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AuthorLang, Serge. author
TitleIntroduction to Arakelov Theory [electronic resource] / by Serge Lang
ImprintNew York, NY : Springer New York : Imprint: Springer, 1988
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Descript X, 187 p. online resource


Arakelov introduced a component at infinity in arithmetic considerations, thus giving rise to global theorems similar to those of the theory of surfaces, but in an arithmetic context over the ring of integers of a number field. The book gives an introduction to this theory, including the analogues of the Hodge Index Theorem, the Arakelov adjunction formula, and the Faltings Riemann-Roch theorem. The book is intended for second year graduate students and researchers in the field who want a systematic introduction to the subject. The residue theorem, which forms the basis for the adjunction formula, is proved by a direct method due to Kunz and Waldi. The Faltings Riemann-Roch theorem is proved without assumptions of semistability. An effort has been made to include all necessary details, and as complete references as possible, especially to needed facts of analysis for Green's functions and the Faltings metrics


I Metrics and Chern Forms -- ยง1. Nรฉron Functions and Divisors -- ยง2. Metrics on Line Sheaves -- ยง3. The Chern Form of a Metric -- ยง4. Chern Forms in the Case of Riemann Surfaces -- II Greenโ{128}{153}s Functions on Rlemann Surface -- ยง1. Greenโ{128}{153}s Functions -- ยง2. The Canonical Greenโ{128}{153}s Function -- ยง3. Some Formulas About the Greenโ{128}{153}s Function -- ยง4. Colemanโ{128}{153}s Proof for the Existence of Greenโ{128}{153}s Function -- ยง5. The Greenโ{128}{153}s Function on Elliptic Curves -- III Intersection on an Arithmetic Surface -- ยง1. The Chow Groups -- ยง2. Intersections -- ยง3. Fibral Intersections -- ยง4. Morphisms and Base Change -- ยง5. Nรฉron Symbols -- IV Hodge Index Theorem and the Adjunction Formula -- ยง1. Arakelov Divisors and Intersections -- ยง2. The Hodge Index Theorem -- ยง3. Metrized Line Sheaves and Intersections -- ยง4. The Canonical Sheaf and the Residue Theorem -- ยง5. Metrizations and Arakelovโ{128}{153}s Adjunction Formula -- V The Faltings Reimann-Roch Theorem -- ยง1. Riemann-Roch on an Arithmetic Curve -- ยง2. Volume Exact Sequences -- ยง3. Faltings Riemann-Roch -- ยง4. An Application of Riemann-Roch -- ยง5. Semistability -- ยง6. Positivity of the Canonical Sheaf -- VI Faltings Volumes on Cohomology -- ยง1. Determinants -- ยง2. Determinant of Cohomology -- ยง3. Existence of the Faltings Volumes -- ยง4. Estimates for the Faltings Volumes -- ยง5. A Lower Bound for Greenโ{128}{153}s Functions -- Appendix by Paul Vojta Diophantine Inequalities and Arakelov Theory -- ยง1. General Introductory Notions -- ยง2. Theorems over Function Fields -- ยง3. Conjectures over Number Fields -- ยง4. Another Height Inequality -- ยง5. Applications -- References -- Frequently Used Symbols

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