Integral Eisenstein cocycles on ${\mathbf{GL}}_n$, II: Shintani’s method

Abstract

We define a cocycle on ${\mathbf{GL}}_n(\mathbf{Q})$ using Shintani's method. This construction is closely related to earlier work of 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 relate the Shintani cocycle to the Sczech cocycle by showing that the two differ by the sum of an explicit coboundary and a simple "polar" cocycle. This generalizes a result of Sczech and Solomon in the case $n=2$.

Finally, we introduce an integral version of our cocycle by smoothing at an auxiliary prime $\ell$. This integral refinement has strong arithmetic consequences. We showed in previous work that certain specializations of the smoothed class yield the $p$-adic $L$-functions of totally real fields. Furthermore, combining our cohomological construction with a theorem of Spiess, one deduces that that the order of vanishing of these $p$-adic $L$-functions is at least as large as the expected one.