On component groups of Jacobians of Drinfeld modular curves

[Sur les groupes de composants des Jacobiennes des courbes modulaires de Drinfeld]

Papikian, Mihran

Annales de l'Institut Fourier, Tome 54 (2004), p. 2163-2199 / Harvested from Numdam

Résumé

Soit $J_0({\mathfrak {n}})$ la variété Jacobienne de la courbe modulaire de Drinfeld $X_0({\mathfrak {n}})$ sur ${\mathbb {F}}_q(t)$, où ${\mathfrak {n}}$ est un idéal de ${\mathbb {F}}_q[t]$. Soit $0 \rightarrow B \rightarrow J_0({\mathfrak {n}}) \rightarrow A \rightarrow 0$ une suite exacte de variétés abéliennes. Supposons que $B$, comme sous-variété de $J_0({\mathfrak {n}})$, est stable sous l’action de l’algèbre de Hecker ${\mathbb {T}} \subset $ End $(J_0({\mathfrak {n}}))$. Nous donnons un critère suffisant pour l’exactitutde de la suite induite $0 \rightarrow \Phi _{B,\infty } \rightarrow \Phi _{J, \infty } \rightarrow \Phi _{A, \infty } \rightarrow 0$ du groupe de composants connexe des modèles de Néron de ces variétés abéliennes sur ${\mathbb {F}}_q [\![{1 \over t}]\!]$. Ce critère est toujours satisfait si $A$ ou $B$ est de dimension $1$. De plus, nous démontrons que la suite des parties de $\ell $-torsion des groupes de composantes connexes est exacte pour tout nombre premier $\ell $ ne divisant pas $(q-1)$. En particulier, cette suite est exacte quand $q=2$.

Abstract

Let $J_0({\mathfrak {n}})$ be the Jacobian variety of the Drinfeld modular curve $X_0({\mathfrak {n}})$ over ${\mathbb {F}}_q(t)$, where ${\mathfrak {n}}$ is an ideal in ${\mathbb {F}}_q[t]$. Let $0\rightarrow B \rightarrow J_0({\mathfrak {n}}) \rightarrow A \rightarrow 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0({\mathfrak {n}})$ , is stable under the action of the Hecke algebra ${\mathbb {T}} \subset $ End $ (J_0({\mathfrak {n}}))$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \rightarrow \Phi _{B, \infty } \rightarrow \Phi _{J, \infty } \rightarrow \Phi _{A, \infty } \rightarrow 0$ of the Néron models of these abelian varieties over ${\mathbb {F}}_q [\![{1 \over t}]\!]$. 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 $\ell $-power torsion for any prime $\ell $ not dividing $(q-1)$. In particular, the sequence is always exact when $q=2$.

Détail

Publié le : 2004-01-01

MR 2139692 | Zbl 1071.11034

DOI : https://doi.org/10.5802/aif.2078
Classification:  11G18,  11G10,  14G22,  11G09
Mots clés: groupe de composants, courbe modulaire de Drinfeld, monodromie

Bibliographie

[1] S. Bosch; W. Lütkebohmert Degenerating abelian varieties, Topology, Tome 30 (1991), pp. 653-698 | Article | MR 1133878 | Zbl 0761.14015

[2] S. Bosch; W. Lütkebohmert Formal and rigid geometry I, Math. Ann., Tome 295 (1993), pp. 291-317 | Article | MR 1202394 | Zbl 0808.14017

[3] S. Bosch; W. Lütkebohmert; M. Raynaud Néron models, Springer (1990) | MR 1045822 | Zbl 0705.14001

[4] B. Conrad Irreducible components of rigid spaces, Ann. Inst. Fourier, Tome 49 (1999), pp. 473-541 | Article | Numdam | MR 1697371 | Zbl 0928.32011

[5] B. Conrad; W. Stein Component groups of purely toric quotients, Math. Research Letters, Tome 8 (2001), pp. 745-766 | MR 1879817 | Zbl 01744231

[6] P. Deligne Formes modulaires et représentations de ${GL}(2)$, Springer (Lecture Notes in Math.) Tome 349 (1973), pp. 55-105 | Zbl 0271.10032

[7] V. Drinfeld Elliptic modules, Math. Sbornik, Tome 94 (1974), pp. 594-627 | MR 384707 | Zbl 0321.14014

[8] M. Emerton Optimal quotients of modular Jacobians, Math. Ann., Tome 327 (2003), pp. 429-458 | Article | MR 2021024 | Zbl 1061.11018

[9] J. Fresnel; M. Van Der Put Géométrie analytique rigide et applications, Birkhäuser (1981) | MR 644799 | Zbl 0479.14015

[10] E.-U. Gekeler Automorphe Formen über ${\bb F}_q(T)$ mit kleinem Führer, Abh. Math. Sem. Univ. Hamburg, Tome 55 (1985), pp. 111-146 | Article | MR 831522 | Zbl 0564.10026

[11] E.-U. Gekeler Über Drinfeld'sche Modulkurven vom Hecke-Typ, Comp. Math., Tome 57 (1986), pp. 219-236 | Numdam | MR 827352 | Zbl 0599.14032

[12] E.-U. Gekeler Analytic construction of Weil curves over function fields, J. Th. nombres Bordeaux, Tome 7 (1995), pp. 27-49 | Article | Numdam | MR 1413565 | Zbl 0846.11037

[13] E.-U. Gekeler Improper Eisenstein series on Bruhat-Tits trees, Manuscripta Math., Tome 86 (1995), pp. 367-391 | Article | MR 1323798 | Zbl 0884.11025

[14] E.-U. Gekeler On the cuspidal divisor group of a Drinfeld modular curve, Doc. Math. J. DMV, Tome 2 (1997), pp. 351-374 | MR 1487469 | Zbl 0895.11024

[15] E.-U. Gekeler; U. Nonnengardt Fundamental domains of some arithmetic groups over function fields, Internat. J. Math., Tome 6 (1995), pp. 689-708 | Article | MR 1351161 | Zbl 0858.11025

[16] E.-U. Gekeler; M. Reversat Jacobians of Drinfeld modular curves, J. reine angew. Math., Tome 476 (1996), pp. 27-93 | Article | MR 1401696 | Zbl 0848.11029

[17] S. Gelbart Automorphic forms on adele groups, Princeton Univ. Press (1975) | MR 379375 | Zbl 0329.10018

[18] L. Gerritzen; M. Van Der Put Schottky groups and Mumford curves, Springer, Lecture Notes in Math., Tome 817 (1980) | MR 590243 | Zbl 0442.14009

[19] A. Grothendieck Groupes de type mulitplicatif: homomorphismes dans un schéma en groupes, SGA 3, Tome exposé IX (1970)

[20] A. Grothendieck Modèles de Néron et monodromie, SGA 7, Tome exposé IX (1972) | Zbl 0248.14006

[21] A. Grothendieck; J. Dieudonné Étude cohomologique des faisceaux cohérents : EGA III, Publ. Math. IHÉS, Tome 11 (1962) | Numdam | Zbl 0118.36206

[21] A. Grothendieck; J. Dieudonné Étude cohomologique des faisceaux cohérents : EGA III, Publ. Math., Inst. Hautes Étud. Sci., Tome 17 (1963) | Numdam | MR 163911 | Zbl 0122.16102

[22] L. Illusie Réalisation $\ell$-adique de l'accouplement de monodromie d'après A. Grothendieck, Astérisque, Tome 196-197 (1991), pp. 27-44 | MR 1141455 | Zbl 0781.14011

[23] B. Mazur Modular curves and the Eisenstein ideal, Publ. Math. IHÉS, Tome 47 (1977), pp. 33-186 | Numdam | MR 488287 | Zbl 0394.14008

[24] D. Mumford Abelian varieties, Oxford Univ. Press (1970) | MR 282985 | Zbl 0326.14012

[25] D. Mumford An analytic construction of degenerating curves over complete local rings, Comp. Math., Tome 24 (1972), pp. 129-174 | Numdam | MR 352105 | Zbl 0228.14011

[26] M. Reversat Sur les revêtements de Schottky des courbes modulaires de Drinfeld, Arch. Math., Tome 66 (1996), pp. 378-387 | Article | MR 1383902 | Zbl 0853.14014

[27] K. Ribet Letter to J.-F. Mestre (1987) (available at xxx.lanl.gov)

[28] K. Ribet On the modular representations of $Gal(\overline{\bb Q}/{\bb Q})$ arising from modular forms, Invent. Math., Tome 100 (1990), pp. 431-476 | Article | MR 1047143 | Zbl 0773.11039

[29] J-P. Serre Trees, Springer (1980) | MR 607504 | Zbl 0548.20018

[30] W. Stein The refined Eisenstein conjecture (1999) (Preprint)

[31] A. Tamagawa The Eisenstein quotient of the Jacobian variety of a Drinfeld modular curve, Publ. RIMS, Kyoto Univ., Tome 31 (1995), pp. 204-246 | Article | MR 1329480 | Zbl 1045.11510

[32] M. Van Der Put A note on $p$-adic uniformization, Proc. Nederl. Akad. Wetensch., Tome 90 (1987), pp. 313-318 | MR 914089 | Zbl 0624.32018

[33] M. Van Der Put Discrete groups, Mumford curves and theta functions, Ann. Fac. Sci. Toulouse, Tome 1 (1992), pp. 399-438 | Article | Numdam | MR 1225666 | Zbl 0789.14020

[34] D. Zagier Modular parametrizations of elliptic curves, Canad. Math. Bull., Tome 28 (1985), pp. 372-384 | Article | MR 790959 | Zbl 0579.14027