Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods

Abstract

In this paper, we introduce the notion of a bi-$\overline{\mathbb{Q}}$-structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi-$\overline{\mathbb{Q}}$-structure decomposes into the direct sum of $1$-dimensional bi-$\overline{\mathbb{Q}}$-subspaces, and make this decomposition explicit for the moduli space of abelian varieties ${\mathbb{A}}_g$.
We propose an analytic subspace conjecture, which is the analogue of the Wüstholz’s analytic subgroup theorem in this context. We show that this conjecture, applied to ${\mathbb{A}}_g$, implies that all quadratic $\overline{\mathbb{Q}}$-relations among the holomorphic periods of CM abelian varieties arise from elementary ones.