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

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

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

Abstract (VO)
Résumé

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 $q = 2$ .

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 $q = 2$ .

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Zbl MR

DOI : 10.5802/aif.2078

Classification : 11G18, 11G10, 14G22, 11G09
Keywords: Component groups, Drinfeld modular curves, monodromy pairing
Mots-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)

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

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     title = {On component groups of {Jacobians} of {Drinfeld} modular curves},
     journal = {Annales de l'Institut Fourier},
     pages = {2163--2199},
     year = {2004},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
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     zbl = {1071.11034},
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