For an arbitrary ring A, we study the abelianization of the elementary group $\mathit{{\rm E}}_2(A)$. In particular, we show that for a commutative ring A there exists an exact sequence

\begin{equation*}{\rm K}_2(2,A)/{\rm C}(2,A) \rightarrow A/M \rightarrow \mathit{{\rm E}}_2(A)^{\rm ab} \rightarrow 1,\end{equation*}

where ${\rm C}(2,A)$ is the central subgroup of the Steinberg group $\mathit{{\rm St}}(2,A)$ generated by the Steinberg symbols and M is the additive subgroup of A generated by $x(a^2-1)$ and $3(b+1)(c+1)$, with $x\in A, a,b,c \in {A^\times}$.

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.