The modified diagonal cycles of Hypergeometric curves

TL;DR

This paper investigates the properties of modified diagonal cycles on hypergeometric curves, demonstrating nontrivial Abel-Jacobi images for prime degree curves and torsion properties for degree three curves.

Contribution

It establishes the nontriviality of the Abel-Jacobi images of modified diagonal cycles for prime degree hypergeometric curves and shows torsion properties for degree three cases.

Findings

Nontrivial Abel-Jacobi images for $p \\geq 3$ primes.

Modified diagonal cycle is torsion for degree 3 curves.

Connections to hypergeometric functions and Fermat curves.

Abstract

For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,\lambda}\}_{\lambda \in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ 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 $\lambda$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,\lambda}$ 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 $X_{3,\lambda}$ is torsion in the Chow group for every $\lambda$ and every choice of a base point.