Soit G un groupe algébrique réductif connexe sur un corps global F de caractéristique nulle. Nous introduisons la notion de famille évanescente de sous-groupes compacts K de G sur les adèles finis et l'utilisons pour calculer asymptotiquement les nombres de Lefschetz et (conjecturalement) le nombre de points des variétés de Shimura (attachées à G et K) sur les corps finis. De cette étude, nous tirons un cadre général donnant naissance à des familles de courbes de Shimura atteignant la borne de Drinfeld–Vlăduţ.
Let G be an algebraic, connected, reductive group over a global field F of characteristic zero. We introduce a notion of vanishing family of compact subgroups K of G over the finite adeles and use it to compute asymptotically Lefschetz numbers and (at least conjecturally) the number of points of Shimura varieties (attached to G and K) over finite fields. We deduce a general setting giving families of Shimura curves reaching the Drinfeld–Vlăduţ bound.
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@article{CRMATH_2006__342_12_899_0,
author = {Sauvageot, Fran\c{c}ois},
title = {Asymptotique des vari\'et\'es de {Shimura}},
journal = {Comptes Rendus. Math\'ematique},
pages = {899--902},
publisher = {Elsevier},
volume = {342},
number = {12},
year = {2006},
doi = {10.1016/j.crma.2006.04.012},
language = {fr},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2006.04.012/}
}
TY - JOUR AU - Sauvageot, François TI - Asymptotique des variétés de Shimura JO - Comptes Rendus. Mathématique PY - 2006 SP - 899 EP - 902 VL - 342 IS - 12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2006.04.012/ DO - 10.1016/j.crma.2006.04.012 LA - fr ID - CRMATH_2006__342_12_899_0 ER -
Sauvageot, François. Asymptotique des variétés de Shimura. Comptes Rendus. Mathématique, Tome 342 (2006) no. 12, pp. 899-902. doi : 10.1016/j.crma.2006.04.012. http://archive.numdam.org/articles/10.1016/j.crma.2006.04.012/
[1] The -Lefschetz numbers of Hecke operators, Invent. Math., Volume 97 (1989), pp. 257-290
[2] Number of points of an algebraic curve, Funktsional. Anal. i Prilozhen., Volume 17 (1983), pp. 68-69
[3] Discrete series characters and the Lefschetz formula for Hecke operators, Duke Math. J., Volume 89 (1997), pp. 477-554
[4] Correction to “Discrete series characters and the Lefschetz formula for Hecke operators”, Duke Math. J., Volume 92 (1998), pp. 665-666
[5] Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981), pp. 721-724
[6] Stable trace formula: elliptic singular terms, Math. Ann., Volume 275 (1984), pp. 365-399
[7] Shimura varieties and λ-adic representations (Clozel, L.; Milne, J.S., eds.), Automorphic Forms, Shimura Varieties and L-Functions, Perspectives in Mathematics, vol. I, Academic Press, 1988, pp. 161-209
[8] The points on a Shimura variety modulo a prime of good reduction (Langlands, R.P.; Ramakrishnan, D., eds.), The Zeta Functions of Picard Modular Surfaces, Les publications CRM, Montréal, 1992, pp. 151-253
[9] The Semi-Simple Zeta Function of Quaternionic Shimura Varieties, Lecture Notes in Mathematics, vol. 1657, Springer-Verlag, 1997
[10] Répartition asymptotique des valeurs propres de l'opérateur de Hecke , J. Amer. Math. Soc., Volume 10 (1997), pp. 75-102
[11] Modular curves, Shimura curves and Goppa codes, better than Varshamov–Gilbert bound, Math. Nachr., Volume 109 (1982), pp. 21-28
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