(References in brackets are to my papers and preprints. Click for more details)
In 1976, Takuro Shintani gave explicit formulae for the values at negative integers of the partial ray-class zeta-functions modulo a conductor of a totally real field k. These can be interpreted in terms of generating functions associated to a `cone decomposition' of a fundamental domain for the action of a unit group on a lattice in R^n.
The initial aim of [8] was to make the cone decomposition more canonical for k real quadratic by considering the sequence of conductors fp^nO_k (for fixed f and prime p) which corresponds to a Z_p tower by class field theory. By examining the behaviour of the associated (2-variable) generating functions under `juxtaposition of cones', I proved that, as n increases, they tend to a limit in an appropriate p-adic normed space. The limiting function enjoys special properties and generates partial zeta-values at all levels in the tower. Article [9] uses the same juxtaposition property of generating functions (now associated to abstract plane lattices and cones) to give an algebraic construction of certain 1-cocycles on PGL_2(Q) which I called `Shintani Cocycles'. This leads to new algebraic proofs of the Petersson-Knopp identities and a reciprocity law for highly generalised Dedekind sums originally proved analytically by Halbritter. Article [10] establishes the non-triviality of these cocycles and examines some cohomologous variants. One is identical with the cocycles constructed by G. Stevens using an extension of modular symbols to the Borel-Serre compactification of the upper half-plane. Another variant is shown to be $p$-adically interpolable (after `smoothing') and so has applications to the construction of $p$-adic partial zeta-functions. previously, R. Sczech had also used also analytic methods to construct higher dimensional cocycles which, in dimension 2, are closely related to Stevens'. The joint paper [11] with S. Hu attempts to generalise the construction in dimension n>2 to produce Shintani (n-1)-cocycles on PGL_n(Q). We succeed for n=3 by dualising the parametrisation of the cones. However we obtain only a partial result for n>3 because of certain degenerate configurations. Subsequently R. Chapman produced reciprocity laws for higher-dimensional Dedekind using techniques inspired by our method. More recently R. Hill developed it considerably and succeeded in constructing higher-dimensional Shintani cocycles by using `algebraic infinitesimals' to cope with the degenerations (unpublished).