| Preface |
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xi | |
| Introduction |
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1 | (10) |
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1 Hermitian Vector Bundles over Arithmetic Curves |
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11 | (14) |
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1.1 Definitions and Basic Operations |
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11 | (2) |
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1.2 Direct Images. The Canonical Hermitian Line Bundle U0Kn over SpecCV |
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13 | (1) |
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1.3 Arakelov Degree and Slopes |
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14 | (2) |
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1.4 Morphisms and Extensions of Hermitian Vector Bundles |
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16 | (9) |
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2 θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves |
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25 | (24) |
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26 | (2) |
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2.2 The θ-Invariants h0θ and h1θ and the Poisson-Riemann-Roch Formula |
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28 | (1) |
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2.3 Positivity and Monotonicity |
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29 | (4) |
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2.4 The Functions τ and η |
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33 | (2) |
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2.5 The θ-invariants of direct sums of Hermitian line bundles over Spec Z |
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35 | (1) |
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2.6 The Theta Function θE and the First Minimum λ1(E) |
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36 | (5) |
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2.7 Application to Hermitian Line Bundles |
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41 | (3) |
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2.8 Subadditivity of h0θ and h1θ |
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44 | (5) |
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3 Geometry of Numbers and θ-Invariants |
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49 | (28) |
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3.1 Comparing h0θ and h0&Ar |
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50 | (3) |
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3.2 Banaszczyk's Estimates and θ-Invariants |
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53 | (6) |
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3.3 Subadditive Invariants of Euclidean Lattices |
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59 | (3) |
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3.4 The Asymptotic Invariant hoar(E,t) |
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62 | (8) |
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3.5 Some Consequences of Siegel's Mean Value Theorem |
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70 | (7) |
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4 Countably Generated Projective Modules over Dedekind Rings |
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77 | (30) |
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4.1 Countably Generated Projective A-Modules |
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77 | (3) |
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4.2 Linearly Compact Tate Spaces with Countable Basis |
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80 | (6) |
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4.3 The Duality Between CPA and CTCA |
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86 | (4) |
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4.4 Strict Morphisms, Exactness and Duality |
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90 | (10) |
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4.5 Localization and Descent Properties |
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100 | (1) |
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101 | (6) |
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5 Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves |
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107 | (30) |
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108 | (5) |
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5.2 Hilbertizable Ind- and Pro-vector Bundles |
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113 | (1) |
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5.3 Constructions as Inductive and Projective Limits |
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114 | (3) |
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5.4 Morphisms Between Ind- and Pro-Hermitian Vector Bundles over Ok |
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117 | (4) |
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5.5 The Duality Between Ind- and Pro-Hermitian Vector Bundles |
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121 | (7) |
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5.6 Examples - I. Formal Series and Holomorphic Functions on Disks |
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128 | (2) |
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5.7 Examples - II. Injectivity and Surjectivity of Morphisms |
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130 | (4) |
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5.8 Examples - III. Subgroups of Pre-Hilbert Spaces |
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134 | (3) |
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6 θ-Invariants of Infmite-Dimensional Hermitian Vector Bundles |
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137 | (18) |
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6.1 Limits of θ-Invariants |
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137 | (2) |
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6.2 Upper and Lower θ-Invariants |
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139 | (4) |
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143 | (4) |
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147 | (8) |
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7 Summable Projective Systems of Hermitian Vector Bundles |
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155 | (22) |
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155 | (1) |
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156 | (3) |
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7.3 Summable Projective Systems and Associated Measures |
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159 | (3) |
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7.4 Proof of Theorem 7.3.4 - I. The Equality h0θ(E) = limi→+θ h0θ(Ei) |
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162 | (2) |
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7.5 Proof of Theorem 7.3.4 - II. Convergence of Measures |
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164 | (3) |
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167 | (3) |
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7.7 Strongly Summable and θ-Finite Pro-Hermitian Vector Bundles |
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170 | (7) |
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8 Exact Sequences of Infinite-Dim. Hermitian Vector Bundles |
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177 | (42) |
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8.1 Short Exact Sequences of Infinite-Dimensional Hermitian Vector Bundles |
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178 | (2) |
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8.2 Short Exact Sequences and θ-Invariants of Pro-Hermitian Vector Bundles |
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180 | (8) |
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8.3 Strongly Summable Pro-Hermitian Vector Bundles |
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188 | (6) |
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8.4 A Vanishing Criterion |
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194 | (6) |
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8.5 The Category pro Vectok as an Exact Category |
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200 | (19) |
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9 Infinite-Dimensional Vector Bundles over Smooth Projective Curves |
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219 | (18) |
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9.1 Pro-vector Bundles over Smooth Curves |
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220 | (4) |
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9.2 The Invariants h0(C,E) and h0(C, E) |
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224 | (5) |
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9.3 Successive Extensions and Wild Pro-vector Bundles over Projective Curves |
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229 | (5) |
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9.4 A Vanishing Criterion |
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234 | (3) |
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10 Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization |
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237 | (68) |
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1.1 An Algebraicity Criterion for Smooth Formal Curves over Q |
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237 | (5) |
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1.2 Sections of Line Bundles and Algebraization |
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242 | (9) |
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1.3 Arithmetically Ample Hermitian Line Bundles and θ-Invariants |
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251 | (8) |
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1.4 Pointed Smooth Formal Curves |
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259 | (3) |
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1.5 Green's Functions, Capacitary Metrics and Schwarz Lemma |
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262 | (11) |
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1.6 Smooth Formal-Analytic Surfaces over Spec Ok |
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273 | (10) |
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1.7 Arithmetic Pseudo-concavity and Finiteness |
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283 | (6) |
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1.8 Arithmetic Pseudo-concavity and Algebraization |
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289 | (5) |
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1.9 The Isogeny Theorem for Elliptic Curves over Q |
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294 | (11) |
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A Large Deviations and Cramer's Theorem |
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305 | (24) |
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A.1 Notation and Preliminaries |
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306 | (2) |
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A.2 Lanford's Inequalities |
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308 | (4) |
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312 | (3) |
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A.4 An Extension of Cramer's Theorem |
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315 | (3) |
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A.5 Reformulation and Complements |
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318 | (11) |
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B Continuity of Linear Forms on Prodiscrete Modules |
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329 | (4) |
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B.1 Preliminary: Maximal Ideals, Discrete Valuation Rings, and Completions |
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329 | (2) |
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B.2 Continuity of Linear Forms on Prodiscrete Modules |
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331 | (2) |
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C Measures on Countable Sets and Their Projective Limits |
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333 | (8) |
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C.1 Finite Measures on Countable Sets |
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333 | (2) |
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C.2 Finite Measures on Projective Limits of Countable Sets |
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335 | (6) |
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341 | (6) |
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D.1 Definitions and Basic Properties |
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341 | (2) |
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D.2 The Derived Category of an Exact Category |
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343 | (4) |
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E Homorphic Sections of Line Bundles over Compact Complex Manifolds |
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347 | (8) |
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E.1 Spaces of Sections of Analytic Line Bundles and Multiplicity Bounds |
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347 | (2) |
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E.2 Proof of Proposition E. 1.2 |
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349 | (2) |
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E.3 Proof of Proposition E. 1.3 |
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351 | (4) |
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F John Ellipsoids and Finite-Dimensional Normed Spaces |
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355 | (4) |
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F.1 John Ellipsoids and John Euclidean Norms |
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355 | (1) |
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F.2 Properties of the John Norm |
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355 | (1) |
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F.3 Application to Lattices in Normed Real Vector Spaces |
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356 | (3) |
| Bibliography |
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359 | (10) |
| Index |
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369 | |
Lisainfo e-raamatute kohta