Higher Ramanujan equations and periods of abelian varieties
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| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, RI
American Mathematical Society
2023
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| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470460198, 147046019X |
| Online Access: | Get full text |
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Table of Contents:
- 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
- 6.2. Integral solution of the higher Ramanujan equations -- Siegel case -- 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 ̂_{ }
- 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}