Eisenstein cocycles in motivic cohomology

Several authors have studied homomorphisms from first homology groups of modular curves to K₂ (X), with X either a cyclotomic ring or a modular curve. 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 GL₂ (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.