The $p$-adic Shintani cocycle

TL;DR

This paper provides an explicit criterion for when the Shintani cocycle can be p-adically interpolated, enabling the construction of p-adic L-functions for totally real fields, building on and recovering classical results.

Contribution

It introduces a new explicit criterion for p-adic interpolation of the Shintani cocycle, facilitating the construction of p-adic L-functions for totally real fields.

Findings

Established a criterion for p-adic interpolation of the Shintani cocycle.

Recovered classical results on p-adic L-functions for totally real fields.

Connected cohomological interpretations with p-adic analytic properties.

Abstract

The Shintani cocycle on $\GL_n(\Q)$, as constructed by Hill, gives a cohomological interpretation of special values of zeta functions for totally real fields of degree $n$. We give an explicit criterion for a specialization of the Shintani cocycle to be $p$-adically interpolable. As a corollary, we recover the results of Deligne-Ribet, Cassou Nogu\`es and Barsky on the construction of $p$-adic $L$-functions attached to totally real fields.