On the Inertia Conjecture for Alternating group covers

Abstract

The wild part of Abhyankar's Inertia Conjecture for a product of certain Alternating groups is shown for any algebraically closed field of odd characteristic. For d a multiple of the characteristic of the base field, a newétale A d -cover of the affine line is obtained using an explicit equation and it is shown that it has the minimal possible upper jump.Conjecture 1.1. Let G be a finite quasi p-group.Wild Part: A p-subgroup P of G occurs as the inertia group of a G-Galois cover of P 1 branched only at ∞ if and only if the conjugates of P generate G. Tame Part: Assume that a p-subgroup P of G occurs as the inertia group of a G-Galoisétale cover of the affine line. Let β be an element of G of prime-to-p order contained in the normalizer of P in G. Let I be an extension of β by P in G. Then there is a G-Galois cover φ : Y → P 1 branched only at ∞ with inertia group I at a point in Y over ∞.By Abhyankar's Lemma ([18, XIII, Proposition 5.2]), one can reduce the tame part of the inertia group. On the other hand, Harbater has shown ([6, Theorem 2]) that the wild part of the inertia group can be increased. As a consequence, the inertia conjecture

Galois Covers of Singular Curves in Positive Characteristics

“…Similarly, since π 1 (Y) (p) for a smooth projective k-curve Y is a free group on s Y (= p-rank of Y) generators ( [22]), (2) follows. Now we prove (3). By Theorem 3.3, G is generated by its subgroups H 1 , .…”

Towards the Purely Wild Inertia Conjecture

“…[10, Section 2]) produces a connected G-Galois étale cover of the affine line with I as an inertia group above ∞. We use similar arguments with a variation of the Lefschetz type principle (also used in [4,Lemma 3.3]) to obtain the required isomorphism of the local extensions.…”

“…We will use this to prove the realization of certain inertia groups in the context of a perfect wreath product (Proposition 6.1) where P 1 and P 2 both have order p, and G is generated by G 1 and G 2 . A special case of this result is [4,Theorem 5.2] where G = G 1 × G 2 , a product of perfect quasi p-groups, P 1 is a cyclic p-group and P 2 has order p. The long proof of the Theorem is broken into several steps for the ease of reading. In the first step, we use patching technique to obtain two Galois étale covers of the affine line with P 1 × P 2 as an inertia group above ∞ for both covers, and such that 1 occurs as a lower jump (or equivalently, an upper jump; see Equation (A.1)) in the corresponding ramification filtration.…”

“…To prove this, we first apply a formal patching result Lemma 3.2 (which is very similar to [19,Theorem 2.3.7]) to a cover obtained in [4,Theorem 5.2] and a suitable Harbater-Katz-Gabber cover to obtain a connected S rp -Galois cover of P 1 that is étale away from {0, ∞}. Taking a pullback under a Kummer cover, we obtain the result.…”

“…For a summary on these works and other important developments, see [14,Section 4]. Recent works from [4], [5], [7] shed light on several systematic ways to construct the Galois étale covers of the affine line. As a consequence, a larger class of groups provide evidence towards the above conjectures.…”