A conjecture on arithmetic fundamental groups

The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth 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. This material is based upon work supported by the National Science Foundation under Grant No. 9970049.