Effective Chabauty for symmetric powers of curves

While we know by Faltings’ theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, Coleman showed that Chabauty’s method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus $g 1$. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of $Sym^d(X)$ that are not parametrized by a projective space or a coset of an abelian variety, where $X$ is a curve of genus $g d$, when the Mordell-Weil rank of the Jacobian of the curve is at most $g-d$.