Abstract:
In this paper the following description of the congruence kernel
$C(\mathrm{SL}_2,\mathcal O)$ is given, where $\mathcal O$ is the coordinate ring of an affine curve obtained by removing a point from a projective curve over a finite
field $k_0$.
Theorem.
{\it
$C(\mathrm{SL}_2,\mathcal O)=(*_{x\in X}H_x)*P$ is the free profinite product over a separable space $X$ of groups $H_x$ that are isomorphic to the direct product
$\prod\mathbb{Z}/p\mathbb{Z}$ of a continuum of groups of order
$p=\operatorname{char}(k_0)$, and a separable projective group $P$ each of whose open subgroups is free.
}
The proof uses a general result on normal subgroups of free profinite products.
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