The étale fundamental group is a generalization of the topological fundamental group and that of the Galois group. We begin the talk with a brief recollection of this generalization.
In the talk, we focus on the abelianized étale fundamental group of a variety \(X\) and denote it by \(\pi_1^{ab}(X)\). One advantage of working with the abelianized étale fundamental group is that, just as in topology, it is the dual of a cohomology group \(H^1(X, \mathbb{Q}/\mathbb{Z})\), which makes it a bit easier to do calculations. It is known that the
torsion subgroup of \(\pi_1^{ab}(X)\) is finite when \(X\) is a smooth projective variety over a finite field or a local field.
In the talk, we sketch a proof of the finiteness of the torsion subgroup of \(\pi_1^{ab}(X)\) when \(X\) is a regular (not necessarily smooth) geometrically integral projective variety over a positive characteristic local field.
The talk is based on a joint work with Dr. Jitendra Rathore.
