Gross-Kudla-Schoen Modified Diagonal 1-Cycles

• Gross–Kudla–Schoen modified diagonal 1-cycles are homologically trivial algebraic cycles on triple products of curves, constructed by correcting the small diagonal with partial diagonals and a base point.
• They connect closely to Ceresa cycles and Beauville decompositions, providing a framework for studying Beilinson–Bloch heights, normal functions, and arithmetic properties on modular and Shimura curves.
• Recent research extends this structure to higher modified diagonals, establishing equivalences with corresponding Beauville components and revealing new vanishing, torsion, and nonvanishing phenomena in arithmetic geometry.

Searching arXiv for recent and foundational papers on Gross–Kudla–Schoen modified diagonal cycles, Ceresa cycles, and related Beauville decomposition. Gross–Kudla–Schoen modified diagonal $1$-cycles are homologically trivial algebraic cycles attached to a smooth projective connected curve together with a degree-$1$ divisor or base point. In their classical form they live on the triple product $C^3$ and are obtained by correcting the small diagonal by partial diagonals and coordinate faces determined by the chosen degree-$1$ class. They are closely linked to the Ceresa cycle in the Jacobian, to Beauville’s decomposition of the curve class, to Beilinson–Bloch heights and normal functions, and to arithmetic constructions on modular and Shimura curves; more recent work shows that the same formalism extends to higher modified diagonals $\Gamma^n(C,e)$ and identifies them with higher Beauville components of the embedded curve class in its Jacobian (Lagarde et al., 8 Oct 2025).

1. Definition and basic geometric form

Let $C$ be a smooth projective connected curve, and let $e$ be a degree-$1$ divisor class on $C$. For each nonempty subset $I\subset \{1,\dots,n\}$, one defines

$1$0

where $1$1 is the small diagonal in $1$2. The $1$3-th modified diagonal cycle is then

$1$4

For $1$5, this is the classical Gross–Schoen cycle, also written $1$6 in much of the literature (Lagarde et al., 8 Oct 2025).

In the triple-product case one has the explicit alternating sum

$1$7

Equivalent formulas appear throughout the literature, with $1$8 denoting the partial diagonals and $1$9 the coordinate curves obtained by fixing two coordinates at $C^3$0. On the threefold $C^3$1 this is simultaneously a codimension-$C^3$2 cycle and a $C^3$3-cycle (0812.0371).

The construction is an inclusion–exclusion refinement of the small diagonal. In the terminology of recent work, it measures how far the small diagonal is from being “explained” by the base point $C^3$4. The same pattern persists for higher $C^3$5, where the triple-product cycle is the first nontrivial case beyond $C^3$6 and

$C^3$7

(Lagarde et al., 8 Oct 2025).

2. Jacobians, Ceresa cycles, and Beauville components

Let $C^3$8, and let $C^3$9 determine the Abel–Jacobi embedding

$1$0

The resulting curve class is

$1$1

Its antisymmetrization

$1$2

is the Ceresa cycle. With Beauville’s decomposition

$1$3

one writes $1$4 for the $1$5-th Beauville component of $1$6, and then

$1$7

Accordingly, the Ceresa cycle is governed by the odd Beauville pieces of the embedded curve class (Lagarde et al., 8 Oct 2025).

A central theorem recalled and extended in recent work is the equivalence

$1$8

This is the $1$9 case of a general statement: for every $\Gamma^n(C,e)$0,

$\Gamma^n(C,e)$1

Thus the sequence of modified diagonals $\Gamma^n(C,e)$2 is matched to the Beauville pieces $\Gamma^n(C,e)$3. The proof proceeds through an intermediate cycle $\Gamma^n(C,e)$4, which is a motivic projection of the modified diagonal, and through the interaction of Fourier transform and addition morphisms on the Jacobian (Lagarde et al., 8 Oct 2025).

The same work also proves propagation statements. If

$\Gamma^n(C,e)$5

then

$\Gamma^n(C,e)$6

Equivalently,

$\Gamma^n(C,e)$7

A weaker consequence is

$\Gamma^n(C,e)$8

In the classical $\Gamma^n(C,e)$9 situation, vanishing also forces the canonical condition

$C$0

so the relevant degree-$C$1 class is $C$2 in the rational sense (Lagarde et al., 8 Oct 2025).

At the integral level, the relation between Ceresa and modified diagonal cycles can be sharpened. If $C$3 is an integral representative of the canonical class and

$C$4

then

$C$5

Conversely, if $C$6 is $C$7-torsion, then $C$8 is killed by $C$9, with $e$0 defined explicitly in the paper (Lagarde et al., 8 Oct 2025).

3. Vanishing, torsion, and nonvanishing phenomena

Several basic vanishing patterns are classical. Gross and Schoen showed that the modified diagonal is rationally equivalent to $e$1 when the curve is rational or elliptic, and that if the curve is hyperelliptic and $e$2 is a Weierstrass point, then the modified diagonal is torsion of order $e$3 (Iyer et al., 2014). In later terminology, all subhyperelliptic curves—rational, elliptic, or hyperelliptic—have vanishing modified diagonal cycles for a suitable degree-$e$4 class (Qiu, 2023).

Vanishing is not confined to the hyperelliptic range. A sufficient criterion for the distinguished class

$e$5

is

$e$6

This criterion produces infinitely many non-hyperelliptic examples, including the Fricke–Macbeath curve, the Bring curve, and two $e$7-dimensional families parameterized by Hurwitz spaces (Qiu et al., 2022). On the Shimura-curve side, one finds vanishing results for triple products and their motivic direct summands, including examples of curves with trivial automorphism groups and vanishing modified diagonal cycles (Qiu et al., 2023).

For Shimura curves, vanishing is also studied through automorphic trilinear forms. A Shimura curve is called good if the automorphic representations appearing in $e$8 admit no nonzero diagonal invariant trilinear forms; good curves have vanishing modified diagonal cycles. This leads to finiteness conjectures and explicit classifications, including an analog of Ogg’s classification and new non-hyperelliptic examples such as $e$9 and $1$0 (Qiu, 2023).

Nonvanishing results are equally prominent. For the modular curves

$1$1

if there exists a weight-$1$2 normalized newform

$1$3

then the associated Ceresa cycle is nontrivial and the Gross–Kudla–Schoen modified diagonal cycle is of infinite order in $1$4 (Kerr et al., 2024). For the hypergeometric family $1$5, a different pattern occurs: if $1$6 is prime, then for every $1$7 and every cuspidal base point $1$8, the Griffiths Abel–Jacobi image of $1$9 is nontrivial; on the other hand, for $C$0 and every $C$1, the modified diagonal cycle, and hence the Ceresa cycle, is torsion for every base point (Eskandari et al., 8 Aug 2025).

A recurring misconception is that vanishing should be equivalent to hyperellipticity. The available examples show otherwise: non-hyperelliptic vanishing occurs in highly symmetric curves, in certain modular and Shimura settings, and in motivic direct summands, while nonvanishing can persist uniformly in large arithmetic families (Qiu et al., 2022).

4. Heights, normal functions, and degeneration

Over a global field, the modified diagonal has a canonical height pairing. For a curve $C$2 of genus $C$3, with admissible dualizing sheaf $C$4 and

$C$5

Zhang’s formula takes the form

$C$6

Here $C$7 is a local invariant, archimedean or non-archimedean, normalized so that good reduction gives $C$8. In the canonical case $C$9, this expresses $I\subset \{1,\dots,n\}$0 in terms of the Gross–Schoen height and local corrections (0812.0371).

A Hodge-theoretic reformulation packages this height as the degree of a line bundle. For a smooth family $I\subset \{1,\dots,n\}$1 with relative degree-$I\subset \{1,\dots,n\}$2 divisor $I\subset \{1,\dots,n\}$3, one has a canonical isomorphism, up to sign,

$I\subset \{1,\dots,n\}$4

whose hermitian norm is

$I\subset \{1,\dots,n\}$5

Here $I\subset \{1,\dots,n\}$6 is Bloch’s pairing line bundle, $I\subset \{1,\dots,n\}$7 and $I\subset \{1,\dots,n\}$8 are Deligne pairings, and $I\subset \{1,\dots,n\}$9 is the archimedean invariant in Zhang’s formula. The same work proves that the Chern form of $1$00 is non-negative, hence the bundle is nef, and computes its class on $1$01: $1$02 It also relates the Gross–Schoen pairing to the Ceresa cycle through a canonical isometry

$1$03

(Jong, 2012).

Under degeneration, the modified diagonal can specialize to higher Chow classes. For a non-hyperelliptic genus-$1$04 curve degenerating to an irreducible nodal curve whose normalization $1$05 is a hyperelliptic genus-$1$06 curve, the specialization of the Gross–Schoen cycle produces

$1$07

and this class is indecomposable with nonzero regulator (Iyer et al., 2014). More generally, a specialization formalism for higher Chow groups shows that the limit of the normal function attached to the modified diagonal is the Abel–Jacobi image of the specialized higher cycle; in the genus-$1$08 case this yields a nonzero limit normal function and implies that the modified diagonal is nontorsion in the Griffiths group for very general fibers (Angel et al., 2017).

5. Arithmetic realizations on modular and Shimura curves

On triple products of modular curves, the Gross–Kudla–Schoen cycle can be isolated by a projector. For $1$09 and a rational base point $1$10, the projector

$1$11

acts on the small diagonal to produce a null-homologous codimension-$1$12 cycle

$1$13

For normalized cuspidal eigenforms $1$14 of weight $1$15 and prime level $1$16, writing $1$17, one has the sign condition

$1$18

If $1$19, then the complex Abel–Jacobi image of the $1$20-isotypic component of $1$21 is torsion in the corresponding Hecke eigenspace of the Griffiths intermediate Jacobian, and the same holds for the étale Abel–Jacobi image. As an application, Chow–Heegner points pushed forward to elliptic curves with split multiplicative reduction at $1$22 are torsion (Lilienfeldt, 2021).

The Shimura-curve literature replaces the basepoint-corrected cycle by the diagonal cycle on a triple product of Shimura curves, but preserves the same Gross–Kudla–Schoen arithmetic. On a triple product $1$23 of Shimura curves, the diagonal class

$1$24

maps under the étale Abel–Jacobi map to cohomology classes whose localizations at admissible primes satisfy explicit reciprocity laws against Gross–Kudla period integrals. In the unramified setting this yields a bipartite Euler system for the symmetric cube motive of a modular form and evidence for the rank-one case of Bloch–Kato; in the ramified setting it gives a reciprocity law at a bad reduction prime and a rank-zero result for the symmetric cube motive under the paper’s hypotheses (Wang, 2020). This suggests a broader Gross–Kudla–Schoen framework in which basepoint-corrected cycles on curves and diagonal cycles on Shimura varieties play parallel arithmetic roles (Wang, 2020).

6. Higher modified diagonals and broader generalizations

The triple-product cycle is the first instance of a larger hierarchy. For a smooth projective variety $1$25 and a degree-$1$26 zero-cycle $1$27, O’Grady’s higher modified diagonals are

$1$28

obtained by alternating genuine diagonals with copies of $1$29. In this formalism, $1$30 is the Gross–Schoen modified small diagonal. Voisin proved that for any smooth projective connected variety and any degree-$1$31 zero-cycle $1$32, $1$33 for $1$34 sufficiently large, proved O’Grady’s conjecture for double covers and its extension to finite covers, and established the vanishing

$1$35

for $1$36, with $1$37 a $1$38 surface and $1$39 (Voisin, 2014).

The vanishing of modified diagonals also governs linear relations among small diagonals. If $1$40 denotes the $1$41-th modified diagonal in the sense of Gross and Schoen, then the first vanishing $1$42 determines the $1$43-linear relations among the small $1$44-diagonals in $1$45: they are generated by $1$46-hyperoctahedral relations. The same mechanism extends from diagonal classes to arbitrary symmetric classes in the Chow ring (Spink, 2018).

Several constructions transplant the Gross–Schoen idea outside $1$47. For $1$48, with $1$49 a curve and $1$50 a surface, one can define an arithmetic diagonal cycle $1$51 attached to an embedding $1$52; when $1$53 and $1$54 is the diagonal embedding, this recovers the classical Gross–Schoen cycle exactly (Zhang, 2021). On products of modular curves, generalized or $1$55-adic-family versions of Gross–Kudla–Schoen diagonal cycles underlie anticyclotomic Euler systems for Rankin–Selberg and higher-weight motives, with applications to Bloch–Kato and Iwasawa–Greenberg type statements (Alonso et al., 2021). A further refinement constructs an anticyclotomic Euler system for $1$56 from a corrected diagonal class on $1$57, where the quotient by diagonal diamond operators removes an extraneous $1$58-factor and yields the exact Euler factor in the tame norm relation (Castella et al., 2023).

Within the curve case itself, the most precise current synthesis is the Beauville dictionary of higher modified diagonals. The equivalence

$1$59

shows that modified diagonal cycles are not merely analogous to the Beauville decomposition of the curve class in the Jacobian; they encode it with a shift by $1$60. In that sense, Gross–Kudla–Schoen modified diagonal $1$61-cycles form both a concrete cycle-theoretic object on $1$62 and the first layer of a broader structure connecting Chow groups, motives, Hodge theory, automorphic forms, and arithmetic $1$63-values (Lagarde et al., 8 Oct 2025).