# Effective Chabauty

One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if $X / {\mathbf Q}$ is an algebraic curve of genus at least 2, then the set $X({\mathbf Q})$ of rational points on $X$ is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set $X({\mathbf Q})$. The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on $X$, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of $X({\mathbf Q})$ provided that the Mordell-Weil rank of the Jacobian of $X$ is not too large.  (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)