Deciding finiteness for matrix groups over function fields

Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n, F(t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n, F), for which the results of [1] can be applied. When F is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more efficient randomized algorithms.