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Component groups of abelian varieties and Grothendieck’s duality conjecture. (English) Zbl 0919.14026
\(K\) is the field of fractions of a complete discrete valuation ring \(R\) and \(A_K\) an abelian variety over \(K\). Write \(A'_K\) for its dual, \(A\) and \(A'\) for the Néron models of \(A_K\) and \(A'_K\) and \(\phi _A\), \(\phi _{A'}\) for the corresponding component groups. Grothendieck has constructed a pairing \(\phi _A\times \phi _{A'}\rightarrow {\mathbb Q}/{\mathbb Z}\) and conjectured that this pairing is perfect. This conjecture is already known in some cases, e.g., semi-stable reduction. The paper proves the conjecture in the case that the residue field of \(R\) is perfect and \(A\) has potentially multiplicative reduction. There is a hint for a possible proof in a more general situation. The highly technical proof uses arithmetic theory of abelian varieties, sheaves for the étale or smooth topology and a rigid analytic description of the groups \(\phi _A\), \(\phi _{A'}\) obtained from rigid uniformization of the abelian variety \(A_K\).

MSC:
14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
13F30 Valuation rings
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