no. 5
Component groups of abelian varieties and Grothendieck's duality conjecture
Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1257-1287.

Nous étudions l’accouplement de Grothendieck sur les groupes des composantes des variétés abéliennes, en utilisant le point de vue de l’uniformisation rigide. Supposant que l’accouplement est parfait, nous démontrons que les filtrations, introduites par Lorenzini et d’une manière plus générale par Bosch et Xarles, sont duales l’une de l’autre. Les méthodes appliquées permettent de progresser sur le problème de la perfection de l’accouplement, surtout pour les variétés abéliennes avec réduction potentiellement multiplicative.

We investigate Grothendieck’s pairing on component groups of abelian varieties from the viewpoint of rigid uniformization theory. Under the assumption that the pairing is perfect, we show that the filtrations, as introduced by Lorenzini and in a more general way by Bosch and Xarles, are dual to each other. Furthermore, the methods yield some progress on the perfectness of the pairing itself, in particular, for abelian varieties with potentially multiplicative reduction.

@article{AIF_1997__47_5_1257_0,
     author = {Bosch, Siegfried},
     title = {Component groups of abelian varieties and {Grothendieck's} duality conjecture},
     journal = {Annales de l'Institut Fourier},
     pages = {1257--1287},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {5},
     year = {1997},
     doi = {10.5802/aif.1599},
     zbl = {0919.14026},
     mrnumber = {98k:14061},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1599/}
}
TY  - JOUR
AU  - Bosch, Siegfried
TI  - Component groups of abelian varieties and Grothendieck's duality conjecture
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 1257
EP  - 1287
VL  - 47
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1599/
DO  - 10.5802/aif.1599
LA  - en
ID  - AIF_1997__47_5_1257_0
ER  - 
%0 Journal Article
%A Bosch, Siegfried
%T Component groups of abelian varieties and Grothendieck's duality conjecture
%J Annales de l'Institut Fourier
%D 1997
%P 1257-1287
%V 47
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1599/
%R 10.5802/aif.1599
%G en
%F AIF_1997__47_5_1257_0
Bosch, Siegfried. Component groups of abelian varieties and Grothendieck's duality conjecture. Annales de l'Institut Fourier, Tome 47 (1997) no. 5, pp. 1257-1287. doi : 10.5802/aif.1599. https://aif.centre-mersenne.org/articles/10.5802/aif.1599/

[1] M. Artin, Grothendieck topologies. Notes on a seminar by M. Artin, Harvard University (1962). | Zbl

[2] L. Bégueri, Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. Fr., 108, fasc. 4 (1980). | Numdam | MR | Zbl

[3] S. Bosch, W. Lütkebohmert, Degenerating abelian varieties, Topology, 30 (1991), 653-698. | MR | Zbl

[4] S. Bosch, W. Lütkebohmert, Formal and rigid geometry II, Flattening techniques, Math. Ann., 296 (1993), 403-429. | MR | Zbl

[5] S. Bosch, W. Lütkebohmert, M. Raynaud, Formal and rigid geometry III. The relative maximum principle, Math. Ann., 302 (1995), 1-29. | Zbl

[6] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models. Ergebnisse der Math. 3. Folge, Bd. 21, Springer (1990). | MR | Zbl

[7] S. Bosch, K. Schlöter, Néron models in the setting of formal and rigid geometry. Math. Ann., 301 (1995), 339-362. | MR | Zbl

[8] S. Bosch, X. Xarles, Component groups of Néron models via rigid uniformization, Math. Ann., 306 (1996), 459-486. | MR | Zbl

[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). | Numdam | Zbl

[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

[13] R. Kiehl, Analytische Familien affinoider Algebren, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 2. Abh. (1967). | Zbl

[14] U. Köpf, Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen, Schriftenreihe Math. Inst. Münster, 2. Serie, Heft 7 (1974). | Zbl

[15] D. Lorenzini, On the group of components of a Néron model. J. reine angew. Math., 445 (1993), 109-160. | MR | Zbl

[16] W. Mccallum, Duality theorems for Néron models, Duke Math. J., 53 (1986), 1093-1124. | MR | Zbl

[17] J. S. Milne, Étale Cohomology, Princeton Math. Series 33, Princeton University Press, Princeton (1980). | MR | Zbl

[18] A. Ogus, F-isocrystals and de Rham cohomology II: Convergent isocrystals, Duke Math. Journal, 51 (1984), 765-850. | MR | Zbl

[19] M. Raynaud, Variétés abéliennes et géométrie rigide, Actes du congrès international de Nice 1970, tome 1, 473-477. | MR | Zbl

[20] J.-P. Serre, Corps Locaux, Hermann, Paris, 1962. | MR | Zbl

[21] A. Werner, On Grothendieck's pairing of component groups in the semistable reduction case, J. reine angew. Math., 486 (1997), 205-215. | MR | Zbl

[22] X. Xarles, The scheme of connected components of the Néron model of an algebraic torus, J. reine angew. Math., 437 (1993), 167-179. | MR | Zbl

Cité par Sources :