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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.
The aim of this work is to provide a new approach for constructing 
$n$-dimensional Steinberg symbols on discrete valuation fields from 
$\left( n\,+\,1 \right)$-cocycles and to study reciprocity laws on curves related to these symbols.
