Arithmetic differential equations in several variables
[Équations différentielles arithmétiques à plusieurs variables]
Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 2685-2708.

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.

DOI : 10.5802/aif.2504
Classification : 11G07, 35G20, 11F03
Keywords: Differential equations, elliptic curves, $p$-adic numbers, modular forms
Mot clés : équations différentielles, courbes elliptiques, nombres $p$-adiques, formes modulaires
Buium, Alexandru 1 ; Simanca, Santiago R. 1

1 University of New Mexico Department of Mathematics and Statistics Albuquerque, NM 87131 (USA)
@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},
     pages = {2685--2708},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     doi = {10.5802/aif.2504},
     zbl = {1198.12003},
     mrnumber = {2649337},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2504/}
}
TY  - JOUR
AU  - Buium, Alexandru
AU  - Simanca, Santiago R.
TI  - Arithmetic differential equations in several variables
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 2685
EP  - 2708
VL  - 59
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2504/
DO  - 10.5802/aif.2504
LA  - en
ID  - AIF_2009__59_7_2685_0
ER  - 
%0 Journal Article
%A Buium, Alexandru
%A Simanca, Santiago R.
%T Arithmetic differential equations in several variables
%J Annales de l'Institut Fourier
%D 2009
%P 2685-2708
%V 59
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2504/
%R 10.5802/aif.2504
%G en
%F 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) no. 7, pp. 2685-2708. doi : 10.5802/aif.2504. http://www.numdam.org/articles/10.5802/aif.2504/

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

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

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

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

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

[6] Buium, A. Differential characters of Abelian varieties over p-adic fields, Invent. Math., Volume 122 (1995), pp. 309-340 | DOI | EuDML | MR | Zbl

[7] Buium, A. Geometry of p-jets, Duke J. Math., Volume 82 (1996) no. 2, pp. 349-367 | DOI | MR | Zbl

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

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

[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)

[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)

[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., Volume 220 (2009), pp. 246-277 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

[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., Volume 22, Izv. Akad. Nauk SSSR, 1958, pp. 737-756 AMS translations Series 2, 37 (1964), pp. 59-78 | Zbl

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

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

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

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

[30] Wilkerson, C. Lambda-rings, binomial domains, and vector bundles over (), Comm. Algebra, Volume 10 (1982) no. 3, pp. 311-328 | DOI | MR | Zbl

Cité par Sources :