Arithmetic differential equations in several variables

Buium, Alexandru; Simanca, Santiago R.

Arithmetic differential equations in several variables
[Équations différentielles arithmétiques à plusieurs variables]

Annales de l'Institut Fourier, Tome 59 (2009), p. 2685-2708 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On présente des résultats récents sur les analogues arithmétiques des équations différentielles ordinaires et aux dérivées partielles.

We survey recent work on arithmetic analogues of ordinary and partial differential equations.

Publié le : 2009-01-01

MR 2649337 | Zbl 1198.12003

DOI : https://doi.org/10.5802/aif.2504
Classification: 11G07, 35G20, 11F03
Mots clés: équations différentielles, courbes elliptiques, nombres

p

-adiques, formes modulaires

@article{AIF_2009__59_7_2685_0,
     author = {Buium, Alexandru and Simanca, Santiago R.},
     title = {Arithmetic differential equations in several variables},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {2685-2708},
     doi = {10.5802/aif.2504},
     zbl = {1198.12003},
     mrnumber = {2649337},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_7_2685_0}
}

Buium, Alexandru; Simanca, Santiago R. Arithmetic differential equations in several variables. Annales de l'Institut Fourier, Tome 59 (2009) pp. 2685-2708. doi : 10.5802/aif.2504. http://gdmltest.u-ga.fr/item/AIF_2009__59_7_2685_0/

Bibliographie

[1] Barcau, M. Isogeny covariant differential modular forms and the space of elliptic curves up to isogeny, Compositio Math., Tome 137 (2003), pp. 237-273 | Article | MR 1988499 | Zbl 1074.14027

[2] Borger, J. The basic geometry of Witt vectors (arXiv:math/0801.1691) | Zbl 1276.13018

[3] Borger, J.; Wieland, B. Plethystic algebra, Adv. Math., Tome 194 (2005) no. 2, pp. 246-283 | Article | MR 2139914 | Zbl 1098.13033

[4] Buium, A. Intersections in jet spaces and a conjecture of S. Lang, Ann. of Math., Tome 136 (1992), pp. 583-593 | Article | MR 1189865 | Zbl 0817.14021

[5] Buium, A. Differential Algebra and Diophantine Geometry, Hermann, Paris (1994) | MR 1487891 | Zbl 0870.12007

[6] Buium, A. Differential characters of Abelian varieties over $p$ -adic fields, Invent. Math., Tome 122 (1995), pp. 309-340 | Article | MR 1358979 | Zbl 0841.14037

[7] Buium, A. Geometry of $p$ -jets, Duke J. Math., Tome 82 (1996) no. 2, pp. 349-367 | Article | MR 1387233 | Zbl 0882.14007

[8] Buium, A. Differential modular forms, J. reine angew. Math., Tome 520 (2000), pp. 95-167 | Article | MR 1748272 | Zbl 1045.11025

[9] Buium, A. Arithmetic Differential Equations, Math. Surveys and Monographs, American Mathematical Society, Providence, RI, Tome 118 (2005), pp. xxxii+310 | MR 2166202 | Zbl 1088.14001

[10] Buium, A.; Poonen, B. Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: global results (Duke Math. J., to appear) | Zbl pre05532606

[11] Buium, A.; Poonen, B. Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results (Compositio Math., to appear) | Zbl pre05572293

[12] Buium, A.; Saha, A. Igusa differential modular forms (preprint)

[13] Buium, A.; Simanca, S. R. Arithmetic partial differential equations (arXiv:math/0605107)

[14] Buium, A.; Simanca, S. R. Arithmetic partial differential equations II: modular curves (arXiv:0804.4856)

[15] Buium, A.; Simanca, S. R. Arithmetic Laplacians, Adv. Math., Tome 220 (2009), pp. 246-277 | Article | MR 2462840 | Zbl 1172.14027

[16] Deninger, C. Local $L$ -factors of motives and regularized determinants, Invent. Math., Tome 107 (1992) no. 1, pp. 135-150 | Article | MR 1135468 | Zbl 0762.14015

[17] Dwork, B.; Gerotto, G.; Sullivan, F. J. An introduction to $G$ -functions, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, Tome 133 (1994), pp. xxii+323 | MR 1274045 | Zbl 0830.12004

[18] Greenberg, M. Schemata over local rings, Ann. of Math., Tome 73 (1961), pp. 624-648 | Article | MR 126449 | Zbl 0115.39004

[19] Grothendieck, A. La théorie des classes de Chern, Bull. Soc. Math. France, Tome 86 (1958), pp. 137-154 | Numdam | MR 116023 | Zbl 0091.33201

[20] Hurlburt, C. Isogeny covariant differential modular forms modulo $p$ , Compositio Math., Tome 128 (2001) no. 1, pp. 17-34 | Article | MR 1847663 | Zbl 1039.11036

[21] Joyal, A. $δ$ -anneaux et $λ$ -anneaux, C.R. Math. Rep. Acad. Sci Canada, Tome 4 (1985), pp. 227-232 | MR 802319 | Zbl 0583.13004

[22] Kac, V.; Cheung, P. Quantum calculus, Universitext, Springer-Verlag, New York (2002) (x+112 pp.) | MR 1865777 | Zbl 0986.05001

[23] Kolchin, E. R. Differential algebra and algebraic groups, Pure and Applied Mathematics, Academic Press, New York-London, Tome 54 (1973), pp. xviii+446 | MR 568864 | Zbl 0264.12102

[24] Manin, Yu. I. Cyclotomy and analytic geometry over $𝔽_{1}$ (arXiv:0809.1564)

[25] Manin, Yu. I. Algebraic curves over fields with differentiation, Ser. Mat., Izv. Akad. Nauk SSSR, Tome 22 (1958), pp. 737-756 (AMS translations Series 2, 37 (1964), pp. 59-78) | Zbl 0151.27601

[26] Nekovár, J.; Schappacher, N. On the asymptotic behaviour of Heegner points, Turkish J. Math., Tome 23 (1999), pp. 549-556 | MR 1780940 | Zbl 0977.11025

[27] Raynaud, M. Courbes sur une variété abélienne et points de torsion, Invent. Math., Tome 71 (1983), pp. 207-235 | Article | MR 688265 | Zbl 0564.14020

[28] Soulé Les variétés sur le corps à un élément, Moscow Math. J., Tome 4 (2004) no. 1, pp. 217-244 | MR 2074990 | Zbl 1103.14003

[29] Vojta, P.; Zannier, U. Jets via Hasse-Schmidt derivations, Diophantine geometry, Sc. Norm., Pisa (CRM Series) Tome 4 (2007), pp. 335-361 | MR 2349665 | Zbl pre05263295

[30] Wilkerson, C. Lambda-rings, binomial domains, and vector bundles over $ℂ ℙ (\infty)$ , Comm. Algebra, Tome 10 (1982) no. 3, pp. 311-328 | Article | MR 651605 | Zbl 0492.55004