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

Ziyang Gao, Emmanuel Ullmo, Andrei Yafaev

TL;DR

The paper develops a framework of bi-$\overline{\mathbb{Q}}$-structures on the tangent spaces at CM points of Shimura varieties, proving a splitting into 1-dimensional subspaces and providing explicit period descriptions, notably on the Siegel moduli space $\mathbb{A}_g$. It introduces the Analytic Subspace Conjecture (HASC), an analogue of Wüstholz’s Analytic Subgroup Theorem in the Shimura setting, and shows that HASC implies that all quadratic relations among holomorphic CM periods arise from elementary ones. The work connects de Rham–Betti comparisons, Kodaira–Spencer maps, and root-space decompositions to period data, yielding concrete expressions for CM periods as products $\theta_j\theta_{j'}/\pi$ and establishing their transcendence properties. It also discusses consequences for Lang’s conjecture, CM Weyl points, and Grothendieck’s Period Conjecture, offering a cohesive program to understand algebraic relations among CM periods via the geometry of Shimura varieties. Overall, the results provide a precise, representation-theoretic pathway to control and predict transcendence and algebraic relations among CM periods through Shimura-theoretic structures and conjectural transcendence theorems.

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.

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

TL;DR

The paper develops a framework of bi--structures on the tangent spaces at CM points of Shimura varieties, proving a splitting into 1-dimensional subspaces and providing explicit period descriptions, notably on the Siegel moduli space . It introduces the Analytic Subspace Conjecture (HASC), an analogue of Wüstholz’s Analytic Subgroup Theorem in the Shimura setting, and shows that HASC implies that all quadratic relations among holomorphic CM periods arise from elementary ones. The work connects de Rham–Betti comparisons, Kodaira–Spencer maps, and root-space decompositions to period data, yielding concrete expressions for CM periods as products and establishing their transcendence properties. It also discusses consequences for Lang’s conjecture, CM Weyl points, and Grothendieck’s Period Conjecture, offering a cohesive program to understand algebraic relations among CM periods via the geometry of Shimura varieties. Overall, the results provide a precise, representation-theoretic pathway to control and predict transcendence and algebraic relations among CM periods through Shimura-theoretic structures and conjectural transcendence theorems.

Abstract

In this paper, we introduce the notion of a bi--structure on the tangent space at a CM point on a locally Hermitian symmetric domain. We prove that this bi--structure decomposes into the direct sum of -dimensional bi--subspaces, and make this decomposition explicit for the moduli space of abelian varieties . 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 , implies that all quadratic -relations among the holomorphic periods of CM abelian varieties arise from elementary ones.
Paper Structure (48 sections, 41 theorems, 103 equations)

This paper contains 48 sections, 41 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Prologue and motivation
  3. Bi-$\overline{\mathbb{Q}}$-structure associated with Shimura varieties
  4. The arithmetic $\overline{\mathbb{Q}}$-structure on $T_o X$
  5. The geometric $\overline{\mathbb{Q}}$-structure on $T_o X$
  6. The Hyperbolic Analytic Subspace Conjecture
  7. Bi-$\overline{\mathbb{Q}}$-spaces and preliminary examples
  8. Bi-$\overline{{\mathbb Q}}$-structures on a complex vector space
  9. Example: Algebraic tori
  10. Example: CM abelian varieties
  11. A first discussion on products of periods of CM abelian varieties
  12. Bi-$\overline{\mathbb{Q}}$-spaces arising from Shimura varieties: Definition
  13. Shimura data and Shimura varieties
  14. Arithmetic $\overline{\mathbb{Q}}$-structure on $T_o X$
  15. Geometric $\overline{\mathbb{Q}}$-structure on $T_o X$
  16. ...and 33 more sections

Key Result

Theorem 1.4

The bi-$\overline{{\mathbb Q}}$-structure on $T_o X$ defined above is split, i.e. we have a decomposition where each ${\mathbb C}(\alpha_j)$ is a $1$-dimensional complex vector space endowed with the restriction of the bi-$\overline{{\mathbb Q}}$-structures on $T_o X$ and $\alpha_j\in {\mathbb C}/\overline{{\mathbb Q}}^*$ compares the two $\overline{{\mathbb Q}}$-structures on ${\mathbb C}(\alpha

Theorems & Definitions (74)

  • Definition 1.2
  • Theorem 1.4: Theorem \ref{['ThmBiIQbarShimuraSplit']}
  • Theorem 1.5: Theorem \ref{['ThmPeriodsSiegel']}
  • Proposition 1.6: Proposition \ref{['PropLangQuestion']}
  • Theorem 1.7: WAST, WAST
  • Conjecture 1.8: HASC; Conjecture \ref{['ConjAnalyticSubspace']}
  • Proposition 1.9: Proposition \ref{['PropConsequenceAnalyticSubspaceQuadraticRelations']}
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • ...and 64 more