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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On ordinary linear $ p$-adic differential equations


Authors: B. Dwork and P. Robba
Journal: Trans. Amer. Math. Soc. 231 (1977), 1-46
MSC: Primary 14G20
MathSciNet review: 0447247
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Abstract: We study the solutions of ordinary linear differential equations whose coefficients are analytic elements. As one application we show nonexistence of index for certain linear differential operators with rational function coefficients.


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  • [1] James Ax, Zeros of polynomials over local fields—The Galois action, J. Algebra 15 (1970), 417–428. MR 0263786
  • [2] D. N. Clark, A note on the 𝑝-adic convergence of solutions of linear differential equations, Proc. Amer. Math. Soc. 17 (1966), 262–269. MR 0186895, 10.1090/S0002-9939-1966-0186895-8
  • [3] B. Dwork, On 𝑝-adic differential equations. II. The 𝑝-adic asymptotic behavior of solutions of ordinary linear differential equations with rational function coefficients, Ann. of Math. (2) 98 (1973), 366–376. MR 0572253
  • [4] -, On p-adic differential equations. III, Invent. Math. 20 (1973), 34-45.
  • [5] Artur Erdélyi, Über die Integration der Whittakerschen Differentialgleichung in geschlossener Form, Monatsh. Math. Phys. 46 (1937), no. 1, 1–9 (German). MR 1550723, 10.1007/BF01792661
  • [6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, Vol. I, McGraw-Hill, New York, 1953. MR 15, 419.
  • [7] Elhanan Motzkin, La décomposition d’un élément analytique en facteurs singuliers, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 1, ix, 67–82 (French, with English summary). MR 0470247
  • [8] -, Sur des ensembles analytiques p-adiques définis à l'aide de fonctions analytiques bornées, Groupe d'Étude d'Analyse Ultramétrique (Amice, Robba), 1975/76.
  • [9] E. Poole, Introduction to the theory of linear differential equations, Oxford, 1936.
  • [10] Philippe Robba, Fonctions analytiques sur les corps valués ultramétriques complets, Prolongement analytique et algèbres de Banach ultramétriques, Société Mathématique de France, Paris, 1973, pp. 109–218, 219–220. Astérisque, No. 10 (French, with English summary). MR 0357841
  • [11] P. Robba, On the index of 𝑝-adic differential operators. I, Ann. of Math. (2) 101 (1975), 280–316. MR 0364243
  • [12] P. Robba, On the index of 𝑝-adic differential operators. II, Duke Math. J. 43 (1976), no. 1, 19–31. MR 0389910
  • [13] Philippe Robba, Prolongement des solutions d’une équation différentielle 𝑝-adique, C. R. Acad. Sci. Paris Sér. A 279 (1974), 153–154 (French). MR 0349643
  • [14] O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776
  • [15] Marius van der Put, The non-archimedean corona problem, Table Ronde d’Analyse Non Archimédienne (Paris, 1972) Soc. Math. France, Paris, 1974, pp. 287–317. Bull. Soc. Math. France, Mém. No. 39-40. MR 0364685

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0447247-8
Keywords: Nonarchimedean valuation, completion field, analytic element, analytic function, generic disk, differential equation
Article copyright: © Copyright 1977 American Mathematical Society