On the Inertia Conjecture for Alternating group covers

Abstract

Let G1 and G2 be perfect groups such that there exist connected G1-Galois and G2-Galois étale covers of the affine line over an algebraically closed field of characteristic p>0 with the cyclic p-groups P1 and P2 as the inertia groups above ∞, respectively. Then we show that there is a connected G1×G2-Galois étale cover of the affine line with an inertia group I above ∞ where I is a cyclic subgroup of P1×P2 of index p. As a consequence, it is shown that the wild part of the Inertia Conjecture is true for any product of Alternating groups, each of degree p or coprime to p. For d a multiple of p, a new étale Ad-cover of the affine line is obtained using an explicit equation, and it is shown that this cover has the minimal possible upper jump.