Shintani Cocycles on $\GL_{n}$

TL;DR

This paper constructs an n-1-cocycle on GL_n(Q) valued in distributions on finite adèles, enabling evaluation of Artin L-functions at negative integers, generalizing Solomon's work for n=2.

Contribution

It introduces a new cocycle on GL_n(Q) with distribution values, extending Solomon's n=2 case to higher dimensions for L-function evaluations.

Findings

Defines an n-1-cocycle on GL_n(Q) with distribution values.

Provides a method to evaluate Artin L-functions at negative integers.

Generalizes Solomon's cocycle from n=2 to arbitrary n.

Abstract

The aim of this paper is to define an n-1-cocycle $\sigma$ on $\GL_{n}(\Q)$ with values in a certain space $\hD$ of distributions on $\A_f^{n}\setminus\{0\}$. Here $\A_f$ denotes the ring of finite ad\`{e}les of $\Q$, and the distributions take values in the Laurent series $\C((z_{1},...,z_{n}))$. This cocycle can be used to evaluate special values of Artin L-functions on number fields at negative integers. The construction generalizes that of Solomon in the case n=2.