Ceresa Cycle: Algebraic & Arithmetic Insights

• Ceresa cycle is the antisymmetric difference of a curve’s Abel–Jacobi embedding and its (-1) pullback on the Jacobian, defining a key cycle in algebraic geometry.
• Recent research employs étale Abel–Jacobi maps, normal functions, and height pairings to analyze its non-torsion and torsion behaviors across diverse families.
• The study connects rational equivalence, Beauville decomposition, and tropical and symplectic analogues, offering comprehensive insights into both arithmetic and geometric dimensions.

The Ceresa cycle is the canonical algebraic $1$-cycle attached to a smooth projective curve $C$ of genus $g\ge 3$ after embedding $C$ into its Jacobian $J=\operatorname{Jac}(C)$ by the Abel–Jacobi map $\iota_e:x\mapsto [x-e]$: it is the antisymmetric difference

$[\iota_e(C)]-(-1)_*[\iota_e(C)].$

It is homologically trivial by construction, but Ceresa’s theorem says that for a generic curve of genus $g\ge 3$, the Ceresa cycle is not algebraically equivalent to zero. Recent work treats not only the Chow-theoretic cycle itself, but also its étale Abel–Jacobi image, its normal function over moduli, its height pairings in degenerating families, and tropical and symplectic analogues (Ellenberg et al., 2024, Corey et al., 2020).

1. Definition, notation, and basic variants

For a smooth projective geometrically integral curve $C$ and a degree-$1$ divisor $C$0, the Abel–Jacobi embedding

$C$1

defines a curve class $C$2. The Ceresa cycle is

$C$3

and papers in the recent literature also write this class as $C$4, $C$5, or $C$6. The underlying issue is not homology, since the cycle is homologically trivial, but rational equivalence, algebraic equivalence, and the behavior of Abel–Jacobi realizations (Ellenberg et al., 2024).

Several distinct objects are routinely separated. One is the actual Chow class in $C$7. A second is its image in the Griffiths group

$C$8

which measures nontriviality modulo algebraic equivalence. A third is an Abel–Jacobi or étale realization, often also called a Ceresa class in modern arithmetic papers. This distinction is essential because torsion in an intermediate Jacobian, torsion in the Griffiths group, and torsion in Chow are different conditions (Lilienfeldt et al., 2021).

The hyperelliptic case provides the basic vanishing phenomenon. If $C$9 is hyperelliptic and the base point is a Weierstrass point, then the hyperelliptic involution acts as $g\ge 3$0 on the Jacobian image, so $g\ge 3$1 is identified with $g\ge 3$2 and the Ceresa cycle vanishes. This remains the model for several tropical and topological vanishing statements (Corey et al., 2020).

A standard generalization replaces the embedded curve by the image of a symmetric product. If $g\ge 3$3 is the image of the $g\ge 3$4-th symmetric product of $g\ge 3$5 in $g\ge 3$6, then the higher Ceresa cycles are

$g\ge 3$7

For $g\ge 3$8 this is the classical Ceresa cycle, while for $g\ge 3$9 these higher cycles have their own Abel–Jacobi invariants and infinitesimal invariants (Nemoto, 2024).

2. Modified diagonals, Beauville decomposition, and vanishing criteria

A central comparison is with the Gross–Kudla–Schoen modified diagonal cycle on $C$0. For a base point $C$1, this is

$C$2

and Zhang’s theorem identifies its torsion behavior with that of the Ceresa cycle when one uses the canonical degree-$C$3 divisor $C$4 satisfying

$C$5

This equivalence is one of the basic structural facts in the subject (Kerr et al., 2024).

The Jacobian side is organized by Beauville’s decomposition. Writing

$C$6

for the Beauville components of the embedded curve class, the Ceresa cycle is the odd part

$C$7

Zhang’s theorem identifies the classical Ceresa condition with the first odd component: $C$8 Recent work extends this pattern to all higher Beauville components: for every $C$9,

$J=\operatorname{Jac}(C)$0

where $J=\operatorname{Jac}(C)$1 is the $J=\operatorname{Jac}(C)$2-nd modified diagonal cycle on $J=\operatorname{Jac}(C)$3. The same work proves successive vanishing statements, for example

$J=\operatorname{Jac}(C)$4

(Lagarde et al., 8 Oct 2025).

Automorphisms supply another vanishing mechanism. If

$J=\operatorname{Jac}(C)$5

then $J=\operatorname{Jac}(C)$6 vanishes in $J=\operatorname{Jac}(C)$7. If

$J=\operatorname{Jac}(C)$8

and the Hodge conjecture holds for the relevant abelian varieties, then the Ceresa cycle vanishes modulo algebraic equivalence. These criteria are especially effective for curves with large automorphism groups, and they identify Picard curves as the first interesting case where vanishing modulo algebraic equivalence occurs without automatic vanishing in Chow (Laga et al., 2024).

3. Étale realizations, normal functions, and height theory

Arithmetic work often isolates the étale Abel–Jacobi image rather than the Chow cycle itself. For a curve $J=\operatorname{Jac}(C)$9 over a number field and a degree-$\iota_e:x\mapsto [x-e]$0 divisor $\iota_e:x\mapsto [x-e]$1, the Ceresa class is defined as

$\iota_e:x\mapsto [x-e]$2

the image of the homologically trivial Ceresa cycle under the étale Abel–Jacobi map. The modified diagonal class

$\iota_e:x\mapsto [x-e]$3

maps to $\iota_e:x\mapsto [x-e]$4, and torsion or non-torsion of these two objects is equivalent for the torsion questions addressed in recent algorithmic work (Ellenberg et al., 2024).

There is also a topological and $\iota_e:x\mapsto [x-e]$5-adic formulation. Hain–Matsumoto’s $\iota_e:x\mapsto [x-e]$6-adic Ceresa class

$\iota_e:x\mapsto [x-e]$7

depends only on the unpointed curve and can be defined even if $\iota_e:x\mapsto [x-e]$8. For curves over $\iota_e:x\mapsto [x-e]$9, this class is controlled by the monodromy multitwist, and in that local-field setting the resulting Ceresa class is always torsion (Corey et al., 2020).

Over moduli, the Ceresa cycle defines a normal function

$[\iota_e(C)]-(-1)_*[\iota_e(C)].$0

and its rank measures how nontrivially the Ceresa cycle varies in families. For every genus $[\iota_e(C)]-(-1)_*[\iota_e(C)].$1, the rank of this normal function is maximal, namely $[\iota_e(C)]-(-1)_*[\iota_e(C)].$2. In genus $[\iota_e(C)]-(-1)_*[\iota_e(C)].$3, the Green–Griffiths invariant is a nonzero multiple of the Teichmüller modular form $[\iota_e(C)]-(-1)_*[\iota_e(C)].$4, and along the hyperelliptic locus the rank is exactly $[\iota_e(C)]-(-1)_*[\iota_e(C)].$5 (Hain, 2024).

Archimedean height theory gives a second layer of structure. For degenerating families, the height jump of the Ceresa cycle equals the slope of the polarized dual graph of the stable limit, and the jump vanishes exactly for tree-like curves and for the special two-rational-component $[\iota_e(C)]-(-1)_*[\iota_e(C)].$6-node curves. Over a function field over $[\iota_e(C)]-(-1)_*[\iota_e(C)].$7, the height of the Ceresa cycle satisfies

$[\iota_e(C)]-(-1)_*[\iota_e(C)].$8

so vanishing of the height is equivalent to vanishing of the curvature current of the Hain–Reed line bundle (Jong et al., 2021). In genus $[\iota_e(C)]-(-1)_*[\iota_e(C)].$9, the limit of archimedean height pairings of Ceresa cycles in a degenerating family can be identified with the archimedean height pairing of naturally associated algebraic cycles on the central fiber (Goswami et al., 2 Apr 2026).

4. Torsion, non-torsion, and explicit families

The expectation that non-hyperelliptic curves should always have Ceresa cycle of infinite order is no longer tenable. Beauville exhibited the non-hyperelliptic genus-$g\ge 3$0 plane quartic

$g\ge 3$1

with torsion Ceresa class in the intermediate Jacobian, using an automorphism of order $g\ge 3$2 and a fixed-point argument on the intermediate Jacobian. Beauville and Schoen then showed that the same curve has Ceresa cycle torsion modulo algebraic equivalence in $g\ge 3$3, giving the first example of a non-hyperelliptic curve with this property (Beauville, 2021, Beauville et al., 2021).

A particularly explicit one-parameter family is formed by the bielliptic Picard curves

$g\ge 3$4

For this family,

$g\ge 3$5

This gives infinitely many smooth plane quartics over $g\ge 3$6 with torsion Ceresa cycle, and over $g\ge 3$7 it gives infinite order for all $g\ge 3$8. The same paper proves that the Beilinson–Bloch height of $g\ge 3$9 is proportional to the Néron–Tate height of $C$0 (Laga et al., 2023).

For the larger Picard family

$C$1

the Ceresa cycle is torsion in the Griffiths group for every Picard curve, and it is torsion in Chow exactly when the explicit point

$C$2

is torsion on the elliptic curve

$C$3

As a byproduct, this yields infinitely many plane quartic curves over $C$4 with torsion Ceresa cycle (Laga et al., 2024).

Modular curves provide two different large-scale non-torsion results. For the family

$C$5

the Ceresa cycle is nontrivial in $C$6 with respect to any base point whenever there exists a weight-$C$7 normalized newform $C$8 with $C$9, and this covers a large concrete set of $1$0 (Kerr et al., 2024). For the complete modular curves $1$1, prime level admits a complete description: if $1$2 is not hyperelliptic, then its Ceresa cycle is non-torsion; for general $1$3, only finitely many $1$4 can have torsion Ceresa cycle (Lupoian et al., 23 Jan 2025).

Arithmetic experiments on cyclic Fermat quotients produce further torsion phenomena. The curves

$1$5

have torsion complex Abel–Jacobi Ceresa invariant, and for $1$6 the Ceresa class is torsion in the Griffiths group. The same study computes that the relevant $1$7-function for $1$8 has nonzero central value, matching the Beilinson–Bloch prediction of rank zero (Lilienfeldt et al., 2021).

5. Arithmetic and algorithmic methods

A recent algorithmic approach certifies that the Ceresa class of an explicit curve over a number field is non-torsion by comparing an upper bound from Frobenius action on

$1$9

with lower bounds coming from computable reduction-theoretic “shadows.” For a good prime $C$00, one defines

$C$01

If $C$02 were torsion of order $C$03, then $C$04 for every auxiliary good prime. On the other hand, a computable shadow order $C$05 divides $C$06 up to powers of the residue characteristic. This yields the criterion

$C$07

Under the hypothesis that the image of $C$08 on $C$09 has finite index in $C$10, the algorithm eventually terminates whenever the Ceresa class is non-torsion (Ellenberg et al., 2024).

A complementary method for covers of curves associates to a ramified cover $C$11 the relative canonical shadow

$C$12

a divisor class constructed from the ramification divisor and a canonical divisor on the base. If $C$13 has infinite order, then the Ceresa cycle of $C$14 has infinite order in the Chow group. By combining this with Manin–Mumford and relative Manin–Mumford, one proves that the Ceresa cycle of a very general ramified cover of a fixed curve $C$15 of genus $C$16 is non-torsion, and one obtains explicit $C$17- and $C$18-dimensional families of genus-$C$19 curves where the torsion locus is Zariski closed of positive codimension (Bhatnagar et al., 2 Mar 2026).

For modular curves, the non-vanishing mechanism passes through Chow–Heegner divisors. If the Ceresa cycle vanishes, every shadow $C$20 attached to a Jacobian endomorphism $C$21 must be torsion. On $C$22, taking $C$23 to be a Hecke operator or an Atkin–Lehner involution turns $C$24 into an explicit linear combination of CM divisors and cusps, and infinite order of these Chow–Heegner points forces the Ceresa cycle to be non-torsion (Lupoian et al., 23 Jan 2025).

6. Tropical, topological, and symplectic analogues

The tropical and topological theories isolate a degeneration shadow of the Ceresa construction. For multitwists in mapping class groups, one gets a topological Ceresa class in $C$25; for tropical curves one gets a corresponding notion of Ceresa triviality; and for smooth curves over $C$26, the $C$27-adic Ceresa class is always torsion. In the same framework, hyperelliptic tropical curves are Ceresa trivial, while the graph $C$28 gives a Ceresa nontrivial example, with order $C$29 in the equilateral case (Corey et al., 2020).

Two graph-theoretic refinements sharpen this picture. The Ceresa–Zharkov class of a graph or tropical curve is trivial if and only if the graph is of hyperelliptic type, equivalently if it has no $C$30 or $C$31 minor (Corey et al., 2022). The Ceresa period $C$32, defined through tropical homology, satisfies the same vanishing criterion: $C$33 again with forbidden minors $C$34 and $C$35 (Ritter, 2024).

A general tropical Abel–Jacobi theory now places the tropical Ceresa cycle in an intermediate Jacobian. For a tropical curve $C$36, the tropical Ceresa cycle

$C$37

is null-homologous in $C$38, and its Abel–Jacobi image is a class

$C$39

The resulting formula is completely combinatorial, depending on a graph model, a spanning tree, edge lengths, and tropical monodromy (Amini et al., 19 Apr 2025).

Mirror-symmetric work even supplies a symplectic analogue. For a generic tropical genus-$C$40 curve of type $C$41, the Lagrangian Ceresa cycle $C$42 in the associated symplectic six-torus has infinite order in the oriented algebraic Lagrangian cobordism group, providing a symplectic counterpart of classical Ceresa non-torsion (Corradini, 22 Jan 2025).

The contemporary theory therefore treats the Ceresa cycle simultaneously as a Chow class on a Jacobian, a modified diagonal phenomenon on powers of the curve, a normal function over moduli, an arithmetic class in Galois cohomology, a height-theoretic object in degeneration theory, and a tropical or symplectic obstruction. Explicit torsion examples, vanishing criteria from automorphisms, modular and covering constructions of non-torsion, and algorithmic certification results show that the subject now extends well beyond Ceresa’s original generic nontriviality theorem.