A family of \'etale coverings of the affine line

Abstract

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.