Perfectness of Grothendieck's Component-Group Pairing

Definitions and mathematical statement

Let $K$ be a complete discrete valuation field, let $\mathcal O_K$ be its valuation ring, and let $k$ be its perfect residue field of characteristic $p>0$. Let $A$ be an abelian variety over $K$, and let $B=A^\vee$ be its dual abelian variety. Write $\mathcal A$ and $\mathcal B$ for their Néron models over $\mathcal O_K$, and set $A_x=\mathcal A\times_{\mathcal O_K}k$ and $B_x=\mathcal B\times_{\mathcal O_K}k$.

The supplied statement asks whether Grothendieck's canonical component-group pairing \[ \pi_0(A_x)\times\pi_0(B_x)\longrightarrow{\mathbb{Q}}/{\mathbb{Z}} \] is perfect for every such $K$, $A$, and $B$. The record does not specify whether $\pi_0$ denotes a component group scheme, its geometric points, or its $k$-rational points, and it does not construct the pairing.

Under the geometric-component interpretation, choose an algebraic closure $\bar k$ of $k$ and define the finite abelian groups \[ \Phi_A=\left(A_x/A_x^0\right)(\bar k),\qquad \Phi_B=\left(B_x/B_x^0\right)(\bar k), \] where $A_x^0$ and $B_x^0$ are the identity components. The intended question under this interpretation is whether the named pairing $\langle\ ,\ \rangle_G:\Phi_A\times\Phi_B\to{\mathbb{Q}}/{\mathbb{Z}}$ always induces isomorphisms \[ \begin{aligned} \Phi_A&\longrightarrow\operatorname{Hom}_{{\mathbb{Z}}}(\Phi_B,{\mathbb{Q}}/{\mathbb{Z}}), & a&\longmapsto\bigl(b\longmapsto\langle a,b\rangle_G\bigr),\\ \Phi_B&\longrightarrow\operatorname{Hom}_{{\mathbb{Z}}}(\Phi_A,{\mathbb{Q}}/{\mathbb{Z}}), & b&\longmapsto\bigl(a\longmapsto\langle a,b\rangle_G\bigr). \end{aligned} \] This explicit formulation is conditional on the geometric-component interpretation; the supplied evidence does not settle that convention or provide a construction identifying the canonical pairing. No algebraic closedness of $k$, restriction on the characteristic of $K$, or reduction-type hypothesis is imposed by the supplied statement.

Short English statement

For an abelian variety and its dual over a complete discretely valued field with perfect residue field of positive characteristic, the question asks whether Grothendieck's pairing between their Néron component groups is perfect. Under the geometric-component interpretation, each component group should be the full ${\mathbb{Q}}/{\mathbb{Z}}$-valued dual of the other. The component-group convention, pairing construction, and conflicting source qualifications require human review.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004163$, source problem code catalogue entry, titled "Perfectness of Grothendieck's Component-Group Pairing." No individual proposer is supplied. The title attributes the pairing to Grothendieck.
• Supplied report citation: Algebraic K-theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/29, pp. 1737--1790. https://doi.org/10.4171/owr/2019/29. The supplied literature-sources entry labels this same reference "OWR primary report."
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the record open, says that the extraction stops after introducing the local-duality setup without stating the pairing or precise assertion, and says that a related conjecture has later proofs. These are supplied dated curation claims, not independently verified conclusions. No reference identifying those later proofs is supplied.
• The supplied record also labels its statement corrected_verified and reports a manual semantic, typographic, and LaTeX review finding the clean statement self-contained and faithful. This supplied verification claim conflicts with the incompleteness qualification in the dated assessment; neither claim has been independently checked here.