On the Inertia Conjecture and its generalizations

We study the two-point branched Galois covers of the projective line given by explicit equations and with prescribed branched data. We also obtain several results using formal patching techniques that are useful in realizing new Galois covers. As a consequence, we prove the Inertia Conjecture for the alternating groups A p +1 , A p +3 , A p +4 when p ≡ 2 (mod 3) is an odd prime and for the group A p +5 when additionally 4 ∤ ( p + 1) and p ≥ 17. 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. 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 × G 2 .