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

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.