Arithmetic differential equations in several variables

Buium, Alexandru; Simanca, Santiago R.

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.

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

[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. $\delta$-anneaux et $\lambda$-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 $ℂ\mathbb{P}\left(\infty \right)$, Comm. Algebra, Volume 10 (1982) no. 3, pp. 311-328 | DOI | MR | Zbl