On component groups of Jacobians of Drinfeld modular curves

Papikian, Mihran

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

Soit $J_0\left(𝔫\right)$ la variété Jacobienne de la courbe modulaire de Drinfeld $X_0\left(𝔫\right)$ sur ${𝔽}_q\left(t\right)$, où $𝔫$ est un idéal de ${𝔽}_q\left[t\right]$. Soit $0\to B\to J_0\left(𝔫\right)\to A\to 0$ une suite exacte de variétés abéliennes. Supposons que $B$, comme sous-variété de $J_0\left(𝔫\right)$, est stable sous l’action de l’algèbre de Hecker $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. Nous donnons un critère suffisant pour l’exactitutde de la suite induite $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ du groupe de composants connexe des modèles de Néron de ces variétés abéliennes sur ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. 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 $q=2$.

Let $J_0\left(𝔫\right)$ be the Jacobian variety of the Drinfeld modular curve $X_0\left(𝔫\right)$ over ${𝔽}_q\left(t\right)$, where $𝔫$ is an ideal in ${𝔽}_q\left[t\right]$. Let $0\to B\to J_0\left(𝔫\right)\to A\to 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0\left(𝔫\right)$ , is stable under the action of the Hecke algebra $𝕋\subset$ End $\left(J_0\left(𝔫\right)\right)$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0\to {\Phi }_{B,\infty }\to {\Phi }_{J,\infty }\to {\Phi }_{A,\infty }\to 0$ of the Néron models of these abelian varieties over ${𝔽}_q\left[\left[\frac{1}{t}\right]\right]$. 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 $q=2$.

Classification : 11G18, 11G10, 14G22, 11G09
Keywords: Component groups, Drinfeld modular curves, monodromy pairing
Mot clés : groupe de composants, courbe modulaire de Drinfeld, monodromie

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

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

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

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

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

[6] P. Deligne Formes modulaires et représentations de $GL\left(2\right)$ (Lecture Notes in Math.), Volume 349 (1973), pp. 55-105 | Zbl

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

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

[9] J. Fresnel; M. van der Put Géométrie analytique rigide et applications, Birkhäuser, 1981 | MR | Zbl

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

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

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

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

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

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

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

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

[18] L. Gerritzen; M. van der Put Schottky groups and Mumford curves, Lecture Notes in Math., 817, Springer, 1980 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[29] J-P. Serre Trees, Springer, 1980 | MR | Zbl

[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., Volume 31 (1995), pp. 204-246 | DOI | MR | Zbl

[32] M. van der Put A note on $p$-adic uniformization, Proc. Nederl. Akad. Wetensch., Volume 90 (1987), pp. 313-318 | MR | Zbl

[33] M. van der Put Discrete groups, Mumford curves and theta functions, Ann. Fac. Sci. Toulouse, Volume 1 (1992), pp. 399-438 | DOI | Numdam | MR | Zbl

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