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.)

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