The guiding principle in this monograph is to develop a new theory of modular forms which encompasses most of the available theory of modular forms in the literature, such as those for congruence groups, Siegel and Hilbert modular forms, many types of automorphic forms on Hermitian symmetric domains, Calabi-Yau modular forms, with its examples such as Yukawa couplings and topological string partition functions, and even go beyond all these cases. Its main ingredient is the so-called 'Gauss-Manin connection in disguise'.
Contents:
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Preface
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Acknowledgments
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Labyrinth of This Book
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Introduction
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Preliminaries in Algebraic Geometry
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Enhanced Schemes
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Topology and Periods
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Foliations on Schemes
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Modular Foliations
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Hodge Cycles and Loci
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Generalized Period Domain
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Elliptic Curves
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Product of Two Elliptic Curves
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Abelian Varieties
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Hypersurfaces
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Calabi-Yau Varieties
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Appendix: A Geometric Introduction to Transcendence Questions on Values of Modular Forms:
- Introduction
- A Biased Overview of Transcendence Theory
- The Theorem of Nesterenko
- Periods
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References
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Index
Readership: Graduate students and researchers.
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