Introduction to Arakelov Theory

1. Front Matter

   Pages i-x

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2. Metrics and Chern Forms

   • Serge Lang

   Pages 1-19

3. Green’s Functions on Riemann Surfaces

   • Serge Lang

   Pages 20-47

4. Intersections on an Arithmetic Surface

   • Serge Lang

   Pages 48-69

5. Hodge Index Theorem and the Adjunction Formula

   • Serge Lang

   Pages 70-101

6. The Faltings Riemann-Roch Theorem

   • Serge Lang

   Pages 102-130

7. Faltings Volumes on Cohomology

   • Serge Lang

   Pages 131-154

8. Back Matter

   Pages 155-187

   PDF

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.

• Divisor
• Grad
• Riemann-Roch theorem
• cohomology
• field

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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