Higher Ramanujan equations II: periods of abelian varieties and transcendence questions

TL;DR

This paper extends the classical Ramanujan equations to higher dimensions, constructing a holomorphic map related to abelian varieties, and explores its implications for periods and transcendence, including a functional form of Grothendieck's conjecture.

Contribution

It introduces higher Ramanujan vector fields on moduli spaces of abelian varieties and links their complex analytic properties to periods and transcendence questions.

Findings

Constructed a holomorphic map satisfying higher Ramanujan equations.

Identified the case g=1 with classical Eisenstein series and Ramanujan's relations.

Proved density of leaves of a foliation, implying a functional form of Grothendieck's periods conjecture.

Abstract

In the first part of this work, we have considered a moduli space $B_g$ classifying principally polarized abelian varieties of dimension $g$ endowed with a symplectic-Hodge basis, and we have constructed the higher Ramanujan vector fields $(v_{kl})_{1\le k\le l \le g}$ on it. In this second part, we study these objects from a complex analytic viewpoint. We construct a holomorphic map $\varphi_g : \mathbf{H}_g \to B_g(\mathbf{C})$, where $\mathbf{H}_g$ denotes the Siegel upper half-space of genus $g$, satisfying the system of differential equations $\frac{1}{2\pi i}\frac{\partial \varphi_g}{\partial \tau_{kl}}=v_{kl}\circ \varphi_g$, $1\le k\le l \le g$. When $g=1$, we prove that $\varphi_1$ may be identified with the triple of Eisenstein series $(E_2,E_4,E_6)$, so that the previous differential equations coincide with Ramanujan's classical relations concerning Eisenstein series. We discuss the relation between the values of $\varphi_g$ and the fields of periods of abelian varieties, and we explain how this relates to Grothendieck's periods conjecture. Finally, we prove that every leaf of the holomorphic foliation on $B_g(\mathbf{C})$ induced by the vector fields $v_{kl}$ is Zariski-dense in $B_{g,\mathbf{C}}$. This last result implies a "functional version" of Grothendieck's periods conjecture for abelian varieties.