On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle
Lucas Lagarde, Mohamed Moakher, Morena Porzio, James Rawson, Fernando Trejos Suárez
TL;DR
The paper extends the established link between the Ceresa cycle and Beauville components to a general degree-1 divisor $e$, proving that the vanishing of the Beauville components $[C]^e_{(s)}$ for any $s\ge1$ is equivalent to the vanishing of the $(s+2)$-nd modified diagonal $\Gamma^{s+2}(C,e)$. It develops a robust framework relating diagonal cycles to Beauville data via motivic projections and Fourier transform, establishing forward and reverse implications that yield a unified equivalence: $[C]^e_{(s)}=0$ iff $\Gamma^{s+2}(C,e)=0$ (hence, successive vanishing). The work also provides a new integral refinement of Zhang’s result, analyzes the $s=2$ case (noting generic non-vanishing of $[C]_{(2)}^{\xi}$ for $g\ge4$), and derives explicit integral torsion bounds connecting $\mathrm{Cer}(C,e)$ and $\Gamma^3(C,e)$. Overall, the results deepen understanding of the Beauville–Fourier structure in the Jacobian and its interaction with diagonal cycles, with applications to nontriviality questions and integral refinements in the Chow ring.
Abstract
Let $C$ be a curve of genus $g \geq 2$, and let $J$ be its Jacobian. The choice of a degree 1 divisor $e$ on $C$ gives an embedding of $C$ into $J$; we denote by $[C]_{}^{e}\in \mathrm{CH}\left( J;\mathbb{Q} \right) $ the class in the Chow group of $J$ defined by its image. It is known that the vanishing of the Ceresa cycle $\mathrm{Cer}(C,e):=[C]^{e} - [-1]_* [C]^e$ is equivalent to both the vanishing of the 1st Beauville component $[C]_{(1)}^e$ and the vanishing of the 3rd Gross--Kudla--Schoen modified diagonal cycle $Γ^3(C,e) \in \mathrm{CH}(C^3;\mathbb{Q})$. We extend this result to show that the vanishing of the $s$-th Beauville component $[C]^e_{(s)}$ for $s \geq 1$ is equivalent to the vanishing of the $(s+2)$-nd modified diagonal cycle $Γ^{s + 2}(C, e) \in \mathrm{CH}(C^{s+2};\mathbb{Q})$. Moreover, we establish "successive vanishing" results for these cycles. We apply our results to study the rational (non)-triviality of $[C]^{e}_{(s)}$ in the special case $s = 2$. Finally in the $s=1$ case, we show an integral refinement to the original statement, relating the order of torsion of $\mathrm{Cer}(C,e) \in \mathrm{CH}(J;\mathbb{Z})$ to that of $Γ^3(C,e) \in \mathrm{CH}(C^3;\mathbb{Z})$.