Vanishing results in Chow groups for the modified diagonal cycles

We prove a sufficient condition for the vanishing of the modified diagonal cycle in the Chow group (with $\mathbb{Q}$-coefficients) of the triple product of a curve over $\mathbb{C}$. We exhibit infinitely many non-hyperelliptic curves, including the Fricke--Macbeath curve, the Bring curve, and two 1-dimensional families parameterized by certain Hurwitz spaces, for which our condition is satisfied.