Uniform bounds for the number of rational points on symmetric squares of curves with low Mordell–Weil rank

A central problem in Diophantine geometry is to uniformly bound the number of $K$-rational points on a smooth curve $X/K$ terms $K$ and its genus $g$. recent paper by Stoll proved uniform bounds for hyperelliptic $X$ provided that rank Jacobian at most $g - 3$. Katz, Rabinoff Zureick-Brown generalized his result arbitrary curves satisfying same condition. In this paper, we prove conditional rational symmetric square outside algebraic special set, $g-4$. We also find rank-favorable (that is, depending Jacobian) case.