On the Inertia Conjecture and its generalizations

Studying two point branched Galois covers of the projective line we prove the Inertia Conjecture for the Alternating groups $A_{p+1}$, $A_{p+3}$, $A_{p+4}$ for any odd prime $p \equiv 2 \pmod{3}$ and for the group $A_{p+5}$ when additionally $4 \nmid (p+1)$ and $p \geq 17$. We obtain a generalization of a patching result by Raynaud which reduces these proofs to showing the realizations of the \'{e}tale Galois covers of the affine line with a fewer candidates for the inertia groups above $\infty$. We also pose a general question motivated by the Inertia Conjecture and obtain some affirmative results. A special case of this question, which we call the Generalized Purely Wild Inertia Conjecture, is shown to be true for the groups for which the purely wild part of the Inertia Conjecture is already established. In particular, we show that if this generalized conjecture is true for the groups $G_1$ and $G_2$ which do not have a common quotient, then the conjecture is also true for the product $G_1 \times G_2$.