Eisenstein cocycles in motivic cohomology

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.

88 pages

References in corpus (3)

• Integral Grothendieck-Riemann-Roch theorem
• On the $K_4$ group of modular curves
• On a conjecture of Sharifi and Mazur's Eisenstein ideal

Cited by in corpus (2)

• Shintani-Barnes cocycles and values of the zeta functions of algebraic number fields
• A Dedekind-Rademacher cocycle for Bianchi groups