The modified diagonal cycles of Hypergeometric curves

Abstract

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

19 pages