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Covers of the Affine Line in Positive Characteristic with Prescribed Ramification

Let k be an algebraically closed field of characteristic p > 0. In this note we consider Galois covers g : Y → P1k which are only branched at t = ∞. We call such covers unramified covers of the affine line. It follows from Abhyankar’s conjecture for the affine line proved by Raynaud ([7]) that such a cover exists for a given group G if and only if G is a quasi-p group, i.e. can be generated by elements of p-power order. Many groups satisfy this property, for example all simple groups of order divisible by p are quasi-p groups. For a given quasi-p group G there are many unramified covers of the affine line. For example, if G = Z/pZ there are infinitely many families of such covers. In this note we fix a quasi-p group G and ask what the minimal genus of an unramified G-Galois cover of the affine line is. This question has been motivated by the work of Muskat–Pries ([6]). Muskat and Pries compute the genus of many alternating-group covers which had been found by Abhyankar ([1]). It turns out that the covers they consider have “small” genus in a sense which can be made precise. We show that for p + 2 ≤ d < 2p the covers of [6] are indeed the unramified Ad-covers of the affine line of minimal genus. Another motivation for studying the existence of covers with given ramification comes from the theory of stable reduction. One method for showing that covers with given tame ramification exist in positive characteristic is to show that not all covers have bad reduction to characteristic p. Associated with a cover with bad reduction is a set of tail covers. Essentially these are the restriction to certain irreducible components of the stable reduction of the cover. A technique due to Wewers ([9]) sometimes allows one to count the number of covers with bad reduction in terms of sufficient knowledge of these tail covers. This technique has been applied in [4] to covers of degree p which are tamely branched at 4 points. To generalize the result of [4] to degree p < d < 2p one needs to know the existence of covers of the affine line of degree d of small genus. The strategy of the proof of our main result is as follows. Rather than considering covers with fixed Galois group, we consider unramified covers f : X → P of the affine line which are non-Galois. Let d be the degree of f . As an application of the Riemann–Hurwitz formula in positive characteristic, we compute the genus of the Galois closure of f in terms of the ramification of f