Memoirs of the American Mathematical Society Series ; Volume 281 [More in this series]
Summary note
"We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series E2, E4, E6. Such "higher Ramanujan equations" are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford's theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing (E2,E4,E6), which are also shown to be defined over Z. This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko's celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck's Period Conjecture. Working in the complex analytic category, we prove "functional" transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations"-- Provided by publisher.
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Includes bibliographical references.
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Contents
Cover
Title page
Introduction
Motivation
Higher Ramanujan equations over
Siegel case
Interlude: Grothendieck's Period Conjecture
Analytic higher Ramanujan equations, periods, and transcendence
The Hilbert-Blumenthal case and an algebraic independence conjecture
Scholia
Terminology and conventions
Acknowledgments
Part 1. The arithmetic theory of the higher Ramanujan equations
Chapter 1. Symplectic vector bundles over schemes
1.1. Symplectic vector bundles
1.2. Lagrangian subbundles
1.3. Symplectic bases
Chapter 2. Symplectic-Hodge bases of principally polarized abelian schemes
2.1. De Rham cohomology of abelian schemes
2.2. Symplectic form associated to a principal polarization
2.3. Symplectic-Hodge bases of ¹_{\dR}( / )
Chapter 3. Abelian schemes with real multiplication
3.1. Symplectic vector bundles with real multiplication
3.2. Principally polarized abelian schemes with real multiplication
3.3. Symplectic-Hodge bases
Chapter 4. The moduli stacks ℬ_{ℊ} and ℬ_{ℱ}
4.1. The moduli stacks _{ℊ} and _{ℱ}
4.2. Definition of the moduli stacks ℬ_{ℊ} and ℬ_{ℱ}
4.3. Siegel parabolic subgroup and proof of Theorem 4.2.2 for ℬ_{ℊ}
4.4. Proof of Theorem 4.2.2 for ℬ_{ℱ}
Chapter 5. The tangent bundles of ℬ_{ℊ} and ℬ_{ℱ}
higher Ramanujan vector fields
5.1. Horizontal subbundles and linear connections
5.2. The Ramanujan subbundle ℛ_{ℊ}⊂ _{ℬ_{ℊ}/ }
5.3. The Ramanujan subbundle ℛ_{ℱ}⊂ _{ℬ_{ℱ}/ }
5.4. Recollections on the Kodaira-Spencer morphism
5.5. The Kodaira-Spencer isomorphism for _{ℊ} and _{ℱ}
5.6. The higher Ramanujan vector fields on ℬ_{ℊ}
5.7. The higher Ramanujan vector fields on ℬ_{ℱ}
Chapter 6. Integral solution of the higher Ramanujan equations
6.1. Higher Ramanujan equations over ℬ_{ℊ}.
6.2. Integral solution of the higher Ramanujan equations
6.3. Higher Ramanujan equations over ℬ_{ℱ}
6.4. Integral solution of the higher Ramanujan equations
Hilbert-Blumenthal case
Chapter 7. Representability of ℬ_{ℊ} and ℬ_{ℱ} by a scheme
7.1. Representability by an algebraic space
7.2. Representability of ℬ_{ℊ, [1/2]} by a quasi-projective scheme _{ }
7.3. _{ } is quasi-affine over [1/2]
Chapter 8. The case of elliptic curves: explicit equations
8.1. Explicit equation for the universal elliptic curve ₁ over ₁ and its universal symplectic-Hodge basis
8.2. Explicit formulas for the Ramanujan vector field
8.3. Explicit formulas for ̂₁
Part 2. The analytic higher Ramanujan equations and periods of abelian varieties
Chapter 9. Analytic families of complex tori, abelian varieties, and their uniformization
9.1. Relative complex tori
9.2. Riemann forms and principally polarized complex tori
9.3. The category ^{\an}_{ℊ} of principally polarized complex tori of relative dimension ℊ
9.4. De Rham cohomology of complex tori
9.5. Relative uniformization of complex abelian schemes
9.6. Principally polarized complex tori with real multiplication
Chapter 10. Analytic moduli spaces of complex abelian varieties with a symplectic-Hodge basis
10.1. Descent of principally polarized complex tori
10.2. Integral symplectic bases over principally polarized complex tori
10.3. Principal (symplectic) level structures
10.4. Symplectic-Hodge bases over complex tori
10.5. The Hilbert-Blumenthal case
Chapter 11. The analytic higher Ramanujan equations
11.1. Definition of _{ } and statement of our main theorem in the Siegel case
11.2. Preliminary results
11.3. Proof of Theorem 11.1.2
11.4. Compatibility of _{ } with ̂_{ }.
11.5. Analytic higher Ramanujan equations over ℬ_{ℱ}
11.6. Compatibility of _{ } and ̂_{ }
Chapter 12. Values of _{ } and _{ }
periods of abelian varieties
12.1. Fields of periods of abelian varieties and statement of our main theorems
12.2. Period matrices
12.3. Auxiliary lemmas
12.4. Proof of Theorem 12.1.3
12.5. Periods of abelian varieties with real multiplication
Chapter 13. An algebraic independence conjecture on the values of _{ }
13.1. Hirzebruch-Zagier divisors and statement of the conjecture
13.2. Periods in the presence of complex multiplication
13.3. Grothendieck's Period Conjecture for abelian surfaces with real multiplication
Chapter 14. Group-theoretic description of the higher Ramanujan vector fields
14.1. Realization of _{ }( ) as an open submanifold of \Sp_{2 }( )\\Sp_{2 }( )
14.2. Explicit analytic description of the higher Ramanujan vector fields ᵢⱼ and of _{ }
14.3. Group-theoretic description of ℬ_{ℱ}, _{ℱ}, and _{ℱ}
Chapter 15. Zariski-density of leaves of the higher Ramanujan foliation
15.1. Characterization of the leaves of the higher Ramanujan foliation
15.2. Auxiliary results
15.3. Statement and proof of our Zariski-density results
15.4. Derivatives of modular functions and _{ }
Appendix A. Gauss-Manin connection on some families of elliptic curves
A.1. The Weierstrass elliptic curve
A.2. The elliptic curve _{/ } over [1/6]
A.3. The universal elliptic curve ₁_{/ ₁} over [1/2]
Bibliography
Index of notation
Back Cover.
ISBN
9781470473242 ((electronic bk.))
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