Hilbert Modular Surfaces - Gerard Van Der Geer

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Résumé :
Over the last 15 years important results have been achieved in the field of IHilbert Modular/I Varieties. Though the main emphasis of this book is on the geometry of Hilbert modular surfaces, both geometric and arithmetic aspects are treated. An abundance of examples - in fact a whole chapter - completes this competent presentation of the subject. This IErgebnisbericht/I will soon become an indispensible tool for graduate students and researchers in this field.

Sommaire:
Notations and Conventions Concerning Quadratic Number Fields.- I. Hilbert's Modular Group.- 1. The Action of the Hilbert Modular Group.- 2. The Distance to the Cusps.- 3. A Fundamental Domain.- 4. The Hurwitz-Maass Extension.- 5. Elliptic Fixed Points.- 6. Hilbert Modular Forms.- 7. The Adelic Version.- II. Resolution of the Cusp Singularities.- 1. The Local Ring at Infinity.- 2. Glueing.- 3. Dividing by the Units.- 4. Digression: the Elliptic r-gon.- 5. Continued Fractions.- 6. Resolution of Cyclic Quotient Singularities.- 7. The Baily-Borel Compactification.- III. Local Invariants.- 1. Local Chern Classes.- 2. Meyer's Theorem.- 3. Extension of Differential Forms.- IV. Global Invariants.- 1. The Volume of ?\?2.- 2. Chern Numbers of Y?.- 3. Inequalities for ? and c12.- 4. Dimensions of Spaces of Cusp Forms.- 5. Representations on Spaces of Cusp Forms.- 6. The Vanishing of the Fundamental Group.- 7. Rigidity.- V. Modular Curves on Modular Surfaces.- 1. The Curves FN and TN.- 2. Intersections with the Cusp Resolutions.- 3. The Components of FN.- 4. The Geometry of SO(2,2).- 5. The Volume of the Modular Curves.- 6. The Intersection Points of the Modular Curves.- 7. Classification of Elliptic Fixed Points.- 8. The Intersection Number of T1 and TN.- 9. The Fixed Points of the Galois Involution.- Appendix: Modular Forms on ?0(D).- VI. The Cohomology of Hilbert Modular Surfaces.- 1. Cohomology and Hilbert Modular Forms.- 2. The Dual of TN.- 3. The Generating Series of the Modular Curves.- 4. The Doi-Naganuma Lifting.- 5. The Intersection Number of TM and TN.- 6. The Action of the Hecke Algebra on the Cohomology.- 7. The Periods of Eigenforms.- 8. The Contribution of an Eigenform to the Picard Number.- VII. The Classification of Hilbert Modular Surfaces.- 1. The RoughClassification of Algebraic Surfaces.- 2. Configurations of Curves on Surfaces.- 3. Classification Theorems.- 4. Exceptional Curves on Hilbert Modular Surfaces.- 5. Estimates for the Numerical Invariants.- 6. Proof of the Classification.- 7. Canonical Divisors.- VIII. Examples of Hilbert Modular Surfaces.- 1. Preliminaries.- 2. The Examples.- IX. Humbert Surfaces.- 1. Modular Embeddings.- 2. Humbert Surfaces.- 3. Examples.- 4. Jacobians with Real Multiplication.- X. Moduli of Abelian Schemes with Real Multiplication.- 1. Abelian Schemes with Real Multiplication.- 2. Modular Stacks.- 3. Hilbert Modular Forms.- 4. The Galois Action on the Set of Components.- XI. The Tate Conjectures for Hilbert Modular Surfaces.- 1. Hodge and Tate Cycles.- 2. Decomposition of the Cohomology and L-Series.- 3. Splitting up the Galois Representation.- 4. The Tate Conjectures.- Table 1. Elliptic Fixed Points.- Table 2. Numerical Invariants.- List of Notations.