P-adic Nori Theory

TL;DR

This paper investigates conditions under which certain p-adic subgroups of GL(n,Z_p) are of bounded index in their Zariski closure, focusing on the case where the Zariski closure lacks a toric component.

Contribution

It provides a new criterion based on mod p reduction to determine when a subgroup has bounded index in its Zariski closure in the p-adic setting.

Findings

Establishes a condition on the mod p reduction of G.

Shows G has bounded index in its Zariski closure under certain conditions.

Applicable for sufficiently large p relative to n.

Abstract

Given a fixed integer n, we consider closed subgroups G of H = GL(n,Z_p) where Z_p denotes the ring of p-adic integers and p is sufficiently large in terms of n. Assuming that the Zariski closure of G has no toric part, we give a condition on the (mod p) reduction of G which guarantees that G is of bounded index in its Zariski closure in H.