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Book Review

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MathSciNet review: 2952710
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Book Information:

Author: Kiran Kedlaya
Title: $p$-adic differential equations
Additional book information: Cambridge Studies in Advanced Mathematics, Vol. 125, Cambridge University Press, Cambridge, 2010, xviii+380 pp., ISBN 978-0-521-76879-5

1. Yves André, Filtrations de type Hasse-Arf et monodromie 𝑝-adique, Invent. Math. 148 (2002), no. 2, 285–317 (French). MR 1906151, https://doi.org/10.1007/s002220100207
2. Yves André, Représentations galoisiennes et opérateurs de Bessel 𝑝-adiques, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 3, 779–808 (French, with English and French summaries). MR 1907387
3. Bruno Chiarellotto, An invitation to -adic differential equations, Arithmetic and Galois theory of differential equations, Séminaires et Congrès, vol. 23, 2011, pp. 115-168.
4. Gilles Christol and Zoghman Mebkhout, Équations différentielles 𝑝-adiques et coefficients 𝑝-adiques sur les courbes, Astérisque 279 (2002), 125–183 (French, with French summary). Cohomologies 𝑝-adiques et applications arithmétiques, II. MR 1922830
5. Philippe Robba and Gilles Christol, Équations différentielles 𝑝-adiques, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1994 (French, with French summary). Applications aux sommes exponentielles. [Applications to exponential sums]. MR 1411447
6. Bernard M. Dwork, Lectures on 𝑝-adic differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 253, Springer-Verlag, New York-Berlin, 1982. With an appendix by Alan Adolphson. MR 678093
7. Kiran S. Kedlaya, A 𝑝-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), no. 1, 93–184. MR 2119719, https://doi.org/10.4007/annals.2004.160.93
8. Kiran S. Kedlaya, Local monodromy of 𝑝-adic differential equations: an overview, Int. J. Number Theory 1 (2005), no. 1, 109–154. MR 2172335, https://doi.org/10.1142/S179304210500008X
9. Kiran S. Kedlaya, Fourier transforms and 𝑝-adic ‘Weil II’, Compos. Math. 142 (2006), no. 6, 1426–1450. MR 2278753, https://doi.org/10.1112/S0010437X06002338
10. Elisabeth Lutz, Sur l'équation dans les corps -adiques, J. Reine Angew. Math. (1937), no. 177, 238-243.
11. Z. Mebkhout, Analogue 𝑝-adique du théorème de Turrittin et le théorème de la monodromie 𝑝-adique, Invent. Math. 148 (2002), no. 2, 319–351 (French). MR 1906152, https://doi.org/10.1007/s002220100208
12. Zoghman Mebkhout, La théorie des équations différentielles 𝑝-adiques et le théorème de la monodromie 𝑝-adique, Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), 2003, pp. 623–665 (French, with English summary). MR 2023201, https://doi.org/10.4171/RMI/363
13. H. L. Turrittin, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27–66. MR 68689, https://doi.org/10.1007/BF02392519

Review Information:

Reviewer: Laurent Berger
Affiliation: UMPA, ENS de Lyon, UMR 5669 du CNRS Université de Lyon, France
Email: laurent.berger@ens-lyon.fr

Journal: Bull. Amer. Math. Soc. 49 (2012), 465-468
MSC (2010): Primary 12H25
DOI: https://doi.org/10.1090/S0273-0979-2012-01371-X
Published electronically: January 25, 2012
Review copyright: © Copyright 2012 American Mathematical Society