Hilbert Modular Surfaces [Paperback]

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.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.Notations and Conventions Concerning Quadratic Number Fields.- I. Hilberts 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. Meyers 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 lÛ