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Higher Ramanujan Equations and Periods of Abelian Varieties

About this Title

Tiago J. Fonseca

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 281, Number 1391
ISBNs: 978-1-4704-6019-8 (print); 978-1-4704-7324-2 (online)
DOI: https://doi.org/10.1090/memo/1391
Published electronically: January 3, 2022
Keywords: Moduli of abelian varieties; algebraic differential equations; integrality; values of derivatives of modular functions; Grothendieck’s period conjecture; functional transcendence

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Table of Contents

Chapters

  • Introduction
  • Terminology and conventions
  • Acknowledgments

1. The arithmetic theory of the higher Ramanujan equations

  • 1. Symplectic vector bundles over schemes
  • 2. Symplectic-Hodge bases of principally polarized abelian schemes
  • 3. Abelian schemes with real multiplication
  • 4. The moduli stacks $\mathcal {B}_g$ and $\mathcal {B}_F$
  • 5. The tangent bundles of $\mathcal {B}_g$ and $\mathcal {B}_F$; higher Ramanujan vector fields
  • 6. Integral solution of the higher Ramanujan equations
  • 7. Representability of $\mathcal {B}_g$ and $\mathcal {B}_F$ by a scheme
  • 8. The case of elliptic curves: explicit equations

2. The analytic higher Ramanujan equations and periods of abelian varieties

  • 9. Analytic families of complex tori, abelian varieties, and their uniformization
  • 10. Analytic moduli spaces of complex abelian varieties with a symplectic-Hodge basis
  • 11. The analytic higher Ramanujan equations
  • 12. Values of $\varphi _g$ and $\varphi _F$; periods of abelian varieties
  • 13. An algebraic independence conjecture on the values of $\varphi _F$
  • 14. Group-theoretic description of the higher Ramanujan vector fields
  • 15. Zariski-density of leaves of the higher Ramanujan foliation
  • A. Gauss-Manin connection on some families of elliptic curves
  • Index of notation

Abstract

We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series $E_2$, $E_4$, $E_6$. 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 $(E_2,E_4,E_6)$, which are also shown to be defined over $\mathbf {Z}$.

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.

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