Kato Cohomology Kernels of Quadratic Pfister Forms

Definitions and mathematical statement

The supplied record asks for the following equality of Kato-cohomology groups. Its characteristic and notation conventions are not explicitly stated. The formulation below uses the conventional characteristic-two interpretation; these conventions, including the ranges of the integer parameters, require confirmation against the cited report.

Let $F$ be a field of characteristic $2$, let $m,n\in{\mathbb{Z}}_{\geq 0}$, let $a_1,\ldots,a_m\in F^\times$, and let $b\in F$. Write $\langle 1,a_i\rangle$ for the symmetric bilinear form with diagonal matrix $\operatorname{diag}(1,a_i)$, and write $[1,b]$ for the quadratic form $(x,y)\mapsto x^2+xy+by^2$. Suppose that the quadratic Pfister form \[ Q=\langle 1,a_1\rangle\otimes\cdots\otimes\langle 1,a_m\rangle\otimes[1,b] \] is anisotropic, meaning that $Q(v)=0$ implies $v=0$. Let $F(Q)$ denote the function field of the projective quadric defined by $Q=0$.

For any field $K$ of characteristic $2$, let $\Omega_K^r=\bigwedge_K^r\Omega^1_{K/{\mathbb{F}}_2}$ be the space of absolute differential $r$-forms, with $\Omega_K^0=K$ and $\Omega_K^{-1}=0$. Write $\mathrm{dlog}(u)={\,\mathrm{d}} u/u$ for $u\in K^\times$. The standard Kato operator \[ \wp_r:\Omega_K^r\longrightarrow\Omega_K^r/{\,\mathrm{d}}\Omega_K^{r-1} \] is the additive map characterized on logarithmic expressions by \[ \wp_r\!\left(x\,\mathrm{dlog}(u_1)\wedge\cdots\wedge\mathrm{dlog}(u_r)\right) =(x^2-x)\,\mathrm{dlog}(u_1)\wedge\cdots\wedge\mathrm{dlog}(u_r) \pmod{{\,\mathrm{d}}\Omega_K^{r-1}}. \] Define \[ \nu_K(r)=\ker(\wp_r),\qquad H_2^{r+1}(K)=\operatorname{coker}(\wp_r). \] In degree zero, the logarithmic product is empty, so $\wp_0(x)=x^2-x$ and $\nu_K(0)={\mathbb{F}}_2$. For an extension $L/K$, define \[ H_2^{r+1}(L/K)=\ker\!\left(H_2^{r+1}(K)\longrightarrow H_2^{r+1}(L)\right), \] where the map is induced by extension of scalars on differential forms.

Under these conventions, is it true for every such choice of data that \[ H_2^{n+1}(F(Q)/F)= \begin{cases} \left\{\left[b\,\mathrm{dlog}(a_1)\wedge\cdots\wedge\mathrm{dlog}(a_m)\wedge\eta\right]:\eta\in\nu_F(n-m)\right\},&n\geq m,\\[4pt] \{0\},&n<m, \end{cases} \] with brackets denoting classes in $H_2^{n+1}(F)$? Empty products are interpreted as $1$ if $m=0$; whether that boundary case is included in the report's intended question is unresolved from the supplied record.

The corrected statement supplied in the registry uses $H_2^{n+1}$. The field named original_statement instead prints $H^{2n+1}$ and leaves the symbol unspecified. This draft follows the supplied corrected version, whose verification is reported by the registry and has not been independently checked here.

Short English statement

For an anisotropic quadratic Pfister form, ask whether the kernel of Kato cohomology after passage to its quadric's function field consists exactly of the displayed differential-symbol classes when $n\geq m$, and vanishes when $n<m$. This draft uses the conventional characteristic-two definitions; the source record omits explicit characteristic and parameter-range hypotheses. The supplied status is open, subject to its dated curation and human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30003044$, title Kato Cohomology Kernels of Quadratic Pfister Forms, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2016$ and names no individual proposer.
• Supplied report citation: Algebraic Cobordism and Projective Homogeneous Varieties (2016), Oberwolfach Reports, pp. 215--265, DOI 10.4171/owr/2016/5. Original OWR report: https://doi.org/10.4171/owr/2016/5. The supplied background describes the reports as records of mathematical workshops and their open-problem discussions.
• Supplied related literature: Aravire--Laghribi--O'Ryan kernel computations, https://doi.org/10.1016/j.jalgebra.2015.11.022. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the report records positive special cases and a conditional theorem, while the displayed equality for general Pfister data remains a question; the related kernel computations do not prove the full statement. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that the official passage was checked and rewritten as a self-contained problem preserving its mathematical content. This is supplied provenance, not an independent verification by this draft.