The Milnor K-theory and the Shintani cocycle

2019/09/08 by Lim, Sung Hyun, Park, Jeehoon
#11F15 #11R70 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1909.03450

Abstract

The goal of this article is to complete the unfinished construction (due to Glenn Stevens in an old preprint) of a certain Milnor K-group valued group cocycle for GL_n(ℚ) where n is a positive integer, which we call the Stevens cocycle. Moreover, we give a precise relationship between the Stevens cocycle and the Shintani cocycle, which encodes key informations on the zeta values of totally real fields of degree n, using the dlog map of K-theory and the Fourier transform of locally constant functions on ℚⁿ with bounded support. Roughly speaking, the Stevens cocycle is a multiplicative version of the Shintani cocyle.

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