Some fundamental groups in arithmetic geometry

Abstract

Deligne’s recently (2012) proved an analogue for smooth projective varieties over a finite field. There are some new complications: you need to have a Cartier divisor D. The result says roughly that there are finitely many irreducible Q` Weil sheaves of rank r with ramification bounded by D, up to twist by Weil characters. In characteristic zero, the proof is by a Lefschetz hyperplane theorem argument to reduce the theorem to a curve, since the theorem tells you that sufficiently “nice” curve C ⊂ X has the property that π1(C) π1(X). Thus, the study of representations of π1(X) reduces to that of the representations of π1(C)> Now this doesn’t work in characteristic p. Drinfeld proved a kind of salvage for the tame part of the etale fundamental group. The proof was not too hard: the main ingredients are Bertini’s Theorem and properties of tameness. This was subsequently enhanced by Kindler to obtain a “full package” Lefschetz Hyperplane Theorem for the tame part of the etale fundamental group. In ’14, Kerz-S. Saito recovered an even stronger version of Lefschetz hyperplane over perfect fields for the abelian part of the fundamental group. There are many technical hypotheses, but the punchline is that for a divisor D supported in X X and Y ⊂ X in “good position” and of dimension at least 2, then