Author
Fonseca, T
Last updated
2020-10-09T14:41:33.92+01:00
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.
Symplectic ID
968732
Publication type
Journal Article
Please contact us with feedback and comments about this page. Created on 04 Feb 2019 - 17:30.