Over the last 15 years important results have been achieved in the field of Hilbert Modular 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 Ergebnisbericht will soon become an indispensible tool for graduate students and researchers in this field.

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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 Rough Classification 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.