Colloquium abstracts

Soumyadip Das
TIFR, Mumbai
October 22, 2020

The Inertia Conjecture and Its Generalizations:  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 etale 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.