Mathematics – Algebraic Geometry
Scientific paper
1993-09-02
Mathematics
Algebraic Geometry
The file is in standard LateX, eight pages
Scientific paper
In the note we construct a family of \'etale coverings of the affine line. More specifically, let $F$ be a finite field of characteristic $p$ and suppose that the cardinality of $F$ is at least 4. Let $A = F[T]$ be the polynomial ring in one variable $T$, $K=F(T)$. Let $K_\infty$ be the completion of $K$ along the valuation given by $1/T$, and let $C$ be the completion of the algebraic closure of $K_\infty$. We prove in this note that there is a continous surjection $$\pi_1^{alg}(\A^1_C) \to \lim_{\leftarrow \atop I} SL_2( A/I )/{\pm 1},$$ where $\pi_1^{alg}(\A^1_C)$ is the algebraic fundamental group of the affine line over $C$, and the inverse limit on the right (above) is taken over all nonzero proper ideals in $A=F[T]$. We use the theory of Drinfel'd modular curves to obtain these coverings.
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