Movasati H. Modular And Automorphic Forms & Beyond

Singapore: World Scientific, 2021. — 323 p.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".Labyrinth of This BookGauss–Manin connection in disguise
Prerequisites
A tale of love and madness
Tupi words
Preliminaries in Algebraic GeometryThe base ring and field
Schemes over integers
Differential forms on schemes
Vector fields
Projective schemes over a ring
Algebraic de Rham cohomology and Hodge filtration
Block matrix notations
Moduli space
Flatness condition
Hilbert schemes
Group schemes and their action
Stable points
Group actions and constant vector fields
Gauss–Manin connection
Infinitesimal variation of Hodge structures
R-varieties
Full Hilbert schemes
Enhanced SchemesA marked projective variety
An algebraic group
Enhanced varieties
Weakly enhanced varieties
Constructing enhanced schemes
Enhanced families with an action of a reductive group
Gauss–Manin connection
Gauss–Manin connection and reductive group
Marked projective scheme
Moduli spaces of enhanced varieties
Other moduli spaces
Compactifications
Topology and PeriodsIntersections in homologies
Monodromy group and covering
Period map
Foliations on SchemesFoliations
Rational first integrals
Leaves
Smooth and reduced algebraic leaves
Singular scheme of a foliation
Classical or general leaves
N-smooth leaves
Foliations and vector fields
Algebraic groups and foliations
F-invariant schemes
Transcendental numbers vs variables
Modular FoliationsA connection matrix
The loci of constant periods
Algebraic groups and modular foliations
A character
Space of leaves
Moduli of modular foliations
Modular foliations and the Lefschetz decomposition
Some other foliations
The foliation F()
Modular vector fields
Constant vector fields
Constant Gauss–Manin connection
Constructing modular vector fields
Modular vector fields and IVHS
Hodge Cycles and LociCattani–Deligne–Kaplan theorem
Hodge cycles and enhanced families
Isolated Hodge cycles
The converse of Cattani–Deligne–Kaplan theorem
Weak absolute cycles
Consequences of Property
Generalized Period DomainPolarized Hodge structures
Generalized period domain
Period maps
τ and t maps
Action of the monodromy group
Period matrix
Griffiths period domain as a quotient of real Lie groups
Gauss–Manin connection matrix
Griffiths transversality distribution
Modular foliations and vector fields
Space of leaves
Transcendental degree of automorphic forms
Elliptic CurvesEnhanced elliptic curves
Modular vector fields
Quasi-modular forms
Halphen property
Modular foliations
Moduli of Hodge decompositions
Algebraic group acting on modular vector fields
Leaves
An alternative proof of Theorem
Halphen differential equation
Product of Two Elliptic CurvesEnhanced elliptic curves
The Gauss–Manin connection and modular vector fields
Modular foliations
Period domain
Character
Modular curves as three-dimensional affine varieties
Eisenstein series and modular curves
Algebraic invariant sets
A vector field in four dimensions
Elliptic curves with complex multiplication
Abelian VarietiesDe Rham cohomologies
Polarization
Algebraic group
Homology group
Generalized period domain
Moduli of enhanced polarized tori
Moduli of enhanced abelian varieties
Modular vector fields
Modular foliations
Differential Siegel modular forms
HypersurfacesFermat hypersurface as marked variety
The algebraic group
Enhanced hypersurfaces
Modular foliations of Hodge type
Moduli of enhanced hypersurfaces
Hypersurfaces with a finite group action
Automorphic forms for hypersurfaces
Period domain
K surfaces
Cubic fourfolds
Calabi–Yau VarietiesPreliminaries
R-varieties and modular vector fields
Universal family of enhanced Calabi–Yau varieties
Modular vector fields for Calabi–Yau varieties
Modular vector fields for K surfaces
Modular vector fields for Calabi–Yau threefolds
Modular vector fields for Calabi–Yau fourfolds
Periods of Calabi–Yau threefolds
Calabi–Yau threefolds with constant Yukawa couplings
Calabi–Yau equations
Appendix: A Geometric Introduction to Transcendence Questions on Values of Modular FormsA biased overview of transcendence theory
First notions
Arithmetic transcendence and Diophantine approximation
Schneider–Lang and Siegel–Shidlovsky
Schneider–Lang
Siegel–Shidlovsky
The theorem of Nesterenko
Nesterenko’s D-property and a zero lemma
Mahler’s theorem and the D-property for the Ramanujan equations
Sketch of Nesterenko’s proof
Periods
Definition
Elliptic periods and values of quasi-modular forms
Open problems