The Drinfeld Modular Jacobian J 1 (n) has connected fibers
Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1217-1252.

We study the integral model of the Drinfeld modular curve X 1 (n) for a prime n𝔽 q [T]. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod n. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order n in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of X 1 (n) which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over n. Thus the mod n component group of J 1 (n) is trivial, i.e. J 1 (n) has connected fibers.

Nous étudions le modèle intégral de la courbe modulaire X 1 (n) de Drinfeld pour un élément irreductible n𝔽 q [T]. Un analogue du corps de fonctions de la théorie des courbes d’Igusa est introduit pour décrire sa réduction mod n. Un résultat décrivant l’anneau universel de déformation d’une paire se composant d’un module de Drinfeld supersingulier et d’un point d’ordre n en termes de l’invariant de Hasse de ce module de Drinfeld est prouvé. Nous appliquons alors la résolution de Jung-Hirzebruch afin que les surfaces arithmétiques produisent un modèle régulier de X 1 (n) qui, après des contractions dans la fibre spéciale, donne un modèle régulier tel que la fibre au-dessus de n est géométriquement intègre. Ainsi, la réduction mod n du groupe des composants de J 1 (n) est triviale, c’est-à-dire les fibres de J 1 (n) sont connexes.

DOI: 10.5802/aif.2292
Classification: 11F52,  14H40,  14L05,  11G09
Keywords: Component groups, Drinfeld modular curves, Igusa curves
Shastry, Sreekar M. 1

1 Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)
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Shastry, Sreekar M. The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers. Annales de l'Institut Fourier, Volume 57 (2007) no. 4, pp. 1217-1252. doi : 10.5802/aif.2292. http://www.numdam.org/articles/10.5802/aif.2292/

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