A new proof of the Artin-Springer theorem in Schur index 2
Anne Quéguiner-Mathieu, Jean-Pierre Tignol
TL;DR
This work proves an Artin–Springer type theorem for groups of type $D$ represented by similitudes over algebras of Schur index $2$: a generalized quadratic form that is anisotropic over a quaternion algebra $Q$ remains anisotropic after generic splitting (and hence over all odd-degree extensions). The authors provide a direct, characteristic-free proof using a degree map on a subring of the split algebra obtained from $Q$ over the function field $F$ of the Severi–Brauer conic, and they establish a Morita equivalence between generalized quadratic spaces over $Q_F$ and quadratic spaces over $F$ via a two-dimensional left ideal with a nonsingular alternating form. A central technical tool is a degree-reduction lemma that iteratively decreases the degree of isotropic vectors, enabling a contradiction if isotropy were to appear over $F$; this is complemented by a Morita-stable transfer and an appendix linking to quadratic pairs. The result extends the known two-step approach (excellence and splitting) to a direct, unified proof and clarifies the quadratic-pair formulation in this noncommutative setting.
Abstract
We provide a new proof of the analogue of the Artin-Springer theorem for groups of type $\mathsf{D}$ that can be represented by similitudes over an algebra of Schur index $2$: an anisotropic generalized quadratic form over a quaternion algebra $Q$ remains anisotropic after generic splitting of $Q$, hence also under odd degree field extensions of the base field. Our proof is characteristic free and does not use the excellence property.