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arXiv · 2401.10488

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

Ziyang Gao·Emmanuel Ullmo·Andrei Yafaev
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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 W\"{u}stholz'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.

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Ziyang Gao, Emmanuel Ullmo, Andrei Yafaev. 2024-01-19. Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods. https://arxiv.org/abs/2401.10488

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