Alternating group covers of the affine line

Abstract

We prove Abhyankar's Inertia Conjecture for the alternating group A_{p+2} on p+2 letters when p = 2 mod 3, by showing that every possible inertia group occurs for a (wildly ramified) A_{p+2}-Galois cover of the projective k-line branched only at infinity where k is an algebraically closed field of characteristic p > 0. More generally, when 1 < s < p and gcd(p-1, s+1)=1, we prove that all but finitely many rational numbers which satisfy the obvious necessary conditions occur as the upper jump in the filtration of higher ramification groups of an A_{p+s}-Galois cover of the projective line branched only at infinity.