Chabauty for symmetric powers of curves

Abstract

Let $C$ be a smooth projective absolutely irreducible curve of genus $g\ge 2$ over a number field $K$, and denote its Jacobian by $J$. Let $d\ge 1$ be an integer and denote the $d$-th symmetric power of $C$ by $C^{\left(d\right)}$. In this paper we adapt the classic Chabauty–Coleman method to study the $K$-rational points of $C^{\left(d\right)}$. Suppose that $J\left(K\right)$ has Mordell–Weil rank at most $g-d$. We give an explicit and practical criterion for showing that a given subset $\mathcal{ℒ}\subseteq C^{\left(d\right)}\left(K\right)$ is in fact equal to $C^{\left(d\right)}\left(K\right)$.