Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911228520038400 |
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| author | Gao, Ziyang Ullmo, Emmanuel Yafaev, Andrei |
| author_facet | Gao, Ziyang Ullmo, Emmanuel Yafaev, Andrei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods Gao, Ziyang Ullmo, Emmanuel Yafaev, Andrei Number Theory Algebraic Geometry 11J81, 11G15, 14K22, 14G35 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. |
| title | Bi-$\overline{\mathbb{Q}}$-structures on Hermitian symmetric spaces and quadratic relations between CM periods |
| topic | Number Theory Algebraic Geometry 11J81, 11G15, 14K22, 14G35 |
| url | https://arxiv.org/abs/2401.10488 |