The Rank of the Normal Functions of the Ceresa and Gross--Schoen Cycles

TL;DR

This paper proves that for genus at least 3, the normal function associated with the Ceresa cycle over the moduli space of curves has maximal rank, with specific results in genus 3 related to Teichmuller modular forms.

Contribution

It establishes the maximal rank of the Ceresa cycle's normal function for genus ≥ 3 and analyzes its behavior in genus 3 using Green--Griffiths invariants.

Findings

Maximal rank of the Ceresa normal function for genus ≥ 3

Green--Griffiths invariant as a Teichmuller modular form in genus 3

Rank exactly 1 along the hyperelliptic locus in genus 3

Abstract

The main result is that when the genus is at least 3, the rank of the normal function function of the Ceresa cycle over the moduli space of curves has maximal rank. This result was proved independently by Z. Gao and S.-W. Zhang (arXiv:2407.01304) by different methods. In genus 3 we show that the Green--Griffiths invariant of this normal function is a Teichmuller modular form of weight (4,0,-1) and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus.