Author | Geer, Gerard van der. author |
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Title | Hilbert Modular Surfaces [electronic resource] / by Gerard van der Geer |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1988 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61553-5 |
Descript | X, 294 p. online resource |
Notations and Conventions Concerning Quadratic Number Fields -- I. Hilbertโs 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. Meyerโs 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