Weakly Induced Torus

Let \(k\) be a Henselian discretely valued field with wild inertia group \(P \subseteq \Theta\), and let \(k_t \subseteq k_s\) be the maximal tamely ramified extension of \(k\), so \(P = \operatorname{Gal}(k_s/k_t)\).

A torus \(T\) over \(k\) is weakly induced if there exists a tamely ramified extension \(\ell/k\) such that \(T_\ell\) is induced. Equivalently:

• \(T_{k_t}\) is induced; equivalently
• \(X^*(T)\) (equivalently \(X_*(T)\)) has a \(\mathbb{Z}\)-basis permuted by the wild inertia group \(P\).

Every induced torus is weakly induced, and every tamely ramified torus is weakly induced (the wild inertia then acts trivially, so any basis is permuted). The class is designed for integral-model theory: it is recorded at the integral-model stage of this topic that the standard integral model of a torus is smooth with well-behaved congruence filtration exactly under this hypothesis, which is why the class — rather than the tame or induced classes — is the natural boundary of the good theory.