Component groups of abelian varieties and Grothendieck's duality conjecture
Bosch, Siegfried
Annales de l'Institut Fourier, Tome 47 (1997), p. 1257-1287 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {47},
     year = {1997},
     pages = {1257-1287},
     doi = {10.5802/aif.1599},
     mrnumber = {98k:14061},
     zbl = {0919.14026},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 1257-1287. doi : 10.5802/aif.1599. http://gdmltest.u-ga.fr/item/AIF_1997__47_5_1257_0/

[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). | Numdam | MR 82k:12019 | Zbl 0502.14016

[3] S. Bosch, W. Lütkebohmert, Degenerating abelian varieties, Topology, 30 (1991), 653-698. | MR 92i:14043 | Zbl 0761.14015

[4] S. Bosch, W. Lütkebohmert, Formal and rigid geometry II, Flattening techniques, Math. Ann., 296 (1993), 403-429. | MR 94e:11070 | 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). | MR 91i:14034 | Zbl 0705.14001

[7] S. Bosch, K. Schlöter, Néron models in the setting of formal and rigid geometry. Math. Ann., 301 (1995), 339-362. | MR 96h:14035 | Zbl 0854.14011

[8] S. Bosch, X. Xarles, Component groups of Néron models via rigid uniformization, Math. Ann., 306 (1996), 459-486. | MR 97f:14022 | 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). | Numdam | 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. | MR 94k:11065 | Zbl 0781.14029

[16] W. Mccallum, Duality theorems for Néron models, Duke Math. J., 53 (1986), 1093-1124. | MR 88c:14062 | Zbl 0623.14023

[17] J. S. Milne, Étale Cohomology, Princeton Math. Series 33, Princeton University Press, Princeton (1980). | MR 81j:14002 | Zbl 0433.14012

[18] A. Ogus, F-isocrystals and de Rham cohomology II: Convergent isocrystals, Duke Math. Journal, 51 (1984), 765-850. | MR 86j:14012 | Zbl 0584.14008

[19] M. Raynaud, Variétés abéliennes et géométrie rigide, Actes du congrès international de Nice 1970, tome 1, 473-477. | MR 55 #360 | Zbl 0223.14021

[20] J.-P. Serre, Corps Locaux, Hermann, Paris, 1962. | MR 27 #133 | 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. | MR 98j:14058 | 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. | MR 94d:14044 | Zbl 0764.14009