The modified diagonal cycles of Hypergeometric curves
Payman Eskandari, Yusuke Nemoto
TL;DR
The paper analyzes the Gross-Kudla-Schoen modified diagonal and Ceresa cycles for the hypergeometric family X_{N,λ}. Using Abel-Jacobi theory and a pushforward construction along an order-2 automorphism, it proves that for any prime p≥3 and cuspidal base point e, the Abel-Jacobi image Φ_1(Δ_GKS,e) is nontrivial for all λ and for all l ≤ (p−1)/2, hence Δ_GKS,e is nontrivial in CH_1(X_{p,λ}^3); in the special case p=3, the cycles are torsion for every λ and choice of e. The proofs blend a Gross-Rohrlich-type pushforward with a de Rham cohomology vanishing criterion (LS2) and a careful analysis of the hypergeometric curve cusps and their divisors. The results provide explicit, nontrivial families of high-genus curves with nontrivial modified diagonal cycles, while also identifying a torsion case in genus 4, and they illuminate how automorphisms and cusp structures govern Chow-theoretic cycle orders in this setting.
Abstract
For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,λ}\}_{λ\in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric functions and the Fermat curve of degree $N$. In this paper, we study the Gross-Kudla-Schoen modified diagonal 1-cycles of these curves. We prove that if $p \ge 3$ is a prime, then for every $λ$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,λ}$ is nontrivial for every cuspidal choice of a base point. On the other hand, we show that the modified diagonal cycle and hence the Ceresa cycle of $X_{3,λ}$ is torsion in the Chow group for every $λ$ and every choice of a base point.