which one associates certain quotients of \(d\)-variable power-series. Shintani proved by complex-analytic methods that, roughly speaking, these quotients act as generating functions for the above-mentioned special values. When \(d\) equals 1 they are the generating functions of the Bernoulli polynomials themselves. These polynomials are well known to give the special values of Hurwitz’ zeta-functions (the case \(d=1\), \(F=\mathbb{Q})\). In the case \(d=2\), Shintani showed how Siegel’s explicit formulae for \(\zeta_K(-k, {\mathfrak a})\) could be recovered from his and the Dedekind-type sums reappear.
A new and unifying element enters the circle in the work of G. Stevens [Théorie des nombres, C. R. Conf. Int., Québec/Can. 1987, 887-927 (1989; Zbl 0684.10028)] and R. Sczech [Comment. Math. Helv. 67, 363-382 (1992; Zbl 0776.11021)]. Both of these author construct ‘universal’ 1-cocycles on the group \(\text{PGL}_2(\mathbb{Q})\). On the one hand their cocycles can be written out explicitly in terms of generalised Dedekind sums. On the other, by specialising them at the appropriate matrix (representing the action of a certain unit), they can be used to calculate the partial zeta-values for any real quadratic field \(K\). Both constructions are analytic in nature. (Stevens uses certain ‘periods’ of Eisenstein series and his theory of ‘modular caps’ and modular symbols, making the connection with zeta-values by means of Siegel’s work. Sczech constructs his cocycles by real-analytic methods). The importance of the cocycle interpretation can be seen from its applications both to the calculation of zeta-values – where it gives rise to highly efficient continued-fraction algorithms – and to generalised Dedekind sums, where it produces very general versions of Dedekind’s ‘Reciprocity Law’ for the sums \(s(h,k)\). These applications are explained in the articles of Stevens and Sczech cited above (see also [D. Hayes, Exp. Math. 8, 137-184 (1990; Zbl 0698.12007)] for the zeta-values at \(s=0)\) although the generalised reciprocity laws are ‘predicted’ rather than written out precisely.
The aim of this paper is to present 1-cocycles on \(\text{PLG}_2 (\mathbb{Q})\) of a similar nature to those of Sczech and Stevens but with two significant differences. Firstly, our ‘Shintani Cocycles’ are ‘parabolic’: they vanish when evaluated at (the images of) matrices of the form \(\left(\begin{smallmatrix} a & b\\ 0 & d\end{smallmatrix}\right)\) in \(\text{PGL}_2 (\mathbb{Q})\). Those of Sczech and Stevens are not parabolic; but then the spaces in which they take values, though similar, are not the same either. Secondly, our construction is completely elementary and essentially algebro-combinatorial. The fundamental tool is a modification of the formal generating functions which were introduced by Shintani and can be attached to certain geometrical data involving cones and lattices in \(\mathbb{R}^2\). The object of Section 2 is to define these Shintani functions and then to elucidate their fundamental properties, not as generators of zeta-values but rather as formal algebraic objects in their own right. A precise connection with Dedekind sums is established in Section 3. It transpires that the coefficients of the Shintani functions are, essentially, the elements of a doubly-infinite sequence of highly general sums which were defined explicitly by U. Halbritter in [Result. Math. 8, 21-46 (1985; Zbl 0577.10011)], although they were already present in Siegel’s formulae. The functional properties of Section 3 consequently lead to new and elementary proofs of certain identitites for these sums, not only Halbritter’s generalisation of the reciprocity law but also the so-called ‘Generalised Petersson-Knopp Identities [see e.g. T. Apostol and T. Vu, J. Number Theory 14, 391-396 (1982; Zbl 0487.10012)]. One could probably devise further applications of a similar nature.
The final section of this paper deals with the construction of cocycles by means of Shintani functions. Thanks to the results of Section 2, very little actually needs to be done once the appropriate framework of \(\text{PGL}_2(\mathbb{Q})\)-modules of distributions on \(\mathbb{R}^2/\mathbb{Z}^2\) has been set up.

MSC:
11F20 Dedekind eta function, Dedekind sums
11R42 Zeta functions and \(L\)-functions of number fields
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