Variétés abéliennes et invariants arithmétiques  [ Abelian varieties and arithmetic class invariants ]
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, p. 277-297
As the sequel to our preceeding works, we study an analogue, for the Néron model of a semi-stable abelian variety defined over a number field, of M. J. Taylor’s class-invariant homomorphism, which allows us to measure Galois module structure of torsors.
Dans la continuité de nos travaux précédents, nous étudions un analogue, pour le modèle de Néron d’une variété abélienne semi-stable sur un corps de nombres, du class-invariant homomorphism introduit par M. J. Taylor, qui nous permet de mesurer la structure galoisienne de certains torseurs.
DOI : https://doi.org/10.5802/aif.2181
Classification:  11G,  11R,  14K
Keywords: Torsors, Galois module structure, elliptic curves, biextensions, duality
@article{AIF_2006__56_2_277_0,
     author = {Gillibert, Jean},
     title = {Vari\'et\'es ab\'eliennes et invariants arithm\'etiques},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     pages = {277-297},
     doi = {10.5802/aif.2181},
     zbl = {1091.11021},
     mrnumber = {2226015},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_2_277_0}
}
Variétés abéliennes et invariants arithmétiques. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 277-297. doi : 10.5802/aif.2181. https://aif.centre-mersenne.org/item/AIF_2006__56_2_277_0/

[1] Agboola, A. A geometric description of the class invariant homomorphism, J. Théor. Nombres Bordeaux, Tome 6 (1994), pp. 273-280 | Article | Numdam | MR 1360646 | Zbl 0833.11055

[2] Agboola, A. Torsion points on elliptic curves and Galois module structure, Invent. Math., Tome 123 (1996), pp. 105-122 | Article | MR 1376248 | Zbl 0864.11055

[3] Agboola, A.; Pappas, G. On arithmetic class invariants, Math. Ann., Tome 320 (2001), pp. 339-365 | Article | MR 1839767 | Zbl 0989.11061

[4] Agboola, A.; Taylor, M. J. Class invariants of Mordell-Weil groups, J. Reine Angew. Math., Tome 447 (1994), pp. 23-61 | Article | MR 1263168 | Zbl 0799.11049

[5] Anantharaman, S. Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Bull. Soc. Math. Fr., Tome 33 (1973), pp. 5-79 (Suppl., Mém.) | Numdam | MR 335524 | Zbl 0286.14001

[6] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron Models, Springer, Berlin-Heidelberg-New York, Ergeb. Math. Grenzgeb. (3), Tome 21 (1990) | MR 1045822 | Zbl 0705.14001

[7] Cassou-Noguès, P.; Taylor, M. J. Structures galoisiennes et courbes elliptiques, J. Théor. Nombres Bordeaux, Tome 7 (1995), pp. 307-331 | Article | Numdam | MR 1413581 | Zbl 0852.11066

[8] Gillibert, J. Invariants de classes : le cas semi-stable, Compositio Mathematica, Tome 141 (2005), pp. 887-901 | Article | MR 2148197 | Zbl 02211030

[9] Grothendieck, A. Groupes de monodromie en géométrie algébrique, Springer, Berlin-Heidelberg-New York, Lecture Notes in Mathematics, Tome 288 (1972) | Zbl 0237.00013

[10] Grothendieck, A.; Artin, M.; Verdier, J. L. Théorie des topos et cohomologie étale des schémas, Springer, Berlin-Heidelberg-New York, Lecture Notes in Mathematics, Tome 269, 270 (1972)

[11] Milne, J. S. Arithmetic Duality Theorems, Academic Press, Boston, MA, Perspectives in Mathematics, Tome 1 (1986) | MR 881804 | Zbl 0613.14019

[12] Mumford, D. Bi-extensions of formal groups, Algebraic Geometry, Oxford University Press (1969), pp. 307-322 (Bombay, 1968) | MR 257089 | Zbl 0216.33101

[13] Pappas, G. On torsion line bundles and torsion points on abelian varieties, Duke Math. J., Tome 91 (1998), pp. 215-224 | Article | MR 1600574 | Zbl 1029.11020

[14] Srivastav, A.; Taylor, M. J. Elliptic curves with complex multiplication and Galois module structure, Invent. Math., Tome 99 (1990), pp. 165-184 | Article | MR 1029394 | Zbl 0705.14031

[15] Taylor, M. J. Mordell-Weil groups and the Galois module structure of rings of integers, Illinois J. Math., Tome 32 (1988), pp. 428-452 | MR 947037 | Zbl 0631.14033

[16] Taylor, M. J. L-functions and Galois modules : Explicit Galois Modules, L-functions and Arithmetic, Cambridge University Press (LMS Lecture Notes) Tome 153 (1991) | MR 1110391 | Zbl 0733.11044

[17] Waterhouse, W. C. Principal homogeneous spaces and group scheme extension, Trans. Am. Math. Soc., Tome 153 (1971), pp. 181-189 | Article | MR 269659 | Zbl 0208.48401

[18] Werner, A. On Grothendieck’s pairing of component groups in the semistable reduction case, J. Reine Angew. Math., Tome 486 (1997), pp. 205-215 | Article | MR 1450756 | Zbl 0872.14037