On component groups of Jacobians of Drinfeld modular curves
Annales de l'Institut Fourier, Volume 54 (2004) no. 7, pp. 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: 10.5802/aif.2078
Classification: 11G18, 11G10, 14G22, 11G09
Keywords: Component groups, Drinfeld modular curves, monodromy pairing
Mot clés : groupe de composants, courbe modulaire de Drinfeld, monodromie
Papikian, Mihran 1

1 Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)
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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://archive.numdam.org/articles/10.5802/aif.2078/

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