We study the integral model of the Drinfeld modular curve for a prime . A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod . A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order 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 which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over . Thus the mod component group of is trivial, i.e. has connected fibers.
Nous étudions le modèle intégral de la courbe modulaire de Drinfeld pour un élément irreductible . Un analogue du corps de fonctions de la théorie des courbes d’Igusa est introduit pour décrire sa réduction mod . 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 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 qui, après des contractions dans la fibre spéciale, donne un modèle régulier tel que la fibre au-dessus de est géométriquement intègre. Ainsi, la réduction mod du groupe des composants de est triviale, c’est-à-dire les fibres de sont connexes.
Keywords: Component groups, Drinfeld modular curves, Igusa curves
Mot clés : Groupes composants, courbes modulaires de Drinfeld, courbes de Igusa
Shastry, Sreekar M. 1
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TY - JOUR AU - Shastry, Sreekar M. TI - The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers JO - Annales de l'Institut Fourier PY - 2007 SP - 1217 EP - 1252 VL - 57 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2292/ DO - 10.5802/aif.2292 LA - en ID - AIF_2007__57_4_1217_0 ER -
%0 Journal Article %A Shastry, Sreekar M. %T The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers %J Annales de l'Institut Fourier %D 2007 %P 1217-1252 %V 57 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2292/ %R 10.5802/aif.2292 %G en %F AIF_2007__57_4_1217_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.2292/
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