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Component groups of Néron models via rigid uniformization. (English) Zbl 0869.14020
Let \(R\) be a discrete valuation ring and \(K=\text{Quot} (R)\). The authors are interested in some aspects of the component group \(\Phi_E\) of the Néron model, if it exists, of an abelian variety \(E_K\). One can suppose \(R\) to be complete, because Néron models well behave with respect to completion, and then one can use rigid techniques as uniformizations of abelian varieties [M. Raynaud, Actes Congr. Internat. Math., Nice 1970, Part I, 473-477 (1971; Zbl 0223.14021) and S. Bosch and W. Lütkebohmert, Invent. Math. 78, 257-297 (1984; Zbl 0554.14015)]. The main results are about a filtration and a dual one of \(\Phi_E\). This has been first introduced by D. J. Lorenzini [J. Reine Angew. Math. 445, 109-160 (1993; Zbl 0781.14029)], only for the \(p\)-part of \(\Phi_E\) \((p=\) residue characteristic of \(R)\). The authors give a filtration for the whole group \(\Phi_E\), with a geometric interpretation of the different terms. Moreover, if \(j:R \hookrightarrow K\) is the natural inclusion, \(E\) can be seen as representing the sheaf \(j_*E_K\) on the small smooth site over \(R\); following this point of view, the authors extend their study to identity components or groups of components for Néron models in the category of sheaves over the small smooth site and even to complexes in the derived category over this site.

14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
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[1] Artin, M.: Grothendieck topologies. Notes on a seminar by M. Artin, Harvard University (1962) · Zbl 0208.48701
[2] Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. Springer, Grundlehren Bd. 261, Berlin, Heidelberg, New York (1984) · Zbl 0539.14017
[3] Bosch, S., Lütkebohmert, W.: Stable reduction and uniformization of abelian varieties II. Invent. Math. 78, 257-297 (1984) · Zbl 0554.14015
[4] Bosch, S., Lütkebohmert, W.: Degenerating abelian varieties. Topology 30, 653-698 (1991) · Zbl 0761.14015
[5] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry I. Rigid spaces. Math. Ann. 295, 291-317 (1993) · Zbl 0808.14017
[6] Bosch, S., Lütkebohmert, W.: Formal and rigid geometry II. Flattening techniques. Math. Ann. 296, 403-429 (1993) · Zbl 0808.14018
[7] Bosch, S., Lütkebohmert, W., Raynaud, M.: Formal and rigid geometry III. The relative maximum principle. Math. Ann. 302, 1-29 (1995) · Zbl 0839.14013
[8] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Math., 3. Folge, Bd. 21, Springer (1990) · Zbl 0705.14001
[9] Bosch, S., Schlöter, K.: Néron models in the setting of formal and rigid geometry. Math. Ann. 301, 339-362 (1995) · Zbl 0854.14011
[10] Chai, C.L., Faltings, G.: Degeneration of Abelian Varieties. Ergebnisse der Math., 3. Folge, Bd. 21, Springer (1990) · Zbl 0744.14031
[11] Demazure, M., Grothendieck, A.: SGA 3II, Schémas en Groupes. Lecture Notes in Mathematics 152, Springer (1970)
[12] Deschamps, D.: Réduction semistable. In ?Séminaire sur les pinceaux de courbes de genre au moins deux?, L. Szpiro, Ed., 1-34 (1981)
[13] Edixhoven, B.: On the prime top-part of the groups of connected components of Néron models. Preprint (1994)
[14] Edixhoven, B., Liu, Q., Lorenzini, D.: Thep-part of the group of components. Preprint (1994)
[15] Grothendieck, A.: SGA 7I, Groupes de Monodromie en Géométrie Algébrique. Lecture Notes in Mathematics 288, Springer (1972)
[16] Hartshorne, R.: Residues and Duality. Lect. Notes in Math. 20, Springer (1966) · Zbl 0212.26101
[17] Köpf, U.: Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen. Schriftenreihe Math. Inst. Univ. Münster, 2. Serie. Heft 7 (1974) · Zbl 0275.14006
[18] Lenstra, H., Oort, F.: Abelian varieties having purely additive reduction. J. Pure Appl. Alg. 36, 281-298 (1985) · Zbl 0557.14022
[19] Lorenzini, D.: On the group of components of a Néron model. J. reine angew. Math. 445, 109-160 (1993) · Zbl 0781.14029
[20] Lütkebohmert, W.: Formal-algebraic and rigid-analytic geometry. Math. Ann. 286, 341-371 (1990) · Zbl 0716.32022
[21] Milne, J.S.: Etale cohomology. Princeton Math. Series 33, Princeton University Press, Princeton (1980) · Zbl 0433.14012
[22] Nart, E., Xarles, X.: Additive reduction of algebraic tori. Arch. Math. 57, 460-466 (1991) · Zbl 0782.14042
[23] Ogus, A.:F-isocrystals and de Rham cohomology II: Convergent isocrystals. Duke Math. Journal 51, 765-850 (1984) · Zbl 0584.14008
[24] Raynaud, M.: Variétés abéliennes et géométrie rigide. Actes du congrès international de Nice 1970, tome 1, 473-477
[25] Raynaud, M.: I-motifs et monodromie géométrique. Astérisque 223, Exp. VII, 295-319 (1994)
[26] Serre, J.-P.: Corps Locaux. Hermann, Paris (1962)
[27] Xarles, X.: The scheme of connected components of the Néron model of an algebraic torus. J. reine angew. Math. 437, 167-179 (1993) · Zbl 0764.14009
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