Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research.
This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.
- Introduction. - Part A Concepts of Arakelov Geometry. - Chapter I:
Arithmetic Intersection. - Chapter II: Minima and Slopes of Rigid Adelic
Spaces.
Chapter III : Introduction aux théorčmes de
Hilbert-Samuel arithmétiques. - Chapter IV: Euclidean Lattices, Theta
Invariants, and Thermodynamic Formalism. - Part B Distribution of Rational
Points and Dynamics. - Chapter V: Beyond Heights: Slopes and Distribution of
Rational Points. - Chapter VI: On the Determinant Method and Geometric
Invariant Theory. - Chapter VII: Arakelov Geometry, Heights,
Equidistribution, and the Bogomolov Conjecture. - Chapter VIII : Autour du
théorčme de Fekete-Szego. - Chapter IX: Some Problems of Arithmetic Origin
in Rational Dynamics. - Part C Shimura Varieties. - Chapter XI: The
Arithmetic RiemannRoch Theorem and the JacquetLanglands Correspondence.
- Chapter XII: The Height of CM Points on Orthogonal Shimura Varieties and
Colmezs Conjecture.
