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

In this paper we show that the rank of the normal function function of the genus g Ceresa cycle over the moduli space of curves has the maximal rank possible, $3g-3$ , provided that g≥ 3 . In genus 3 we show that the Green–Griffiths invariant of this normal function is a Teichmüller 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. We also introduce new techniques and tools for studying the behaviour of normal functions along and transverse to boundary divisors. These include the introduction of residual normal functions and the use of global monodromy arguments to compute them.