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We describe a higher dimensional generalization of Ramanujan's differentialequations satisfied by the Eisenstein series $E_2$, $E_4$, and $E_6$. This willbe obtained geometrically as follows. For every integer $g\ge 1$, we constructa moduli stack $\mathcal{B}_g$ over $\mathbf{Z}$ classifying principallypolarized abelian varieties of dimension $g$ equipped with a suitableadditional structure: a symplectic-Hodge basis of its first algebraic de Rhamcohomology. We prove that $\mathcal{B}_g$ is a smooth Deligne-Mumford stackover $\mathbf{Z}$ of relative dimension $2g^2 + g$ and that$\mathcal{B}_g\otimes \mathbf{Z}[1/2]$ is representable by a smoothquasi-projective scheme over $\mathbf{Z}[1/2]$. Our main result is adescription of the tangent bundle $T_{\mathcal{B}_g/\mathbf{Z}}$ in terms ofthe cohomology of the universal abelian scheme over the moduli stack ofprincipally polarized abelian varieties $\mathcal{A}_g$. We derive from thisdescription 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 fieldassociated to the classical Ramanujan equations.  This geometric framework taking account of integrality issues is mainlymotivated by questions in transcendental number theory. In the upcoming secondpart of this work, we shall relate the values of a particular analytic solutionto the differential equations defined by $v_{ij}$ with Grothendieck's periodsconjecture on abelian varieties.

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