Exponential generation and largeness for compact \(p\)-adic Lie groups
Let \(\mathcal G\) be a closed subgroup of the compact group \(\text{GL}_n(\mathbb Z_p)\). Here \(p\) is a prime that is sufficiently large in relation to the positive integer \(n\). Let \(G\) be the Zariski closure of \(\mathcal G\) in \({\text{GL}_n}_{/{\mathbb Q_p}}\). In the article under review, the author uses scheme-theoretic techniques to obtain a criterion (when the radical of \(G\) is unipotent) in terms of the reduction of \(\mathcal G\) modulo \(p\) that implies that the index \([\text{GL}_n(\mathbb Z_p)\cap G(\mathbb Q_p) : \mathcal G]\) is bounded by a constant depending only on \(n\). More precisely, the author introduces the Nori dimension of \(\mathcal G\). If \(\Gamma\) is a subgroup of \(\text{GL}_n(\mathbb F_p)\) [\textit{M. V. Nori}, Invent. Math. 88, 257--275 (1987; Zbl 0632.20030)], defined the algebraic group \(G_{N(\Gamma)}\) generated by the one-parameter subgroups \(t\mapsto \text{exp}(t \log u)\), where \(u\) ranges over all elements of order \(p\) in \(\Gamma\). The Nori dimension of \(\Gamma\) is defined to be \(\dim G_{N(\Gamma)}\). The author defines the Nori dimension of \(\mathcal G\) to be that of the image of \(\mathcal G\) in \(\text{GL}_n(\mathbb F_p)\). The main result of this paper is that for \(p\) sufficiently large, the Nori dimension of \(\mathcal G\) is at most \(\dim G\), and in the case of equality, (1) \(\mathcal G\) is open in \(G(\mathbb Q_p)\), (2) \(G/G^\circ\) has order prime to \(p\) and has a normal abelian subgroup of index bounded by a constant depending only on \(n\), (3) if the radical of \(G\) is unipotent, \([\text{GL}_n(\mathbb Z_p)\cap G(\mathbb Q_p) : \mathcal G]\) is bounded by a constant depending only on \(n\). This mirrors a bounded-index result due to Nori for subgroups of \(\text{GL}_n(\mathbb F_p)\) generated by elements of order \(p\). It should have applications to monodromy of projective varieties over function fields of positive characteristic.
scientific article; zbMATH DE number 5880072