% %\input{preamble} \begin{thebibliography}{Dwo73a} \bibitem[AS02]{abbes-saito1} A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, \textit{Amer. J. Math.} \textbf{124} (2002), 879--920. \bibitem[AS03]{abbes-saito2} A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, II, \textit{Doc. Math.} Extra Vol. (2003), 5--72. \bibitem[AKR07]{akr} T.G. Abbott, K.S. Kedlaya, and D. Roe, Bounding Picard numbers of surfaces using $p$-adic cohomology, arXiv:math/0601508v2 (2007); to appear in \textit{Arithmetic, Geometry and Coding Theory (AGCT 2005)}, Societ\'e Math\'ematique de France. \bibitem[And01]{andre-diff} Y. Andr\'e, Diff\'erentelles non commutatives et th\'eorie de Galois diff\'erentielle ou aux diff\'erences, \textit{Ann. Sci. \'Ecole Norm. Sup.} (4) \textbf{34} (2001), 685--739. \bibitem[And02]{andre-hasse} Y. Andr\'e, Filtrations de type Hasse-Arf et monodromie $p$-adique, \textit{Invent. Math.} \textbf{148} (2002), 285--317. \bibitem[And07]{andre-dwork} Y. Andr\'e, Dwork's conjecture on the logarithmic growth of solutions of $p$-adic differential equations, \textit{Compos. Math.}, to appear. \bibitem[BdV07]{baldassarri-divizio} F. Baldassarri and L. di Vizio, Continuity of the radius of convergence of $p$-adic differential equations on Berkovich analytic spaces, preprint. \bibitem[BR07]{baker-rumely-book} M. Baker and R. Rumely, \textit{Potential Theory on the Berkovich Projective Line}, book in progress, available at \texttt{http://www.math.gatech.edu/\~{}mbaker/papers.html}. \bibitem[Brg02]{berger-invent} L. Berger, Repr\'esentations $p$-adiques et \'equations diff\'erentielles, \textit{Invent. Math.} \textbf{148} (2002), 219--284. \bibitem[Brg04]{berger-dwork} L. Berger, An introduction to the theory of $p$-adic representations, in \textit{Geometric Aspects of Dwork Theory. Vol I, II}, de Gruyter, Berlin, 2004, 255--292. \bibitem[Brg07a]{berger-b-pairs} L. Berger, Construction de $(\varphi,\Gamma)$-modules: repr\'esentations $p$-adiques et $B$-paires, \textit{Algebra and Num. Theory}, to appear. \bibitem[Brg07b]{berger-adm} L. Berger, \'Equations diff\'erentielles $p$-adiques et $(\phi, N)$-modules filtr\'es, \textit{Astr\'erisque}, to appear. \bibitem[BC07]{berger-colmez-familles} L. Berger and P. Colmez, Familles de repr\'esentations de de Rham et monodromie $p$-adique, \textit{Ast\'erisque}, to appear. \bibitem[Brk90]{berkovich1} V.G. Berkovich, \textit{Spectral Theory and Analytic Geometry over Non-Archimedean Fields}, Math. Surveys and Monographs 33, Amer. Math. Soc., Providence, 1990. \bibitem[Brk93]{berkovich2} V.G. Berkovich, \'Etale cohomology for non-Archimedean analytic spaces, \textit{Publ. Math. IH\'ES} \textbf{78} (1993), 5--161. \bibitem[Brt86]{berthelot-memoirs} P. Berthelot, G\'eom\'etrie rigide et cohomologie des vari\'et\'es alg\'ebriques de caract\'eristique $p$, Introductions aux cohomologies $p$-adiques (Luminy, 1984), \textit{M\'em. Soc. Math. France (N.S.)} \textbf{23} (1986), 7--32. \bibitem[Brt96]{berthelot-fragment} P. Berthelot, Cohomologie rigide et cohomologie rigide \`a support propre. Premi\`ere partie, Pr\'epublication IRMAR 96-03, available at \texttt{www.math.univ-rennes1.fr/\~{}berthelo}. \bibitem[Brt97a]{berthelot-finite} P. Berthelot, Finitude et puret\'e cohomologique en cohomologie rigide (with an appendix in English by A.J. de~Jong), \textit{Invent. Math.} \textbf{128} (1997), 329--377. \bibitem[Brt97b]{berthelot-poincare} P. Berthelot, Dualit\'e de Poincar\'e et formule de K\"unneth en cohomologie rigide, \textit{C.R. Acad. Sci. Paris} \textbf{325} (1997), 493--498. \bibitem[Brt02]{berthelot-intro-dmod} P. Berthelot, Introduction \`a la th\'eorie arithm\'etique des $\mathcal{D}$-modules, Cohomologies $p$-adiques et applications arithm\'etiques, II, \textit{Ast\'erisque} \textbf{279} (2002), 1--80. \bibitem[BO78]{berthelot-ogus} P. Berthelot and A. Ogus, \textit{Notes on Crystalline Cohomology}, Princeton Univ. Press, Princeton, 1978. \bibitem[Bha97]{bhatia} R. Bhatia, \textit{Matrix Analysis}, Graduate Texts in Math. 169, Springer-Verlag, New York, 1997. \bibitem[Bos05]{bosch} S. Bosch, Lectures on formal and rigid geometry, preprint (2005) available at \texttt{http://wwwmath1.uni-muenster.de/sfb/about/publ/bosch.html}. \bibitem[BGR84]{bgr} S. Bosch, U. G\"untzer, and R. Remmert, \textit{Non-Archimedean Analysis}, Grundlehren der Math. Wiss. 261, Springer-Verlag, Berlin, 1984. \bibitem[BC05]{buzzard-calegari} K. Buzzard and F. Calegari, Slopes of overconvergent 2-adic modular forms, \textit{Compos. Math.} \textbf{141} (2005), 591--604. \bibitem[CC98]{cherbonnier-colmez} F. Cherbonnier and P. Colmez, Repr\'esentations $p$-adiques surconvergentes, \textit{Invent. Math.} \textbf{133} (1998), 581--611. \bibitem[CT06]{chiarellotto-tsuzuki} B. Chiarellotto and N. Tsuzuki, Logarithmic growth and Frobenius filtrations for solutions of $p$-adic differential equations, preprint (2006). \bibitem[CP07]{chiarellotto-pulita} B. Chiarellotto and A. Pulita, Arithmetic and Differential Swan conductors of rank one representations with finite local monodromy, preprint (2007). \bibitem[Chr83]{christol} G. Christol, \textit{Modules Diff\'erentiels et Equations Diff\'erentielles $p$-adiques}, Queen's Papers in Pure and Applied Math. 66, Queen's Univ., Kingston, 1983. \bibitem[Chr01]{christol-on-tsuzuki} G. Christol, About a Tsuzuki theorem, in \textit{$p$-adic Functional Analysis (Ioannina, 2000)}, Lecture Notes in Pure and Appl. Math., 222, Dekker, New York, 2001, 63--74. \bibitem[CD91]{christol-dwork-eff} G. Christol and B. Dwork, Effective $p$-adic bounds at regular singular points, \textit{Duke Math. J.} \textbf{62} (1991), 689--720. \bibitem[CD92]{christol-dwork-diff} G. Christol and B. Dwork, Differential modules of bounded spectral norm, in \textit{$p$-adic Methods in Number Theory and Algebraic Geometry}, Contemp. Math. 133, Amer. Math. Soc., Providence, 1992. \bibitem[CD94]{christol-dwork} G. Christol and B. Dwork, Modules diff\'erentielles sur les couronnes, \textit{Ann. Inst. Fourier} \textbf{44} (1994), 663--701. \bibitem[CM97]{christol-mebkhout2} G. Christol and Z. Mebkhout, Sur le th\'eor\`eme de l'indice des \'equations diff\'erentielles, II, \textit{Annals of Math.} \textbf{146} (1997), 345--410. \bibitem[CM00]{christol-mebkhout3} G. Christol and Z. Mebkhout, Sur le th\'eor\`eme de l'indice des \'equations diff\'erentielles, III, \textit{Annals of Math.} \textbf{151} (2000), 385--457. \bibitem[CM01]{christol-mebkhout4} G. Christol and Z. Mebkhout, Sur le th\'eor\`eme de l'indice des \'equations diff\'erentielles, IV, \textit{Invent. Math.} \textbf{143} (2001), 629--672. \bibitem[Cla66]{clark} D. Clark, A note on the $p$-adic convergence of solutions of linear differential equations, \textit{Proc. Amer. Math. Soc.} \textbf{17} (1966), 262--269. \bibitem[Coh65]{cohn} R.M. Cohn, \textit{Difference Algebra}, John Wiley \& Sons, New York-London-Sydney, 1965. \bibitem[Col82]{coleman-dilogarithms} R.F. Coleman, Dilogarithms, regulators and $p$-adic $L$-functions, \textit{Invent. Math.} \textbf{69} (1982), 171--208. \bibitem[Col07]{colmez-trianguline} P. Colmez, Repr\'esentations triangulines de dimension 2, preprint (2007), available online at \texttt{http://www.institut.math.jussieu.fr/\~{}colmez/publications.html}. \bibitem[Con07]{conrad} B. Conrad, Several approaches to non-archimedean geometry, lecture notes from the 2007 Arizona Winter School, available online at \texttt{http://swc.math.arizona.edu}. \bibitem[Cre87]{crew-rep} R. Crew, $F$-isocrystals and $p$-adic representations., in \textit{Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985)}, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, 1987, 111--138. \bibitem[Cre98]{crew-finite} R. Crew, Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, \textit{Ann. Sci. \'Ec. Norm. Sup.} \textbf{31} (1998), 717--763. \bibitem[Cre00]{crew-index} R. Crew, Canonical extensions, irregularities, and the Swan conductor, \textit{Math. Ann.} \textbf{316} (2000), 19--37. \bibitem[Csi07]{csima} N.E. Csima, Newton-Hodge filtration for self-dual $F$-crystals, arXiv:0706/2530v1 (2007). \bibitem[dJ98a]{dejong-barsotti} A.J. de Jong, Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, \textit{Invent. Math.} \textbf{134} (1998), 301--333. \bibitem[dJ98b]{dejong-icm} A.J. de Jong, Barsotti-Tate groups and crystals, \textit{Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)}, \textit{Doc. Math.} Extra Vol. II (1998), 259--265. \bibitem[Del70]{deligne} P. Deligne, \textit{Equations Diff\'erentielles \`a Points Singuliers R\'eguliers}, Lecture Notes in Math. 163, Springer-Verlag, Berlin, 1970. \bibitem[DK73]{sga7} P. Deligne and N. Katz (eds.), \textit{Groupes de Monodromie en G\'eom\'etrie Alg\'ebrique II, S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois-Marie 1967--1969 (SGA 7 II)} Lecture Notes in Math. 340, Springer-Verlag, Berlin-New York, 1973. \bibitem[Dem72]{demazure} M. Demazure, \textit{Lectures on $p$-divisible Groups}, Lecture Notes in Math. 302, Springer-Verlag, New York, 1972. \bibitem[DV06]{denef-vercauteren} J. Denef and F. Vercauteren, An extension of Kedlaya's algorithm to hyperelliptic curves in characteristic 2, \textit{J. Cryptology} \textbf{19} (2006), 1--25. \bibitem[Dwo69]{dwork69} B. Dwork, $p$-adic cycles, \textit{Publ. Math. IH\'ES} \textbf{37} (1969), 27--115. \bibitem[Dwo73a]{dwork-pde2} B. Dwork, On $p$-adic differential equations, II. The $p$-adic asymptotic behavior of solutions of ordinary linear differential equations with rational function coefficients, \textit{Ann. of Math.} (2) \textbf{98} (1973), 366--376. \bibitem[Dwo73b]{dwork-pde3} B. Dwork, On $p$-adic differential equations, III. On $p$-adically bounded solutions of ordinary linear differential equations with rational function coefficients, \textit{Invent. Math.} \textbf{20} (1973), 35--45. \bibitem[Dwo74]{dwork-bessel} B. Dwork, Bessel functions as $p$-adic functions of the argument, \textit{Duke Math. J.} \textbf{41} (1974), 711--738. \bibitem[Dwo97]{dwork97} B.M. Dwork, On exponents of $p$-adic differential modules, \textit{J. reine angew. Math.} \textbf{484} (1997), 85--126. \bibitem[DGS94]{dgs} B. Dwork, G. Gerotto, and F. Sullivan, \textit{An Introduction to $G$-Functions}, Annals of Math. Studies 133, Princeton University Press, Princeton, 1994. \bibitem[DR77]{dwork-robba} B. Dwork and P. Robba, On ordinary linear $p$-adic differential equations, \textit{Trans. Amer. Math. Soc.} \textbf{231} (1977), 1--46. \bibitem[DR80]{dwork-robba-eff} B. Dwork and P. Robba, Effective $p$-adic bounds for solutions of homogeneous linear differential equations, \textit{Trans. Amer. Math. Soc.} \textbf{259} (1980), 559--577. \bibitem[Edi06]{edixhoven} B. Edixhoven, Point counting after Kedlaya, course notes (2006) available at \texttt{http://www.math.leidenuniv.nl/\~{}edix/oww/mathofcrypt/carls\_edixhoven/kedlaya.pdf}. \bibitem[Fon94]{fontaine-rep} J.-M. Fontaine, Repr\'esentations $p$-adiques semi-stables, P\'eriodes $p$-adiques (Bures-sur-Yvette, 1988), \textit{Ast\'erisque} \textbf{23} (1994), 113--184. \bibitem[FW79]{fontaine-wintenberger} J.-M. Fontaine and J.-P. Wintenberger, Le ``corps de normes'' de certaines extensions alg\'ebriques de corps locaux, \textit{C.R. Acad. Sci. Paris S\'er. A-B} \textbf{288} (1979), A367--A370. \bibitem[FvdP04]{fresnel-vanderput} J. Fresnel and M. van der Put, \textit{Rigid Analytic Geometry and its Applications}, Progress in Mathematics 218, Birkh\"auser, Boston, 2004. \bibitem[Ful98]{fulton-book} W. Fulton, \textit{Intersection Theory}, second edition, Springer-Verlag, Berlin, 1998. \bibitem[Ful00]{fulton-eigen} W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, \textit{Bull. Amer. Math. Soc. (N.S.)} \textbf{37} (2000), 209--249. \bibitem[Ger07]{gerkmann} R. Gerkmann, Relative rigid cohomology and deformation of hypersurfaces, \textit{Int. Math. Res. Papers} \textbf{2007} (2007), article ID rpm003 (67 pages). \bibitem[Har77]{hartshorne} R. Hartshorne, \textit{Algebraic Geometry}, Graduate Texts in Math. 52, Springer-Verlag, New York, 1977. \bibitem[Har07]{harvey} D. Harvey, Kedlaya's algorithm in larger characteristic, \textit{Int. Math. Res. Notices} \textbf{2007} (2007), article ID rnm095 (29 pages). \bibitem[Her98]{herr} L. Herr, Sur la cohomologie galoisienne des corps $p$-adiques, \textit{Bull. Soc. Math. France} \textbf{126} (1998), 563--600. \bibitem[Her01]{herr-tate} L. Herr, Une approche nouvelle de la dualit\'e locale de Tate, \textit{Math. Ann.} \textbf{320} (2001), 307--337. \bibitem[Hor54]{horn} A. Horn, On the eigenvalues of a matrix with prescribed singular values, \textit{Proc. Amer. Math. Soc.} \textbf{5} (1954), 4--7. \bibitem[Hor62]{horn-sums} A. Horn, Eigenvalues of sums of Hermitian matrices, \textit{Pacific J. Math.} \textbf{12} (1962), 225--241. \bibitem[Hub07]{hubrechts} H. Hubrechts, Point counting in families of hyperelliptic curves, \textit{Found. Comp. Math.}, to appear. \bibitem[Kap42]{kaplansky} I. Kaplansky, Maximal fields with valuations, \textit{Duke Math. J.} \textbf{9} (1942), 303--321. \bibitem[Kat79]{katz-slope} N.M. Katz, Slope filtration of $F$-crystals, Journ\'ees de G\'eom\'etrie Alg\'ebrique de Rennes (Rennes, 1978), Vol. I, \textit{Astérisque} \textbf{63} (1979), 113--163. \bibitem[Kat89]{kato} K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, \textit{Contemp. Math.}, vol. 83, Amer. Math. Soc., Providence, 1989, 101--131. \bibitem[KO68]{katz-oda} N.M. Katz and T. Oda, On the differentiation of de Rham cohomology classes with respect to parameters, \textit{J. Math. Kyoto Univ.} \textbf{8} (1968), 199--213. \bibitem[Ked01]{kedlaya-hyper} K.S. Kedlaya, Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology, \textit{J. Ramanujan Math. Soc.} \textbf{16} (2001), 323-338; errata, \textit{ibid.} \textbf{18} (2003), 417-418. \bibitem[Ked04a]{kedlaya-annals} K.S. Kedlaya, A $p$-adic local monodromy theorem, \textit{Annals of Math.} \textbf{160} (2004), 93--184. \bibitem[Ked04b]{kedlaya-full} K.S. Kedlaya, Full faithfulness for overconvergent $F$-isocrystals, in A. Adolphson et al (eds.), \textit{Geometric Aspects of Dwork Theory (Volume II)}, de Gruyter, Berlin, 2004, 819--835. \bibitem[Ked04c]{kedlaya-ants} K.S. Kedlaya, Computing zeta functions via $p$-adic cohomology, \textit{Algorithmic Number Theory (ANTS VI)}, Lecture Notes in Comp. Sci. 3076, Springer-Verlag, 2004, 1--17. \bibitem[Ked05a]{kedlaya-overview} K.S. Kedlaya, Local monodromy of $p$-adic differential equations: an overview, \textit{Int. J. Number Theory} \textbf{1} (2005), 109--154; errata at \texttt{http://math.mit.edu/\~{}kedlaya/papers}. \bibitem[Ked05b]{kedlaya-revisited} K.S. Kedlaya, Slope filtrations revisited, \textit{Documenta Math.} \textbf{10} (2005), 447-525; errata, \textit{ibid.} \textbf{12} (2007), 361--362. \bibitem[Ked05c]{kedlaya-frob-mod} K.S. Kedlaya, Frobenius modules and de Jong's theorem, \textit{Math. Res. Lett.} \textbf{12} (2005), 303--320. \bibitem[Ked06a]{kedlaya-finite} K.S. Kedlaya, Finiteness of rigid cohomology with coefficients, \textit{Duke Math. J.} \textbf{134} (2006), 15--97. \bibitem[Ked06b]{kedlaya-weilii} K.S. Kedlaya, Fourier transforms and $p$-adic ``Weil II'', \textit{Compos. Math.} \textbf{142} (2006), 1426--1450. \bibitem[Ked07a]{kedlaya-swan1} K.S. Kedlaya, Swan conductors for $p$-adic differential modules, I: A local construction, \textit{Algebra and Number Theory} \textbf{1} (2007), 269--300. \bibitem[Ked07b]{kedlaya-fake} K.S. Kedlaya, The $p$-adic local monodromy theorem for fake annuli, arXiv:math/0507496v4 (2006), to appear in \textit{Rend. Sem. Math. Padova}. \bibitem[Ked07c]{kedlaya-relative} K.S. Kedlaya, Slope filtrations for relative Frobenius, arXiv:math/0609272v2 (2007); to appear in \textit{Ast\'erisque}. \bibitem[Ked07d]{kedlaya-swan2} K.S. Kedlaya, Swan conductors for $p$-adic differential modules, II: Global variation, arXiv:0705.0031v1 (2007). \bibitem[Ked07e]{kedlaya-semi3} K.S. Kedlaya, Semistable reduction for overconvergent $F$-isocrystals, III: local semistable reduction at monomial valuations, arXiv:math/0609645v2 (2007). \bibitem[Ked07f]{kedlaya-journees} K.S. Kedlaya, Some new directions in $p$-adic Hodge theory, arXiv:0709.1970v1 (2007). \bibitem[Kis03]{kisin-fm} M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, \textit{Invent. Math.} \textbf{153} (2003), 373--454. \bibitem[Kis06]{kisin-crystalline} M. Kisin, Crystalline representations and $F$-crystals, in \textit{Algebraic Geometry and Number Theory}, Progress in Math.\ 253, Birkh\"auser, Boston, 2006, 459--496. \bibitem[Kot85]{kottwitz1} R.E. Kottwitz, Isocrystals with additional structure, \textit{Comp. Math.} \textbf{56} (1985), 201--220. \bibitem[Kot97]{kottwitz2} R.E. Kottwitz, Isocrystals with additional structure. II, \textit{Comp. Math.} \textbf{109} (1997), 255--339. \bibitem[Kot03]{kottwitz3} R.E. Kottwitz, On the Hodge-Newton decomposition for split groups, \textit{Int. Math. Res. Notices} \textbf{26} (2003), 1433--1447. \bibitem[Kru32]{krull} W. Krull, Allgemeine Bewertungstheorie, \textit{J. f\"ur Math.} \textbf{167} (1932), 160--196. \bibitem[Lan56]{lang} S. Lang, Algebraic groups over finite fields, \textit{Amer. J. Math.} \textbf{78} (1956), 555--563. \bibitem[Lau04]{lauder} A.G.B. Lauder, Deformation theory and the computation of zeta functions, \textit{Proc. London Math. Soc.} \textbf{88} (2004), 565--602. \bibitem[Laz62]{lazard} M. Lazard, Les z\'eros d'une fonction analytique d'une variable sur un corps valu\'e complet, \textit{Publ. Math. IH\'ES} \textbf{14} (1962), 47--75. \bibitem[leS07]{lestum} B. le Stum, \textit{Rigid Cohomology}, Cambridge Tracts in Math. 172, Cambridge Univ. Press, 2007. \bibitem[Liu07]{liu} R. Liu, Cohomology and duality for $(\phi, \Gamma)$-modules over the Robba ring, \textit{Int. Math. Res. Notices}, to appear. \bibitem[Loe96]{loeser} F. Loeser, Exposants $p$-adiques et th\'eor\`emes d'indice pour les \'equations diff\'erentielles $p$-adiques (d'apr\`es G. Christol et Z. Mebkhout), S\'eminaire Bourbaki, Vol. 1996/97, \textit{Ast\'erisque} \textbf{245} (1997), 57--81. \bibitem[Lut37]{lutz} E. Lutz, Sur l'\'equation $y^2 = x^3 + Ax + B$ sur les corps $\mathfrak{p}$-adiques, \textit{J. reine angew Math.} \textbf{177} (1937), 238--247. \bibitem[Mal74]{malgrange} B. Malgrange, Sur les points singuliers des \'equations diff\'erentielles, \textit{Enseign. Math.} \textbf{20} (1974), 147--176. \bibitem[Man63]{manin} Yu. I. Manin, The theory of commutative formal groups over fields of finite characteristic (Russian), \textit{Usp. Math.} 18 (1963), 3--90; English translation, \textit{Russian Math. Surveys} \textbf{18} (1963), 1--80. \bibitem[Mar04]{marmora} A. Marmora, Irr\'egularit\'e et conducteur de Swan $p$-adiques, \textit{Doc. Math.} \textbf{9} (2004), 413--433. \bibitem[Mat02]{matsuda} S. Matsuda, Katz correspondence for quasi-unipotent overconvergent isocrystals, \textit{Comp. Math.} \textbf{134} (2002), 1--34. \bibitem[Mat04]{matsuda-dwork} S. Matsuda, Conjecture on Abbes-Saito filtration and Christol-Mebkhout filtration, in \textit{Geometric Aspects of Dwork Theory. Vol. I, II}, de Gruyter, Berlin, 2004, 845--856. \bibitem[Meb02]{mebkhout-mono} Z. Mebkhout, Analogue $p$-adique du th\'eor\`eme de Turrittin et le th\'eor\`eme de la monodromie $p$-adique, \textit{Invent. Math.} \textbf{148} (2002), 319--351. \bibitem[Nag62]{nagata} M. Nagata, \textit{Local Rings}, Interscience Tracts in Pure and Applied Math. 13, John Wiley \& Sons, New York, 1962. \bibitem[Ore33]{ore} O. Ore, Theory of non-commutative polynomials, \textit{Annals of Math.} \textbf{34} (1933), 480--508. \bibitem[Pil90]{pila} J. Pila, Frobenius maps of abelian varieties and finding roots of unity in finite fields, \textit{Math. Comp.} \textbf{55} (1990), 745--763. \bibitem[Poo93]{poonen} B. Poonen, Maximally complete fields, \textit{Enseign. Math.} (2) \textbf{39} (1993), 87--106. \bibitem[Ram84]{ramis} J.-P. Ramis, Th\'eor\`emes d'indices Gevrey pour les \'equations diff\'erentielles ordinaires, \textit{Mem. Amer. Math. Soc.} \textbf{48} (1984). \bibitem[Rib99]{ribenboim} P. Ribenboim, \textit{The theory of classical valuations}, Springer-Verlag, New York, 1999. \bibitem[Rit50]{ritt} J.F. Ritt, \textit{Differential Algebra}, Colloq. Pub. XXXIII, Amer. Math. Soc., New York, 1950. \bibitem[Rob80]{robba-hensel} P. Robba, Lemmes de Hensel pour les op\'erateurs diff\'erentiels. Application \`a la r\'eduction formelle des \'equations diff\'erentielles, \textit{Enseign. Math.} (2) \textbf{26} (1980), 279--311. \bibitem[Rob00]{robert} A.M. Robert, \textit{A Course in $p$-adic Analysis}, Graduate Texts in Math. 198, Springer-Verlag, New York, 2000. \bibitem[Sch02]{schneider} P. Schneider, \textit{Nonarchimedean Functional Analysis}, Springer-Verlag, Berlin, 2002. \bibitem[Sch85]{schoof} R. Schoof, Elliptic curves over finite fields and the computation of square roots mod $p$, \textit{Math. Comp.} \textbf{44} (1985), 483--494. \bibitem[Ser79]{serre} J.-P. Serre, \textit{Local Fields}, Graduate Texts in Math. 67, Springer-Verlag, 1979. \bibitem[Sil91]{silverman} J.H. Silverman, \textit{The Arithmetic of Elliptic Curves}, second printing, Graduate Texts in Math. 106, Springer-Verlag, New York, 1991. \bibitem[SvdP97]{singer-vanderput-difference} M.F. Singer and M. van der Put, \textit{Galois Theory of Difference Equations}, Lecture Notes in Math. 1666, Springer-Verlag, Berlin, 1997. \bibitem[SvdP03]{singer-vanderput-differential} M.F. Singer and M. van der Put, \textit{Galois Theory of Linear Differential Equations}, Grundlehren der Math. Wiss. 328, Springer-Verlag, Berlin, 2003. \bibitem[Thu05]{thuillier} A. Thuillier, Th\'eorie du potentiel sur les courbes en g\'eom\'etrie analytique non archim\'edienne. Applications \`a la theorie d'Arakelov, thesis, Universit\'e de Rennes 1, 2005. \bibitem[Tsu96]{tsuzuki-etale} N. Tsuzuki (as T. Nobuo), The overconvergence of morphisms of \'etale $\phi$-$\nabla$-spaces on a local field, \textit{Compos. Math.} \textbf{103} (1996), 227--239. \bibitem[Tsu98a]{tsuzuki} N. Tsuzuki, Finite local monodromy of overconvergent unit-root $F$-isocrystals on a curve, \textit{Amer. J. Math.} \textbf{120} (1998), 1165--1190. \bibitem[Tsu98b]{tsuzuki-index} N. Tsuzuki, The local index and the Swan conductor, \textit{Comp. Math.} \textbf{111} (1998), 245--288. \bibitem[Tsu98c]{tsuzuki-slope} N. Tsuzuki, Slope filtration of quasi-unipotent overconvergent $F$-isocrystals, \textit{Ann. Inst. Fourier (Grenoble)} \textbf{48} (1998), 379--412. \bibitem[Tsu02]{tsuzuki-duke} N. Tsuzuki, Morphisms of $F$-isocrystals and the finite monodromy theorem for unit-root $F$-isocrystals, \textit{Duke Math. J.} \textbf{111} (2002), 385--419. \bibitem[vdP86]{vdp-mw} M. van der Put, The cohomology of Monsky and Washnitzer, Introductions aux cohomologies $p$-adiques (Luminy, 1984), \textit{M\'em. Soc. Math. France} \textbf{23} (1986), 33--59. \bibitem[vR78]{van-rooij} A.C.M. van Rooij, \textit{Non-Archimedean Functional Analysis}, Monographs and Textbooks in Pure and Applied Math. 51, Marcel Dekker, New York, 1978. \bibitem[Wey12]{weyl12} H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), \textit{Math. Ann.} \textbf{71} (1912), 441--479. \bibitem[Wey49]{weyl} H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, \textit{Proc. Nat. Acad. Sci. USA} \textbf{35} (1949), 408--411. \bibitem[Xia07]{xiao} L. Xiao, On Abbes-Saito's ramification filtrations and $p$-adic differential equations, I, in preparation. \bibitem[You92]{young} P.T. Young, Radii of convergence and index for $p$-adic differential operators, \textit{Trans. Amer. Math. Soc.} \textbf{333} (1992), 769--785. \end{thebibliography} %\end{document}