Essential dimension of sequences of quadratic Pfister forms
Fatma Kader Bingöl, Adam Chapman, Ahmed Laghribi
TL;DR
The paper investigates the essential dimension of $m$-tuples of quadratic $n$-fold Pfister forms whose Witt class lies in $I_q^{n+1}F$, focusing on the cases $(n,m)=(n,3)$ and $(2,4)$. It develops a descent framework using parametrizations of Pfister forms and analyzes hyperbolicity of the associated Witt sums to derive tight bounds, linking the problem to quaternion algebras and Spin groups in characteristic 2. The main results are: (i) $\operatorname{ed}(\mathcal{F}_{n,3}) = n+1$ for all $n\ge 2$, with equality exactly when the Witt sum $\Sigma_S$ is not hyperbolic, (ii) $\operatorname{ed}(\mathcal{F}_{2,4})\in\{4,5\}$ in characteristic $2$, and (iii) a suite of lower and upper bounds for $\operatorname{ed}(\mathcal{F}_{2,m})$ in various characteristics, including a canonical relation to $\mathcal{G}_m$ and $\operatorname{Spin}_{2m+1}$. These results advance understanding of how Pfister-form data constrain essential dimension and shed light on characteristic-2 phenomena via explicit descent techniques.
Abstract
We study the essential dimension of the set of isometry classes of $m$-tuples $(\varphi_1,...,\varphi_m)$ of quadratic $n$-fold Pfister forms over a field $F$ such that the Witt class of $\varphi_1 \perp \ldots \perp \varphi_m$ lies in $I_q^{n+1}F$. We show that the essential dimension is equal to $n+1$, when $m=3$, and is either $4$ or $5$, when $n=\text{char} F=2$, $m=4$.