Heights of Ceresa and Gross-Schoen cycles

TL;DR

This paper investigates the heights of Ceresa and Gross-Schoen cycles in families of algebraic curves, establishing lower bounds and Northcott property for these heights over a dense subset of the moduli space.

Contribution

It constructs a Zariski open dense subset of the moduli space where the heights of these cycles are bounded below and satisfy Northcott's property, advancing understanding of their arithmetic behavior.

Findings

Heights of Ceresa and Gross-Schoen cycles have lower bounds.

These heights satisfy Northcott's property over the constructed subset.

The results apply to all genus g ≥ 3.

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.