Effective Chabauty for symmetric powers of curves

Abstract

Faltings' theorem states that curves of genus $g \geq 2$ have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than $g$, can be used to give a good effective bound on the number of rational points of curves of genus $g > 1$. We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of $d$-th symmetric power of $X$, where $X$ is a curve of genus $g > d$, when the Mordell-Weil rank of the Jacobian of the curve is at most $g-d$ and the curve further satisfies certain rigid analytic conditions.