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Mathematics, Algebraic Geometry, Number Theory
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.
