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

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.