A family of étale coverings of the affine line

This note was inspired by a colloquium talk given by S. S. Abhyankar at the Tata Institute, on the work of Abhyankar, Popp and Seiler (see [2]). It was pointed out in this talk that classical modular curves can be used to construct (by specialization) coverings of the affine line in positive characteristic. In this “modular” optic it seemed natural to consider Drinfel’d modular curves for constructing coverings of the affine line. This note is a direct outgrowth of this idea. While it is trivial to see that affine line in characteristic zero has no non-trivial étale coverings, in [1] it was shown that the situation in positive characteristic is radically different and far more interesting. Let us, for the sake of definiteness, work over a field k of characteristic p > 0. In [1], Abhyankar conjectured that any finite group whose order is divisble by p and which is generated by its p-Sylow subgroups (such a finite group is sometimes called a “quasi-p-group”), occurs as quotient of the algebraic fundamental group of the affine line. It is customary to write π 1 (A 1 k) to denote the algebraic fundamental group of the affine line. While Abhyankar’s conjecture indicates that the algebraic fundamental group of the affine line is quite complicated, our result perhaps illustrates its cyclopean proportions. The result we prove (see Theorem 3 below) is the analogue of the following wellknown classical result, which falls out of the theory of elliptic modular curves over the field of complex numbers: there is a continous quotient π 1 (P 1 C − {0, 1728,∞}) −→ ∏