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Properties of Higher‐Dimensional Shintani Generating Functions and Cocycles on Pgl 3 (Q)
Author(s) -
Hu Shubin,
Solomon David
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012612
Subject(s) - mathematics , reciprocity law , pure mathematics , dimension (graph theory) , algebraic number , hyperplane , algebra over a field , discrete mathematics , combinatorics , mathematical analysis
In an earlier paper the second author used the formal, algebraic properties of 2‐dimensional Shintani generating functions to construct a 1‐cocycle on PGL 2 {Q}. We aim to generalise these results by using such functions in dimension n to obtain an ( n −1)‐cocycle on PGL n {Q}, presumably related to the Bernoulli and Eisenstein cocycles of R. Sczech. By improving our methods we achieve this goal for n =3. For n >3 we encounter obstacles related to degenerate configurations of hyperplanes in n ‐space. Nevertheless, we obtain partial results closely connected to reciprocity laws for certain n ‐dimensional Dedekind sums. 1991 Mathematics Subject Classification : 11F20, 11F75.