p-adic Differential Equations (Cambridge Studies in Advanced Mathematics, Vol. 125) (Cambridge Studies in Advanced Mathematics, Series Number 125)

Main subject categories: • Number theory • Differential equations • p-adic differential equations • Tools of p-adic analysis • Norms on algebraic structures • Newton polygons • Ramification theory • Matrix analysis • Differential algebra • p-adic differential equations on discs and annuli • Difference algebra and Frobenius modules • Frobenius structures • The p-adic local monodromy theorem • Global theoryOver the last 50 years the theory of p-adic differential equations has grown into an active area of research in its own right, and has important applications to number theory and to computer science. This book, the first comprehensive and unified introduction to the subject, improves and simplifies existing results as well as including original material. Based on a course given by the author at MIT, this modern treatment is accessible to graduate students and researchers. Exercises are included at the end of each chapter to help the reader review the material, and the author also provides detailed references to the literature to aid further study. Cover......Page 1 Half-title......Page 3 Series-title......Page 4 Title......Page 5 Copyright......Page 6 Contents......Page 7 Preface......Page 15 0.1 Why p-adic differential equations?......Page 21 0.2 Zeta functions of varieties......Page 23 0.3 Zeta functions and p-adic differential equations......Page 25 0.4 A word of caution......Page 27 Notes......Page 28 Exercises......Page 29 Part I Tools of p-adic Analysis......Page 31 1.1 Norms on abelian groups......Page 33 1.2 Valuations and nonarchimedean norms......Page 36 1.3 Norms on modules......Page 37 1.4 Examples of nonarchimedean norms......Page 45 1.5 Spherical completeness......Page 48 Notes......Page 51 Exercises......Page 53 2.1 Introduction to Newton polygons......Page 55 2.2 Slope factorizations and a master factorization theorem......Page 58 2.3 Applications to nonarchimedean field theory......Page 61 Notes......Page 62 Exercises......Page 63 3 Ramification theory......Page 65 3.1 Defect......Page 66 3.2 Unramified extensions......Page 67 3.3 Tamely ramified extensions......Page 69 3.4 The case of local fields......Page 72 Notes......Page 73 Exercises......Page 74 4 Matrix analysis......Page 75 4.1 Singular values and eigenvalues (archimedean case)......Page 76 4.2 Perturbations (archimedean case)......Page 80 4.3 Singular values and eigenvalues (nonarchimedean case)......Page 82 4.4 Perturbations (nonarchimedean case)......Page 88 4.5 Horn's inequalities......Page 91 Notes......Page 92 Exercises......Page 94 Part II Differential Algebra......Page 95 5.1 Differential rings and differential modules......Page 97 5.2 Differential modules and differential systems......Page 100 5.3 Operations on differential modules......Page 101 5.4 Cyclic vectors......Page 104 5.5 Differential polynomials......Page 105 5.7 Cyclic vectors: a mixed blessing......Page 107 5.8 Taylor series......Page 110 Exercises......Page 111 6.1 Spectral radii of bounded endomorphisms......Page 113 6.2 Spectral radii of differential operators......Page 115 6.3 A coordinate-free approach......Page 122 6.4 Newton polygons for twisted polynomials......Page 124 6.5 Twisted polynomials and spectral radii......Page 125 6.6 The visible decomposition theorem......Page 127 6.7 Matrices and the visible spectrum......Page 129 6.8 A refined visible decomposition theorem......Page 132 6.9 Changing the constant field......Page 134 Notes......Page 136 Exercises......Page 137 7 Regular singularities......Page 138 7.1 Irregularity......Page 139 7.2 Exponents in the complex analytic setting......Page 140 7.3 Formal solutions of regular differential equations......Page 143 7.4 Index and irregularity......Page 146 7.5 The Turrittin–Levelt–Hukuhara decomposition theorem......Page 147 Notes......Page 149 Exercises......Page 150 Part III p-adic Differential Equations on Discs and Annuli......Page 153 8 Rings of functions on discs and annuli......Page 155 8.1 Power series on closed discs and annuli......Page 156 8.2 Gauss norms and Newton polygons......Page 158 8.3 Factorization results......Page 160 8.4 Open discs and annuli......Page 163 8.5 Analytic elements......Page 164 8.6 More approximation arguments......Page 167 Notes......Page 169 Exercises......Page 170 9 Radius and generic radius of convergence......Page 171 9.1 Differential modules have no torsion......Page 172 9.2 Antidifferentiation......Page 173 9.3 Radius of convergence on a disc......Page 174 9.4 Generic radius of convergence......Page 175 9.5 Some examples in rank 1......Page 177 9.6 Transfer theorems......Page 178 9.7 Geometric interpretation......Page 180 9.9 Another example in rank 1......Page 182 9.10 Comparison with the coordinate-free definition......Page 184 Notes......Page 185 Exercises......Page 186 10.1 Why Frobenius descent?......Page 188 10.2 pth powers and roots......Page 189 10.3 Frobenius pullback and pushforward operations......Page 190 10.4 Frobenius antecedents......Page 192 10.5 Frobenius descendants and subsidiary radii......Page 194 10.6 Decomposition by spectral radius......Page 196 10.7 Integrality of the generic radius......Page 200 10.8 Off-center Frobenius antecedents and descendants......Page 201 Notes......Page 202 Exercises......Page 203 11 Variation of generic and subsidiary radii......Page 204 11.1 Harmonicity of the valuation function......Page 205 11.2 Variation of Newton polygons......Page 206 11.3 Variation of subsidiary radii: statements......Page 209 11.4 Convexity for the generic radius......Page 210 11.5 Measuring small radii......Page 211 11.6 Larger radii......Page 213 11.7 Monotonicity......Page 215 11.8 Radius versus generic radius......Page 217 11.9 Subsidiary radii as radii of optimal convergence......Page 218 Notes......Page 219 Exercises......Page 220 12 Decomposition by subsidiary radii......Page 221 12.1 Metrical detection of units......Page 222 12.2 Decomposition over a closed disc......Page 223 12.3 Decomposition over a closed annulus......Page 227 12.4 Decomposition over an open disc or annulus......Page 229 12.5 Partial decomposition over a closed disc or annulus......Page 230 12.6 Modules solvable at a boundary......Page 231 12.7 Solvable modules of rank 1......Page 232 12.8 Clean modules......Page 234 Exercises......Page 236 13.1 p-adic Liouville numbers......Page 238 13.2 p-adic regular singularities......Page 241 13.3 The Robba condition......Page 242 13.4 Abstract p-adic exponents......Page 243 13.5 Exponents for annuli......Page 245 13.6 The p-adic Fuchs theorem for annuli......Page 251 13.7 Transfer to a regular singularity......Page 254 Notes......Page 257 Exercises......Page 258 Part IV Difference Algebra and Frobenius Modules......Page 261 14.1 Difference algebra......Page 263 14.2 Twisted polynomials......Page 266 14.3 Difference-closed fields......Page 267 14.4 Difference algebra over a complete field......Page 268 14.5 Hodge and Newton polygons......Page 274 14.6 The Dieudonné–Manin classification theorem......Page 276 Notes......Page 278 Exercises......Page 280 15.1 A multitude of rings......Page 282 15.2 Frobenius lifts......Page 284 15.3 Generic versus special Frobenius lifts......Page 286 15.4 A reverse filtration......Page 289 Notes......Page 291 Exercises......Page 292 16.1 Frobenius modules on open discs......Page 293 16.2 More on the Robba ring......Page 295 16.3 Pure difference modules......Page 297 16.4 The slope filtration theorem......Page 299 16.5 Proof of the slope filtration theorem......Page 301 Notes......Page 304 Exercises......Page 306 Part V Frobenius Structures......Page 309 17.1 Frobenius structures......Page 311 17.2 Frobenius structures and the generic radius of convergence......Page 314 17.3 Independence from the Frobenius lift......Page 316 17.5 Extension of Frobenius structures......Page 318 Notes......Page 319 Exercises......Page 320 18.1 A first bound......Page 321 18.2 Effective bounds for solvable modules......Page 322 18.3 Better bounds using Frobenius structures......Page 326 18.4 Logarithmic growth......Page 328 Notes......Page 330 Exercises......Page 331 19 Galois representations and differential modules......Page 333 19.1 Representations and differential modules......Page 334 19.2 Finite representations and overconvergent differential modules......Page 336 19.3 The unit-root p-adic local monodromy theorem......Page 338 19.4 Ramification and differential slopes......Page 341 Notes......Page 343 Exercises......Page 345 20.1 Statement of the theorem......Page 346 20.2 An example......Page 348 20.3 Descent of sections......Page 349 20.4 Local duality......Page 352 20.5 When the residue field is imperfect......Page 353 Notes......Page 355 Exercises......Page 357 21.1 Running hypotheses......Page 358 21.2 Modules of differential slope 0......Page 359 21.3 Modules of rank 1......Page 361 21.4 Modules of rank prime to p......Page 362 Notes......Page 363 Exercises......Page 364 Part VI Areas of Application......Page 365 22.1 Origin of Picard–Fuchs modules......Page 367 22.2 Frobenius structures on Picard–Fuchs modules......Page 368 22.3 Relationship to zeta functions......Page 369 Notes......Page 370 23.1 Isocrystals on the affine line......Page 372 23.2 Crystalline and rigid cohomology......Page 373 23.3 Machine computations......Page 374 Notes......Page 375 24.1 A few rings......Page 377 24.2 (phi,gamma)-modules......Page 379 24.3 Galois cohomology......Page 381 24.4 Differential equations from (phi, gamma)-modules......Page 382 24.5 Beyond Galois representations......Page 383 Notes......Page 384 References......Page 385 Notation......Page 394 Index......Page 396 Over The Last 50 Years The Theory Of P-adic Differential Equations Has Grown Into An Active Area Of Research In Its Own Right, And Has Important Applications To Number Theory And To Computer Science. This Book, The First Comprehensive And Unified Introduction To The Subject, Improves And Simplifies Existing Results As Well As Including Original Material. Based On A Course Given By The Author At Mit, This Modern Treatment Is Accessible To Graduate Students And Researchers. Exercises Are Included At The End Of Each Chapter To Help The Reader Review The Material, And The Author Also Provides Detailed References To The Literature To Aid Further Study-- Although The Very Existence Of A Highly Developed Theory Of P-adic Ordinary Differential Equations Is Not Entirely Well Known Even Within Number Theory, The Subject Is Actually Almost 50 Years Old. Here Are Circumstances, Past And Present, In Which It Arises-- Machine Generated Contents Note: Preface; Introductory Remarks; Part I. Tools Of P-adic Analysis: 1. Norms On Algebraic Structures; 2. Newton Polygons; 3. Ramification Theory; 4. Matrix Analysis; Part Ii. Differential Algebra: 5. Formalism Of Differential Algebra; 6. Metric Properties Of Differential Modules; 7. Regular Singularities; Part Iii. P-adic Differential Equations On Discs And Annuli: 8. Rings Of Functions On Discs And Annuli; 9. Radius And Generic Radius Of Convergence; 10. Frobenius Pullback And Pushforward; 11. Variation Of Generic And Subsidiary Radii; 12. Decomposition By Subsidiary Radii; 13. P-adic Exponents; Part Iv. Difference Algebra And Frobenius Modules: 14. Formalism Of Difference Algebra; 15. Frobenius Modules; 16. Frobenius Modules Over The Robba Ring; Part V. Frobenius Structures: 17. Frobenius Structures On Differential Modules; 18. Effective Convergence Bounds; 19. Galois Representations And Differential Modules; 20. The P-adic Local Monodromy Theorem: Statement; 21. The P-adic Local Monodromy Theorem: Proof; Part Vi. Areas Of Application: 22. Picard-fuchs Modules; 23. Rigid Cohomology; 24. P-adic Hodge Theory; References; Index Of Notation; Index. Kiran S. Kedlaya. Includes Bibliographical References (p. 365-373) And Index.