Anisotropic Torus
A torus \(T\) over \(k\) is anisotropic if \[ X_*(T)^{\Theta} \;=\; 0, \] that is, \(T\) admits no non-trivial \(k\)-homomorphism \(\mathbb{G}_m \to T\). This is the working definition throughout the knowledge base.
Equivalent formulations. The following are equivalent:
- \(X_*(T)^{\Theta} = 0\);
- \(X^*(T)^{\Theta} = 0\), i.e. \(T\) has no non-trivial \(k\)-rational character \(T \to \mathbb{G}_m\);
- the maximal split subtorus of \(T\) (see Split-Anisotropic Decomposition of a Torus) is trivial.
Equivalence of (1) and (2). The pairing \(\langle\,,\,\rangle : X^*(T) \times X_*(T) \to \mathbb{Z}\) is perfect and \(\Theta\)-invariant, and \(\Theta\) acts through a finite quotient \(Q\) (see The Minimal Splitting Field of a Torus). Suppose \(\lambda \in X_*(T)^{\Theta}\) is non-zero. Choose \(\chi \in X^*(T)\) with \(\langle \chi, \lambda\rangle \neq 0\) and set \(\chi' = \sum_{q \in Q} q\chi\). Then \(\chi'\) is \(\Theta\)-fixed and \(\langle \chi', \lambda \rangle = \sum_{q}\langle \chi, q^{-1}\lambda\rangle = \lvert Q\rvert \,\langle \chi, \lambda \rangle \neq 0\), so \(\chi' \neq 0\) and \(X^*(T)^{\Theta} \neq 0\). The same argument with the roles of the lattices exchanged gives the converse.
Over a discretely valued Henselian field, anisotropy of \(T\) is further equivalent to boundedness of the group \(T(k)\); that equivalence belongs to the arithmetic stage of this topic and is recorded with the source span above.