Non-torsion algebraic cycles on the Jacobians of Fermat quotients

We study the Abel-Jacobi image of the Ceresa cycle $W_{k,e}-W_{k,e}^{-}$, where $W_{k,e}$ is the image of the $k$th symmetric product of a curve $X$ with a base point $e$ on its Jacobian variety. For certain Fermat quotient curves of genus $g$, we prove that for any choice of the base point and $k\le g-2$, the Abel-Jacobi image of the Ceresa cycle is non-torsion. In particular, these cycles are non-torsion modulo rational equivalence.