On component groups of Jacobians of Drinfeld modular curves

Papikian, Mihran

doi:10.5802/aif.2078

On component groups of Jacobians of Drinfeld modular curves
[Sur les groupes de composants des Jacobiennes des courbes modulaires de Drinfeld]

Papikian, Mihran ¹

¹ Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)

Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2163-2199.

Résumé
Abstract

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 $0 \to B \to J_{0} (𝔫) \to A \to 0$ 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 $𝕋 \subset$ End $(J_{0} (𝔫))$ . Nous donnons un critère suffisant pour l’exactitutde de la suite induite $0 \to Φ_{B, \infty} \to Φ_{J, \infty} \to Φ_{A, \infty} \to 0$ du groupe de composants connexe des modèles de Néron de ces variétés abéliennes sur $𝔽_{q} [[\frac{1}{t}]]$ . Ce critère est toujours satisfait si $A$ ou $B$ est de dimension $1$ . De plus, nous démontrons que la suite des parties de $ℓ$ -torsion des groupes de composantes connexes est exacte pour tout nombre premier $ℓ$ ne divisant pas $(q - 1)$ . En particulier, cette suite est exacte quand $q = 2$ .

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 $0 \to B \to J_{0} (𝔫) \to A \to 0$ 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 $𝕋 \subset$ End $(J_{0} (𝔫))$ . We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \to Φ_{B, \infty} \to Φ_{J, \infty} \to Φ_{A, \infty} \to 0$ of the Néron models of these abelian varieties over $𝔽_{q} [[\frac{1}{t}]]$ . 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 $ℓ$ -power torsion for any prime $ℓ$ not dividing $(q - 1)$ . In particular, the sequence is always exact when $q = 2$ .

MR Zbl

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

Affiliations des auteurs :

Papikian, Mihran ¹

¹ Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)

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     title = {On component groups of {Jacobians} of {Drinfeld} modular curves},
     journal = {Annales de l'Institut Fourier},
     pages = {2163--2199},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {7},
     year = {2004},
     doi = {10.5802/aif.2078},
     zbl = {1071.11034},
     mrnumber = {2139692},
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Papikian, Mihran. On component groups of Jacobians of Drinfeld modular curves. Annales de l'Institut Fourier, Tome 54 (2004) no. 7, pp. 2163-2199. doi : 10.5802/aif.2078. https://aif.centre-mersenne.org/articles/10.5802/aif.2078/

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