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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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