Finiteness of Tori of Bounded Rank

Let \(k\) be a field and \(n \geq 0\). Then:

1. \(\operatorname{GL}_n(\mathbb{Z})\) has only finitely many conjugacy classes of finite subgroups (a form of the Jordan–Zassenhaus theorem: there are finitely many isomorphism classes of \(\mathbb{Z}[G]\)-lattices of rank \(n\) for each finite group \(G\), and finitely many candidate \(G\) embed in \(\operatorname{GL}_n(\mathbb{Z})\) since finite subgroups have order bounded in terms of \(n\) — Minkowski);
2. consequently, for each \(n\) there are only finitely many \(\Gamma\)-lattice types of rank \(n\) for any profinite \(\Gamma\) acting continuously, and hence — by F-Tori Are Classified by Galois Modules in the cocycle form Tori Are Classified by Cocycles in GL_n(Z) — for a fixed finite Galois extension \(E/k\), only finitely many isomorphism classes of rank-\(n\) tori over \(k\) split by \(E\).

Over \(k\) itself the set of rank-\(n\) tori need not be finite (splitting fields vary), but each is determined by a finite-image continuous homomorphism \(\Theta \to \operatorname{GL}_n(\mathbb{Z})\) whose image is one of the finitely many finite subgroups up to conjugacy.

Status. Statement-only: the Jordan–Zassenhaus theorem and the Minkowski bound are recorded here as external inputs with source; their proofs are outside the scope of this knowledge base (verification.proof: gap marks the node as consumed statement-only). Downstream uses: finiteness of the candidate list in rank-one and rank-two classifications.