Nicely produced monograph by a distinguished mathematician provides mature readers with organized review of a fairly esoteric topic of fairly recent origin. Six chapters, plus a relatively more advanced appendix (Diophantine Inequalities and Arakelov Theory) by Paul Vojta, brief bibliography, no exercises. Readers are presumed to understand the motivating objectives of the work. (NW) Annotation copyright Book News, Inc. Portland, Or.
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.
