Kooperativer Bibliotheksverbund

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Format: 1 online resource (xx, 496 pages) : , digital, PDF file(s).

Edition: Second edition.

ISBN: 1-009-27565-8 , 1-009-12768-3

Series Statement: Cambridge studies in advanced mathematics ; 198

Content: Now in its second edition, this volume provides a uniquely detailed study of $P$-adic differential equations. Assuming only a graduate-level background in number theory, the text builds the theory from first principles all the way to the frontiers of current research, highlighting analogies and links with the classical theory of ordinary differential equations. The author includes many original results which play a key role in the study of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge theory, perfectoid spaces, and algorithms for L-functions of arithmetic varieties. This updated edition contains five new chapters, which revisit the theory of convergence of solutions of $P$-adic differential equations from a more global viewpoint, introducing the Berkovich analytification of the projective line, defining convergence polygons as functions on the projective line, and deriving a global index theorem in terms of the Laplacian of the convergence polygon.

Note: Title from publisher's bibliographic system (viewed on 08 Aug 2022). , Cover -- Half-title -- Series information -- Title page -- Copyright information -- Contents -- Preface -- Acknowledgments -- 0 Introductory remarks -- 0.1 Why p-adic differential equations? -- 0.2 Zeta functions of varieties -- 0.3 Zeta functions and p-adic differential equations -- 0.4 A word of caution -- Notes -- Exercises -- Part I Tools of p-adic Analysis -- 1 Norms on algebraic structures -- 1.1 Norms on abelian groups -- 1.2 Valuations and nonarchimedean norms -- 1.3 Norms on modules -- 1.4 Examples of nonarchimedean norms -- 1.5 Spherical completeness -- Notes -- Exercises -- 2 Newton polygons -- 2.1 Newton polygons -- 2.2 Slope factorizations and a master factorization theorem -- 2.3 Applications to nonarchimedean field theory -- Notes -- Exercises -- 3 Ramification theory -- 3.1 Defect -- 3.2 Unramified extensions -- 3.3 Tamely ramified extensions -- 3.4 The case of local fields -- Notes -- Exercises -- 4 Matrix analysis -- 4.1 Singular values and eigenvalues (archimedean case) -- 4.2 Perturbations (archimedean case) -- 4.3 Singular values and eigenvalues (nonarchimedean case) -- 4.4 Perturbations (nonarchimedean case) -- 4.5 Horn's inequalities -- Notes -- Exercises -- Part II Differential Algebra -- 5 Formalism of differential algebra -- 5.1 Differential rings and differential modules -- 5.2 Differential modules and differential systems -- 5.3 Operations on differential modules -- 5.4 Cyclic vectors -- 5.5 Differential polynomials -- 5.6 Differential equations -- 5.7 Cyclic vectors: a mixed blessing -- 5.8 Taylor series -- Notes -- Exercises -- 6 Metric properties of differential modules -- 6.1 Spectral radii of bounded endomorphisms -- 6.2 Spectral radii of differential operators -- 6.3 A coordinate-free approach -- 6.4 Newton polygons for twisted polynomials -- 6.5 Twisted polynomials and spectral radii. , 6.6 The visible decomposition theorem -- 6.7 Matrices and the visible spectrum -- 6.8 A refined visible decomposition theorem -- 6.9 Changing the constant field -- Notes -- Exercises -- 7 Regular and irregular singularities -- 7.1 Irregularity -- 7.2 Exponents in the complex analytic setting -- 7.3 Formal solutions of regular differential equations -- 7.4 Index and irregularity -- 7.5 The Turrittin-Levelt-Hukuhara decomposition theorem -- 7.6 Asymptotic behavior -- Notes -- Exercises -- Part III p-adic Differential Equations on Discs and Annuli -- 8 Rings of functions on discs and annuli -- 8.1 Power series on closed discs and annuli -- 8.2 Gauss norms and Newton polygons -- 8.3 Factorization results -- 8.4 Open discs and annuli -- 8.5 Analytic elements -- 8.6 More approximation arguments -- Notes -- Exercises -- 9 Radius and generic radius of convergence -- 9.1 Differential modules have no torsion -- 9.2 Antidifferentiation -- 9.3 Radius of convergence on a disc -- 9.4 Generic radius of convergence -- 9.5 Some examples in rank 1 -- 9.6 Transfer theorems -- 9.7 Geometric interpretation -- 9.8 Subsidiary radii -- 9.9 Another example in rank 1 -- 9.10 Comparison with the coordinate-free definition -- 9.11 An explicit convergence estimate -- Notes -- Exercises -- 10 Frobenius pullback and pushforward -- 10.1 Why Frobenius? -- 10.2 p-th powers and roots -- 10.3 Moving along Frobenius -- 10.4 Frobenius antecedents -- 10.5 Frobenius descendants and subsidiary radii -- 10.6 Decomposition by spectral radius -- 10.7 Integrality of the generic radius -- 10.8 Off-center Frobenius antecedents and descendants -- Notes -- Exercises -- 11 Variation of generic and subsidiary radii -- 11.1 Harmonicity of the valuation function -- 11.2 Variation of Newton polygons -- 11.3 Variation of subsidiary radii: statements -- 11.4 Convexity for the generic radius. , 11.5 Measuring small radii -- 11.6 Larger radii -- 11.7 Monotonicity -- 11.8 Radius versus generic radius -- 11.9 Subsidiary radii as radii of optimal convergence -- Notes -- Exercises -- 12 Decomposition by subsidiary radii -- 12.1 Metrical detection of units -- 12.2 Decomposition over a closed disc -- 12.3 Decomposition over a closed annulus -- 12.4 Partial decomposition over a closed disc or annulus -- 12.5 Decomposition over an open disc or annulus -- 12.6 Modules solvable at a boundary -- 12.7 Solvable modules of rank 1 -- 12.8 Clean modules -- Notes -- Exercises -- 13 p-adic exponents -- 13.1 p-adic Liouville numbers -- 13.2 p-adic regular singularities -- 13.3 The Robba condition -- 13.4 Abstract p-adic exponents -- 13.5 Exponents for annuli -- 13.6 The p-adic Fuchs theorem for annuli -- 13.7 Transfer to a regular singularity -- 13.8 Liouville partitions -- Notes -- Exercises -- Part IV Difference Algebra and Frobenius Modules -- 14 Formalism of difference algebra -- 14.1 Difference algebra -- 14.2 Twisted polynomials -- 14.3 Difference-closed fields -- 14.4 Difference algebra over a complete field -- 14.5 Hodge and Newton polygons -- 14.6 The Dieudonné-Manin classification theorem -- Notes -- Exercises -- 15 Frobenius modules -- 15.1 A multitude of rings -- 15.2 Substitutions and Frobenius lifts -- 15.3 Generic versus special Frobenius -- 15.4 A reverse filtration -- 15.5 Substitution maps in the Robba ring -- Notes -- Exercises -- 16 Frobenius modules over the Robba ring -- 16.1 Frobenius modules on open discs -- 16.2 More on the Robba ring -- 16.3 Pure difference modules -- 16.4 The slope filtration theorem -- 16.5 Harder-Narasimhan filtrations -- 16.6 Extended Robba rings -- 16.7 Proof of the slope filtration theorem -- Notes -- Exercises -- Part V Frobenius Structures -- 17 Frobenius structures on differential modules. , 17.1 Frobenius structures -- 17.2 Frobenius structures and the generic radius of convergence -- 17.3 Independence from the Frobenius lift -- 17.4 Slope filtrations and differential structures -- 17.5 Extension of Frobenius structures -- 17.6 Frobenius intertwiners -- Notes -- Exercises -- 18 Effective convergence bounds -- 18.1 A first bound -- 18.2 Effective bounds for solvable modules -- 18.3 Better bounds using Frobenius structures -- 18.4 Logarithmic growth -- 18.5 Nonzero exponents -- Notes -- Exercises -- 19 Galois representations and differential modules -- 19.1 Representations and differential modules -- 19.2 Finite representations and overconvergent differential modules -- 19.3 The unit-root p-adic local monodromy theorem -- 19.4 Ramification and differential slopes -- Notes -- Exercises -- Part VI The p-adic local monodromy theorem -- 20 The p-adic local monodromy theorem -- 20.1 Statement of the theorem -- 20.2 An example -- 20.3 Descent of horizontal sections -- 20.4 Local duality -- 20.5 When the residue field is imperfect -- 20.6 Minimal slope quotients -- Notes -- Exercises -- 21 The p-adic local monodromy theorem: proof -- 21.1 Running hypotheses -- 21.2 Modules of differential slope 0 -- 21.3 Modules of rank 1 -- 21.4 Modules of rank prime to .. -- 21.5 The general case -- Notes -- Exercises -- 22 p-adic monodromy without Frobenius structures -- 22.1 The Robba ring revisited -- 22.2 Modules of cyclic type -- 22.3 A Tannakian construction -- 22.4 Interlude on finite linear groups -- 22.5 Back to the Tannakian construction -- 22.6 Proof of the theorem -- 22.7 Relation to Frobenius structures -- Notes -- Exercises -- Part VII Global theory -- 23 Banach rings and their spectra -- 23.1 Banach rings -- 23.2 The spectrum of a Banach ring -- 23.3 Topological properties -- 23.4 Complete residue fields -- Notes -- Exercises. , 24 The Berkovich projective line -- 24.1 Points -- 24.2 Classification of points -- 24.3 The domination relation -- 24.4 The tree structure -- 24.5 Skeleta -- 24.6 Harmonic and subharmonic functions -- Notes -- Exercises -- 25 Convergence polygons -- 25.1 The normalized radius of convergence -- 25.2 Normalized subsidiary radii and the convergence polygon -- 25.3 A constancy criterion for convergence polygons -- 25.4 Finiteness of the convergence polygon -- 25.5 Effect of singularities -- 25.6 Affinoid subspaces -- 25.7 Meromorphic differential equations -- 25.8 Open discs and annuli -- Notes -- 26 Index theorems -- 26.1 The index of a differential module -- 26.2 More on affinoid subspaces of [mathbb(P)][sub(K)] -- 26.3 The Laplacian of the convergence polygon -- 26.4 An index formula for algebraic differential equations -- 26.5 Local analysis on a disc -- 26.6 Local analysis on an annulus -- 26.7 Some nonarchimedean functional analysis -- 26.8 Plus and minus indices -- 26.9 Global analysis on a disc -- 26.10 A global index formula -- Notes -- Exercises -- 27 Local constancy at type-4 points -- 27.1 Geometry around a point of type 4 -- 27.2 Local constancy in the visible range -- 27.3 Local monodromy at a point of type 4 -- 27.4 End of the proof -- Notes -- APPENDICES AND BACK MATTER -- Appendix A Picard-Fuchs modules -- A.1 Picard-Fuchs modules -- A.2 Frobenius structures on Picard-Fuchs modules -- A.3 Relationship with zeta functions -- Notes -- Appendix B Rigid cohomology -- B.1 Isocrystals on the affine line -- B.2 Crystalline and rigid cohomology -- B.3 Machine computations -- Notes -- Appendix C p-adic Hodge theory -- C.1 A few rings -- C.2 (φ, Γ)-modules -- C.3 Galois cohomology -- C.4 Differential equations from (φ, Γ)-modules -- C.5 Beyond Galois representations -- Notes -- References -- Index of notation -- Subject index.

Additional Edition: ISBN 9781009123341

Language: English

URL: lizenzpflichtig

URL: https://doi.org/10.1017/9781009127684