Hensel's lemma for the norm principle for spinor groups

Let $K$ be a complete discretely valued field with residue field $k$ with $char \ k \ne 2$. Assuming that the norm principle holds for spinor groups $Spin(\mathfrak{h})$ for every regular skew-hermitian form $\mathfrak{h}$ over every quaternion algebra $\mathfrak{D}$ (with respect to the canonical involution on $\mathfrak{D}$) defined over any finite extension of $k$, we show that the norm principle holds for spinor groups $Spin(h)$ for every regular skew-hermitian form $h$ over every quaternion algebra $D$ (with respect to the canonical involution on $D$) defined over $K$.

💡 Research Summary

The paper addresses the norm principle for spinor groups over complete discretely valued fields. Let K be such a field with residue field k of characteristic different from 2. The authors assume that the H¹‑variant of the norm principle holds for every spinor group Spin(h) attached to a regular skew‑hermitian form h over any quaternion algebra D (with its canonical involution) defined over any finite extension of k. Under this hypothesis they prove that the same H¹‑norm principle holds for all spinor groups Spin(h) over quaternion algebras defined over K itself.

The work begins by recalling two formulations of the norm principle. The H⁰‑variant (due to Merkurjev) concerns the image of the norm map N_{L/K}:T(L)→T(K) composed with a homomorphism f:G→T, requiring Im(N∘f_L)⊆Im(f_K) for every finite separable extension L/K. The H¹‑variant (due to Gille) is cohomological: for a semisimple group S with central subgroup Z, one asks that any class u∈Ker(H¹(L,Z)→H¹(L,S)) has its corestriction cor_{L/K}(u) lying in Ker(H¹(K,Z)→H¹(K,S)).

Theorem 2.1 proves the equivalence of these two variants for all reductive (resp. semisimple) groups. The proof uses an embedding of the central subgroup into a quasi‑trivial torus, forms a push‑out diagram, and applies the H⁰‑norm principle for the resulting reductive group to deduce the H¹‑statement; the converse direction is similar. Lemmas 2.2 and 2.3 further reduce the H¹‑problem to the full center of a semisimple group and to its simply‑connected covering. Consequently, for spinor groups the relevant central subgroup is μ₂, and the simply‑connected cover is the spin group itself.

The paper’s main object is a regular skew‑hermitian form h over a quaternion algebra D. Via Morita equivalence such a pair (D,h) corresponds to a central simple algebra A of degree 2n with orthogonal involution σ, and Spin(h)=Spin(A,σ) is a group of type Dₙ. The authors aim to lift the norm principle from the residue field k to the valued field K.

Sections 3–5 collect auxiliary results. Section 3 reviews norm principles of Gille, Knebusch, and Scharlau, and provides a new proof of a generalized Scharlau principle that will be used later. Section 4 shows that it suffices to verify the norm principle for separable quadratic extensions L/K rather than arbitrary finite extensions; this reduction is achieved by analyzing the obstruction class u∈Ker(α_L) and using the aforementioned H¹‑equivalence. Section 5 reduces to the case where the skew‑hermitian form h is anisotropic, a necessary condition for the later valuation‑theoretic arguments.

From Section 6 onward the authors work over a complete discretely valued field. They recall Springer’s and Larmour’s theorems, which describe the behavior of quadratic and hermitian forms under unramified and ramified extensions. These results provide explicit formulas for the discriminant and the Clifford invariant of h after scalar extension, allowing precise control of the norm map at the level of cohomology.

Section 7 introduces notation and several explicit formulas for the spinor norm, the reduced norm on D, and the corestriction maps. Sections 8 and 9 treat two distinct reduction scenarios. In Section 8, under the hypothesis that h, D, and a scalar λ are all unramified, the authors prove that the norm of any element from a quadratic extension lies in the image of the spinor norm, using Larmour’s description of the unramified case. Section 9 handles the situation where only λ is unramified, while h and D may be ramified; a careful analysis of the ramified case shows that the obstruction still vanishes.

Section 10 contains a technical lemma concerning the interaction of images and kernels of certain homomorphisms; this lemma is invoked repeatedly in the final proof.

Finally, Section 11 assembles all previous reductions and lemmas to prove the main theorem (Theorem 7.1, restated as Theorem 1.5 in the introduction). The argument proceeds as follows: starting from an arbitrary finite separable extension L/K and an element in the kernel of the relevant cohomology map, one reduces to a quadratic extension, then to an anisotropic unramified (or suitably ramified) form, applies the valuation‑theoretic results to show that the corestriction of the obstruction class is trivial, and finally lifts back to K using the H⁰‑norm principle for the associated reductive group.

Thus the paper establishes a Hensel‑type lifting theorem for the norm principle: if the norm principle holds for all spinor groups over finite extensions of the residue field k, then it automatically holds for all spinor groups over the complete discretely valued field K. This result extends earlier work of Parimala‑Sridharan‑Suresh (which treated the split case, i.e., quadratic forms) to the more general setting of skew‑hermitian forms over quaternion algebras. It provides a powerful tool for studying local‑global principles for type Dₙ groups over fields such as number fields, p‑adic fields, and global function fields of positive characteristic.