The Rank of the Normal Functions of the Ceresa and Gross-Schoen Cycles
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2025-09-08
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In this paper we show that the rank of the normal function function of the genus Ceresa cycle over the moduli space of curves has the maximal rank possible, provided that. In genus 3 we show that the Green-Griffiths invariant of this normal function is a TeichmÜller modular form of weight 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.
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Hain, R (2025). The Rank of the Normal Functions of the Ceresa and Gross-Schoen Cycles. Forum of Mathematics Sigma, 13. 10.1017/fms.2025.10089 Retrieved from https://hdl.handle.net/10161/33435.
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Richard Hain
I am a topologist whose main interests include the study of the topology of complex algebraic varieties (i.e. spaces that are the set of common zeros of a finite number of complex polynomials). What fascinates me is the interaction between the topology, geometry and arithmetic of varieties defined over subfields of the complex numbers, particularly those defined over number fields. My main tools include differential forms, Hodge theory and Galois theory, in addition to the more traditional tools used by topologists. Topics of current interest to me include:
- the topology and related geometry of various moduli spaces, such as the moduli spaces of smooth curves and moduli spaces of principally polarized abelian varieties;
- the study of fundamental groups of algebraic varieties, particularly of moduli spaces whose fundamental groups are mapping class groups;
- the study of various enriched structures (Hodge structures, Galois actions, and periods) of fundamental groups of algebraic varieties;
- polylogarithms, mixed zeta values, and their elliptic generalizations, which occur as periods of fundamental groups of moduli spaces of curves.
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