Abstract
Several authors have studied homomorphisms from first homology groups of modular curves to $K_2(X)$ , 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 $\mathrm {GL}_2(\mathbb {Z})$ 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.
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