The Inertia Conjecture and Its Generalizations

Abstract:

In this talk, I will present Abhyankar's Inertia Conjecture, some of its
generalizations and evidence towards these problems.

In 1957, Abhyankar conjectured that the finite groups that occur as the
Galois groups of the \'{e}tale connected Galois covers of the affine line
over an algebraically closed field of prime characteristic $p$ are
precisely the quasi $p$-groups (groups generated by their Sylow
$p$-subgroups). This is now a Theorem due to Serre and Raynaud. In 2001,
Abhyankar proposed a refined conjecture on the inertia groups that occur
over $\infty$ for such covers, now known as the Inertia Conjecture. The
conjecture remains wide open at the moment. We will see the previously
known evidence and discuss the new ones together with the technique used.
We will also see some generalizations of the conjecture and the evidence
towards them. The talk will be based on the articles `On the Inertia
Conjecture for alternating group covers', J. Pure Appl. Agebra, vol. 224,
9, 2020. https://doi.org/10.1016/j.jpaa.2020.106363. (with Manish Kumar)
and `The Inertia Conjecture and its generalizations', Preprint, 2020.
arXiv:2002.04934(Submitted).