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Properties of higher-dimensional Shintani generating functions and cocycles on \(\text{PGL}_ 3(\mathbb{Q})\). (English) Zbl 1045.11035
From the introduction: This paper is the fruit of an attempt to generalise the two-dimensional results of D. Solomon [Compos. Math. 112, 333–362 (1998; Zbl 0920.11026)] to arbitrary dimension \(n\). The ultimate goal of such a generalisation would be a formal, algebraic construction of \((n-1)\)-cocycles on \(\text{PGL}_n(\mathbb{Q})\) similar, perhaps cohomologous, to those constructed by analytic methods in R. Sczech [Invent. Math. 113, 581–616 (1993; Zbl 0809.11029)], for all \(n\). In the present work we modify and generalise the ‘Shintani Function’-based approach of Solomon (loc. cit.) to construct cocycles for \(n=2\) and 3. Although actual cocycles have not (so far) been obtained in this way for \(n>3\), nevertheless, a large number of our results are valid in any dimension and can, for instance, be applied to ‘reciprocity laws’ for \(n\)-dimensional Dedekind sums. Such laws were derived in the first author’s thesis. S. Hu [‘Shintani cocycles and generalized Dedekind sums’, PhD Thesis, University of Pennsylvania (1997)].
In addition to some minor departures from the definitions and notations of Solomon’s paper, the construction presented in this paper embodies two particular new features of greater importance. Firstly, in his paper the Shintani functions were constructed using simplicial cones in a (two-dimensional) real vector space that were parametrised by means of their generating rays. Here, cones in an \(n\)-dimensional vector space \(W\) are parametrised not by their rays but by the \(n\) elements of the projective dual space \(\mathbb{P}(W^*)\) which define their codimension-1 faces. This change, banal as it may seem, solves certain combinatorial problems that would otherwise appear on the boundaries of the cones in the higher-dimensional ‘generic case’ of the fundamental alternating relation (see Theorem 3.1). The latter relation is essential to the cocycle construction. Note that in two dimensions there is a natural, two-to-one correspondence between individual rays in \(W\) and elements of \(\mathbb{P}(W^*)\), and so the new construction for \(n=2\) is essentially equivalent to that of Solomon’s paper. Unfortunately, there is still a potential stumbling-block to the ‘non-generic’ case in dimensions 4 and higher: it is not clear how (or whether) we can define the cocycle in situations where the corresponding cones degenerate.
The other major innovation here is the introduction of a new, two-step method. In §§2 to 4 of this paper we define Shintani functions with values in a certain ‘abstract’ space \(\mathbb{Q} \langle W \rangle\), as well as the corresponding cocycles. It is only in §5 that we ‘specialise’ these to obtain functions and cocycles with values in spaces of (quotients of) formal power series, similar to those of Solomon’s. Not only does this approach clarify and simplify some of the proofs, it also shows how one might obtain cocycles of a similar nature in other situations, for instance by simply employing another specialisation in the second stage of the method.
Many of the results in this paper were established in a similar form in the PhD thesis of the first author. However, the methods used here differ in certain respects from those of the thesis, and this has led to a number of improvements.

MSC:
11F75 Cohomology of arithmetic groups
11S40 Zeta functions and \(L\)-functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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