A conjecture on arithmetic fundamental groups

Abstract

The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth $$\overline {\mathbb{F}_\ell ((t))} $$ is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings.