Eisenstein cocycles in motivic cohomology

TL;DR

This paper introduces a new construction of Eisenstein cocycles in motivic cohomology, providing explicit maps from modular curve homology to K-theory, with proofs of Hecke equivariance using cocycles.

Contribution

It offers a novel construction of homomorphisms from modular curve homology to K-theory, demonstrating Hecke equivariance via cocycles, extending previous work with a new approach.

Findings

New construction of Eisenstein cocycles in motivic cohomology.

Explicit maps from homology to K-theory with Hecke equivariance.

Proofs using cocycles from GL_2(Z) to K-groups.

Abstract

Several authors have studied homomorphisms from first homology groups of modular curves to the second K-group of a cyclotomic ring or a modular curve X. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a 1-cocycle from GL_2(Z) to the second K-group of the function field of a suitable group scheme over X, from which the maps of interest arise by specialization.