The modified diagonal cycles of Hypergeometric curves

For each N≥ 2, Asakura and Otsubo have recently introduced a smooth family of algebraic curves ,λ_λ∈ P¹ \0, 1, ∞\ 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 ≥ 3 is a prime, then for every $λ$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of Xp,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 X3,λ is torsion in the Chow group for every $λ$ and every choice of a base point.