Higher Ramanujan equations I: moduli stacks of abelian varieties and higher Ramanujan vector fields

Abstract

We describe a higher dimensional generalization of Ramanujan's differential equations satisfied by the Eisenstein series $E_2$, $E_4$, and $E_6$. This will be obtained geometrically as follows. For every integer $g\ge 1$, we construct a moduli stack $\mathcal{B}_g$ over $\mathbf{Z}$ classifying principally polarized abelian varieties of dimension $g$ equipped with a suitable additional structure: a symplectic-Hodge basis of its first algebraic de Rham cohomology. We prove that $\mathcal{B}_g$ is a smooth Deligne-Mumford stack over $\mathbf{Z}$ of relative dimension $2g^2 + g$ and that $\mathcal{B}_g\otimes \mathbf{Z}[1/2]$ is representable by a smooth quasi-projective scheme over $\mathbf{Z}[1/2]$. Our main result is a description of the tangent bundle $T_{\mathcal{B}_g/\mathbf{Z}}$ in terms of the cohomology of the universal abelian scheme over the moduli stack of principally polarized abelian varieties $\mathcal{A}_g$. We derive from this description a family of $g(g+1)/2$ commuting vector fields $(v_{ij})_{1\le i \le j \le g}$ on $\mathcal{B}_g$; these are the higher Ramanujan vector fields. In the case $g=1$, we show that $v_{11}$ coincides with the vector field associated to the classical Ramanujan equations. This geometric framework taking account of integrality issues is mainly motivated by questions in transcendental number theory. In the upcoming second part of this work, we shall relate the values of a particular analytic solution to the differential equations defined by $v_{ij}$ with Grothendieck's periods conjecture on abelian varieties.