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Heights of Ceresa and Gross-Schoen cycles

Ziyang Gao, Shou-Wu Zhang

TL;DR

The paper proves generic positivity and a Northcott property for Beilinson–Bloch heights of Ceresa and Gross–Schoen cycles in families of genus $g\ge3$ curves by constructing a integrable $\mathbb{Q}$-adelic line bundle on the moduli space ${\mathscr{M}}_g$ whose height recovers BB heights and whose curvature equals a semi-positive Betti form. A key volume identity connects the analytic curvature to the arithmetic volume, and a Betti-form non-vanishing result (via mixed Ax–Schanuel for VMHS) ensures the bigness needed for the height lower bounds on an ample locus ${\mathscr{M}}_g^{\mathrm{amp}}$. The work also describes the ample locus as the complement of a Betti-stratum and establishes a robust framework (period maps, MT-domains, and Ax–Schanuel) that could extend to broader families of homologically trivial cycles. These results have implications for diophantine geometry and conjectures relating cycle heights to $L$-functions, offering a path toward effective height bounds and uniform finiteness statements in families. The approach integrates Hodge-theoretic period maps, adelic geometry, and o-minimal techniques to connect geometric positivity with arithmetic heights.

Abstract

We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any $g\ge 3$, a Zariski open dense subset $\mathcal{M}_g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property.

Heights of Ceresa and Gross-Schoen cycles

TL;DR

The paper proves generic positivity and a Northcott property for Beilinson–Bloch heights of Ceresa and Gross–Schoen cycles in families of genus curves by constructing a integrable -adelic line bundle on the moduli space whose height recovers BB heights and whose curvature equals a semi-positive Betti form. A key volume identity connects the analytic curvature to the arithmetic volume, and a Betti-form non-vanishing result (via mixed Ax–Schanuel for VMHS) ensures the bigness needed for the height lower bounds on an ample locus . The work also describes the ample locus as the complement of a Betti-stratum and establishes a robust framework (period maps, MT-domains, and Ax–Schanuel) that could extend to broader families of homologically trivial cycles. These results have implications for diophantine geometry and conjectures relating cycle heights to -functions, offering a path toward effective height bounds and uniform finiteness statements in families. The approach integrates Hodge-theoretic period maps, adelic geometry, and o-minimal techniques to connect geometric positivity with arithmetic heights.

Abstract

We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any , a Zariski open dense subset of , the coarse moduli of curves of genus over , such that the heights of Ceresa cycles and Gross-Schoen cycles over have a lower bound and satisfy the Northcott property.
Paper Structure (41 sections, 30 theorems, 129 equations)

This paper contains 41 sections, 30 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Ceresa cycles and Gross--Schoen cycles
  3. Main results
  4. Sketch of proof
  5. Organization of the paper
  6. Archimedean local height pairing and metrized Poincaré bundle
  7. Height pairing at Archimedean places
  8. Intermediate Jacobian and the normal function associated with $Z$
  9. Betti foliation
  10. Metrized Poincaré bundle and metrized tautological bundle
  11. Betti form and Betti rank
  12. Gross--Schoen and Ceresa cycles
  13. More general intermediate Jacobians
  14. Passing to sub-VHS
  15. Beilinson--Bloch height of Gross--Schoen and Ceresa cycles
  16. ...and 26 more sections

Key Result

Theorem 1.1

For each $g\ge 3$, there exist a Zariski open dense subset $U$ of ${\mathscr {M}}_g$ defined over ${\mathbb {Q}}$ and positive numbers $\epsilon$ and $c$, such that for any $s\in \pi^{-1}(U)(\overline{\mathbb{Q}})$, we have where $h_{\mathrm{Fal}}$ is the Faltings height of $C_s$, and $h_{\mathrm{NT}}$ is the canonical Néron--Tate height on ${\mathrm{Jac}} ({\mathscr{C}}_s)(\overline{\mathbb{Q}})

Theorems & Definitions (65)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Theorem 1.4: Theorem \ref{['ThmZhangBB']} and \ref{['PropTwoCurvatureEqualAdeGS']}
  • Theorem 1.5: Theorem \ref{['PropHSforGS']}
  • Theorem 1.6: Theorem \ref{['ThmBettiRankFormulaMain']} and \ref{['ThmZariskiClosedDegLoci']}
  • Definition 2.1
  • Proposition 2.2: Hain
  • Definition 2.3
  • Definition 2.4
  • ...and 55 more