P-adic Nori Theory

Given a fixed integer n, we consider closed subgroups G of GLn(Zp), where p is sufficiently large in terms of n. Assuming that the Zariski closure of G in GLn has no toric part, we give a condition on the (mod p) reduction of G which guarantees that G is of bounded index in GLn(Zp) ∩ G(Qp). In [No], Nori considered a special class of subgroups of GLn(Fp), namely groups which are generated by elements of order p or, as we shall say, p-generated groups. He showed that if p is sufficiently large in terms of n, there is a correspondence between p-generated groups and a certain class of connected algebraic groups which he called exponentially generated. In particular, every p-generated group Γ is a subgroup of G(Fp) for the corresponding algebraic group G, and [G(Fp) : Γ] is bounded by a constant depending only on n. The p-generated groups are admittedly rather special, but on the other hand, every finite subgroup Γ ⊂ GLn(Fp) contains a p-generated normal subgroup, Γ , of prime-to-p index, which shows that every Γ can be related to a connected algebraic group in a weak sense. This construction can serve in some measure as a substitute for the (identity component of the) Zariski-closure in the setting of finite linear groups, where the actual identity component of the Zariski-closure of Γ is always trivial. In this paper we consider closed subgroups G of the compact p-adic Lie group GLn(Zp). In this setting, of course, Zariski-closure behaves well, so we do not need a substitute. Nevertheless, it turns out that there is an interesting class of groups G for which we can prove a bounded index result analogous to that of Nori. Throughout the paper, n will denote a positive integer and F a field. If F is of characteristic p > 0, we assume p ≥ n, so i! is non-zero for i < n. As every nilpotent element x ∈ Mn(F ) satisfies x n = 0, the 1991 Mathematics Subject Classification. 20G25.