Effective Mordell for curves with enough automorphisms

We prove a completely explicit and effective upper bound for the Néron–Tate height of rational points of curves of genus at least $2$ over number fields, provided that they have enough automorphisms with respect to the Mordell–Weil rank of their jacobian. Our arguments build on Arakelov theory for arithmetic surfaces. Our bounds are practical, and we illustrate this by explicitly computing the rational points of a certain genus $2$ curve whose jacobian has Mordell–Weil rank $2$.