Hensel’s lemma for the norm principle for type Dₙ groups

Classical norm principles examine the behavior of quadratic forms and central simple algebras under the extensions of the base field. Merkurjev reformulated them as a property of algebraic groups, and Gille reformulated them as a property of torsors under algebraic groups using Galois cohomology. After recalling the classical norm principles, we will show that Merkurjev’s formulation of the norm principle for reductive linear algebraic groups is equivalent to Gille’s formulation for torsors under semisimple groups. We will then prove that it is sufficient to show the norm principle for simply connected groups. Among classical groups, the only case for which the norm principle is open, are groups of type Dₙ. Absolutely simple, simply connected, classical groups of type Dₙ are spinor groups of central simple algebras with orthogonal involution. We will reduce the norm principle for the spinor groups to the case that the field extension has degree 2. We then focus on the spinor groups of skew-hermitian forms defined over quaternion algebras and will reduce the question in this case to the case that the skew-hermitian form is anisotropic. Let K be a complete discretely valued field with residue field k with char(k) ≠ 2. Suppose that the norm principle holds for spinor groups Spin(h) for every nonsingular skew-hermitian form h defined over every quaternion algebra with canonical involution defined over finite separable extensions of k. Then we will show that it holds for spinor groups Spin(H) for every nonsingular skew-hermitian form H defined over every quaternion algebra with canonical involution defined over K.