On component groups of Jacobians of Drinfeld modular curves
Annales de l'Institut Fourier, Volume 54 (2004) no. 7, p. 2163-2199

Let J 0 (𝔫) be the Jacobian variety of the Drinfeld modular curve X 0 (𝔫) over 𝔽 q (t), where 𝔫 is an ideal in 𝔽 q [t]. Let 0BJ 0 (𝔫)A0 be an exact sequence of abelian varieties. Assume B, as a subvariety of J 0 (𝔫) , is stable under the action of the Hecke algebra 𝕋 End (J 0 (𝔫)). We give a criterion which is sufficient for the exactness of the induced sequence of component groups 0Φ B, Φ J, Φ A, 0 of the Néron models of these abelian varieties over 𝔽 q [[1 t]]. This criterion is always satisfied when either A or B is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on -power torsion for any prime not dividing (q-1). In particular, the sequence is always exact when q=2.

Soit J 0 (𝔫) la variété Jacobienne de la courbe modulaire de Drinfeld X 0 (𝔫) sur 𝔽 q (t), où 𝔫 est un idéal de 𝔽 q [t]. Soit 0BJ 0 (𝔫)A0 une suite exacte de variétés abéliennes. Supposons que B, comme sous-variété de J 0 (𝔫), est stable sous l’action de l’algèbre de Hecker 𝕋 End (J 0 (𝔫)). Nous donnons un critère suffisant pour l’exactitutde de la suite induite 0Φ B, Φ J, Φ A, 0 du groupe de composants connexe des modèles de Néron de ces variétés abéliennes sur 𝔽 q [[1 t]]. Ce critère est toujours satisfait si A ou B est de dimension 1. De plus, nous démontrons que la suite des parties de -torsion des groupes de composantes connexes est exacte pour tout nombre premier ne divisant pas (q-1). En particulier, cette suite est exacte quand q=2.

DOI : https://doi.org/10.5802/aif.2078
Classification:  11G18,  11G10,  14G22,  11G09
Keywords: Component groups, Drinfeld modular curves, monodromy pairing
@article{AIF_2004__54_7_2163_0,
     author = {Papikian, Mihran},
     title = {On component groups of Jacobians of Drinfeld modular curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {7},
     year = {2004},
     pages = {2163-2199},
     doi = {10.5802/aif.2078},
     zbl = {1071.11034},
     mrnumber = {2139692},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_7_2163_0}
}
Papikian, Mihran. On component groups of Jacobians of Drinfeld modular curves. Annales de l'Institut Fourier, Volume 54 (2004) no. 7, pp. 2163-2199. doi : 10.5802/aif.2078. http://www.numdam.org/item/AIF_2004__54_7_2163_0/

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