Witt invariants of quaternionic forms
Nicolas Garrel
TL;DR
This paper studies Witt invariants of anti-hermitian forms over a quaternion algebra with its canonical involution, focusing on the algebraic group $O(A,\sigma)$ when $A$ has index $2$. It develops a framework of mixed Witt rings and Morita-compatible $\\lambda$-operations to describe all Witt invariants, showing they are generated by $\lambda$-operations with coefficients in the mixed Witt ring $\widetilde{W}^{-1}(Q,\gamma)$, up to the obstruction given by the norm form $n_Q$. The core technique is to pass to a generic splitting field via scalar extension to the Severi–Brauer variety, control residues through valuations at closed points, and then transfer information back to the original field using Morita equivalences. The paper provides explicit presentations of the invariant modules $I_Q^{(r)}$ and $\overline{I}_Q^{(r)}$, including freeness results for $\overline{I}_Q^{(r)}$ with bases given by products of $\overline{\lambda}^d$, and a concrete relation system for $I_Q^{(r)}$ governed by $n_Q$ and binomial coefficients. Overall, it extends Serre’s Witt invariants program to quaternionic settings and offers computable, Morita-stable descriptions of invariants that connect to cohomological invariants via the Milnor conjecture.
Abstract
We describe all Witt invariants of anti-hermitian forms over a quaternion algebra with its canonical involution, and in particular all Witt invariants of orthogonal groups $O(A,σ)$ where $(A,σ)$ is an central simple algebra with orthogonal involution and $A$ has index $2$. They are combinations of appropriately defined $λ$-powers, similarly to the case of quadratic forms, but the module of invariants is no longer free over those operations. The method involves extending the scalars to a generic splitting field of $A$, and controlling the residues of the invariants with respect to valuations coming from closed points in the Severi-Brauer variety.