Integral Eisenstein cocycles on GLn, II : Shintani's method

TL;DR

This paper constructs a new cocycle on GLn using Shintani's method, providing novel proofs of fundamental theorems, relating to Eisenstein cocycles, and connecting to p-adic L-functions and Gross's conjecture.

Contribution

It introduces an integral version of the Shintani cocycle, relates it to Eisenstein cocycles, and links cohomological constructions to p-adic L-functions and number theory conjectures.

Findings

New proof of Diaz y Diaz and Friedman's theorem on signed fundamental domains.

Cohomological reformulation of Shintani's proof of the Klingen-Siegel rationality theorem.

Construction of an integral Shintani cocycle related to p-adic L-functions.

Abstract

We define a cocycle on Gln using Shintani's method. It is closely related to cocycles defined earlier by Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corollary of our result we obtain a new proof of a theorem of Diaz y Diaz and Friedman on signed fundamental domains, and give a cohomological reformulation of Shintani's proof of the Klingen-Siegel rationality theorem on partial zeta functions of totally real fields. Next we prove that the cohomology class represented by our Shintani cocycle is essentially equal to that represented by the Eisenstein cocycle introduced by Sczech. This generalizes a result of Sczech and Solomon in the case n=2. Finally we introduce an integral version of our Shintani cocycle by smoothing at an auxiliary prime ell. Applying the formalism of the first paper in this series, we prove that certain specializations of the smoothed class yield the p-adic L-functions of totally real fields. Combining our cohomological construction with a theorem of Spiess, we show that the order of vanishing of these p-adic L-functions is at least as large as the one predicted by a conjecture of Gross.