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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 \(A_K\).

14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
13F30 Valuation rings
Full Text: DOI Numdam EuDML
[1] M. ARTIN, Grothendieck topologies. Notes on a seminar by M. Artin, Harvard University (1962). · Zbl 0208.48701
[2] L. BÉGUERI, Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. Fr., 108, fasc. 4 (1980). · Zbl 0502.14016
[3] S. BOSCH, W. LÜTKEBOHMERT, Degenerating abelian varieties, Topology, 30 (1991), 653-698. · Zbl 0761.14015
[4] S. BOSCH, W. LÜTKEBOHMERT, Formal and rigid geometry II, Flattening techniques, Math. Ann., 296 (1993), 403-429. · Zbl 0808.14018
[5] S. BOSCH, W. LÜTKEBOHMERT, M. RAYNAUD, Formal and rigid geometry III. The relative maximum principle, Math. Ann., 302 (1995), 1-29. · Zbl 0839.14013
[6] S. BOSCH, W. LÜTKEBOHMERT, M. RAYNAUD, Néron models. Ergebnisse der Math. 3. Folge, Bd. 21, Springer (1990). · Zbl 0705.14001
[7] S. BOSCH, K. SCHLÖTER, Néron models in the setting of formal and rigid geometry. Math. Ann., 301 (1995), 339-362. · Zbl 0854.14011
[8] S. BOSCH, X. XARLES, Component groups of Néron models via rigid uniformization, Math. Ann., 306 (1996), 459-486. · Zbl 0869.14020
[9] R. COLEMAN, The monodromy pairing, preprint (1996).
[10] A. GROTHENDIECK, J. DIEUDONNÉ, Ega iv4. Etude locale des schémas et des morphismes de schémas, Publ. Math. IHES 32 (1967). · Zbl 0153.22301
[11] A. GROTHENDIECK, Schémas en groupes, SGA 3, I, II, III, Lecture Notes in Mathematics 151, 152, 153, Springer (1970).
[12] A. GROTHENDIECK, Sga 7i, Groupes de Monodromie en Géométrie Algébrique, Lecture Notes in Mathematics 288, Springer (1972). · Zbl 0237.00013
[13] R. KIEHL, Analytische familien affinoider algebren, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 2. Abh. (1967). · Zbl 0177.06101
[14] U. KÖPF, Über eigentliche familien algebraischer varietäten über affinoiden Räumen, Schriftenreihe Math. Inst. Münster, 2. Serie, Heft 7 (1974). · Zbl 0275.14006
[15] D. LORENZINI, On the group of components of a Néron model. J. reine angew. Math., 445 (1993), 109-160. · Zbl 0781.14029
[16] W. MCCALLUM, Duality theorems for Néron models, Duke Math. J., 53 (1986), 1093-1124. · Zbl 0623.14023
[17] J. S. MILNE, Étale cohomology, Princeton Math. Series 33, Princeton University Press, Princeton (1980). · Zbl 0433.14012
[18] A. OGUS, F-isocrystals and de Rham cohomology II: Convergent isocrystals, Duke Math. Journal, 51 (1984), 765-850. · Zbl 0584.14008
[19] M. RAYNAUD, Variétés abéliennes et géométrie rigide, Actes du congrès international de Nice 1970, tome 1, 473-477. · Zbl 0223.14021
[20] J.-P. SERRE, Corps locaux, Hermann, Paris, 1962. · Zbl 0137.02601
[21] A. WERNER, On Grothendieck’s pairing of component groups in the semistable reduction case, J. reine angew. Math., 486 (1997), 205-215. · Zbl 0872.14037
[22] X. XARLES, The scheme of connected components of the Néron model of an algebraic torus, J. reine angew. Math., 437 (1993), 167-179. · Zbl 0764.14009
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