We survey recent work on arithmetic analogues of ordinary and partial differential equations.
On présente des résultats récents sur les analogues arithmétiques des équations différentielles ordinaires et aux dérivées partielles.
Accepted:
DOI: 10.5802/aif.2504
Classification: 11G07, 35G20, 11F03
Keywords: Differential equations, elliptic curves, -adic numbers, modular forms
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2504/} }
TY - JOUR TI - Arithmetic differential equations in several variables JO - Annales de l'Institut Fourier PY - 2009 DA - 2009/// SP - 2685 EP - 2708 VL - 59 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2504/ UR - https://zbmath.org/?q=an%3A1198.12003 UR - https://www.ams.org/mathscinet-getitem?mr=2649337 UR - https://doi.org/10.5802/aif.2504 DO - 10.5802/aif.2504 LA - en ID - AIF_2009__59_7_2685_0 ER -
Buium, Alexandru; Simanca, Santiago R. Arithmetic differential equations in several variables. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2685-2708. doi : 10.5802/aif.2504. https://aif.centre-mersenne.org/articles/10.5802/aif.2504/
[1] 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] The basic geometry of Witt vectors (arXiv:math/0801.1691) | Zbl: 1276.13018
[3] Plethystic algebra, Adv. Math., Tome 194 (2005) no. 2, pp. 246-283 | Article | MR: 2139914 | Zbl: 1098.13033
[4] 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] Differential Algebra and Diophantine Geometry, Hermann, Paris, 1994 | MR: 1487891 | Zbl: 0870.12007
[6] Differential characters of Abelian varieties over -adic fields, Invent. Math., Tome 122 (1995), pp. 309-340 | Article | EuDML: 144323 | MR: 1358979 | Zbl: 0841.14037
[7] Geometry of -jets, Duke J. Math., Tome 82 (1996) no. 2, pp. 349-367 | Article | MR: 1387233 | Zbl: 0882.14007
[8] Differential modular forms, J. reine angew. Math., Tome 520 (2000), pp. 95-167 | Article | MR: 1748272 | Zbl: 1045.11025
[9] Arithmetic Differential Equations, Math. Surveys and Monographs, Tome 118, American Mathematical Society, Providence, RI, 2005, xxxii+310 pages | MR: 2166202 | Zbl: 1088.14001
[10] 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] 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] Igusa differential modular forms (preprint)
[13] Arithmetic partial differential equations (arXiv:math/0605107)
[14] Arithmetic partial differential equations II: modular curves (arXiv:0804.4856)
[15] Arithmetic Laplacians, Adv. Math., Tome 220 (2009), pp. 246-277 | Article | MR: 2462840 | Zbl: 1172.14027
[16] Local -factors of motives and regularized determinants, Invent. Math., Tome 107 (1992) no. 1, pp. 135-150 | Article | MR: 1135468 | Zbl: 0762.14015
[17] An introduction to -functions, Annals of Mathematics Studies, Tome 133, Princeton University Press, Princeton, NJ, 1994, xxii+323 pages | MR: 1274045 | Zbl: 0830.12004
[18] Schemata over local rings, Ann. of Math., Tome 73 (1961), pp. 624-648 | Article | MR: 126449 | Zbl: 0115.39004
[19] La théorie des classes de Chern, Bull. Soc. Math. France, Tome 86 (1958), pp. 137-154 | Numdam | MR: 116023 | Zbl: 0091.33201
[20] Isogeny covariant differential modular forms modulo , Compositio Math., Tome 128 (2001) no. 1, pp. 17-34 | Article | MR: 1847663 | Zbl: 1039.11036
[21] -anneaux et -anneaux, C.R. Math. Rep. Acad. Sci Canada, Tome 4 (1985), pp. 227-232 | MR: 802319 | Zbl: 0583.13004
[22] Quantum calculus, Universitext, Springer-Verlag, New York, 2002 (x+112 pp.) | MR: 1865777 | Zbl: 0986.05001
[23] Differential algebra and algebraic groups, Pure and Applied Mathematics, Tome 54, Academic Press, New York-London, 1973, xviii+446 pages | MR: 568864 | Zbl: 0264.12102
[24] Cyclotomy and analytic geometry over (arXiv:0809.1564)
[25] Algebraic curves over fields with differentiation, Ser. Mat., Tome 22, Izv. Akad. Nauk SSSR, 1958, pp. 737-756 (AMS translations Series 2, 37 (1964), pp. 59-78) | Zbl: 0151.27601
[26] On the asymptotic behaviour of Heegner points, Turkish J. Math., Tome 23 (1999), pp. 549-556 | MR: 1780940 | Zbl: 0977.11025
[27] Courbes sur une variété abélienne et points de torsion, Invent. Math., Tome 71 (1983), pp. 207-235 | Article | MR: 688265 | Zbl: 0564.14020
[28] 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] Jets via Hasse-Schmidt derivations, Diophantine geometry (Zannier, U., ed.) (CRM Series) Tome 4, Sc. Norm., Pisa, 2007, pp. 335-361 | MR: 2349665 | Zbl: pre05263295
[30] Lambda-rings, binomial domains, and vector bundles over , Comm. Algebra, Tome 10 (1982) no. 3, pp. 311-328 | Article | MR: 651605 | Zbl: 0492.55004
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