Method of Chabauty-Coleman and the Second Case of Fermat's Last Theorem forRegular Primes

摘要

Let C be a curve of genus g = or > 2, defined over a number field K, and let J bethe Jacobian of C. Colemen, following Chabauty, has shown how to obtain good bounds on the cardinality of C(K) if the rank r of the Mordell-Weil group J(K) is less than g. The key to the method is to construct a logarithm on J(K sub v), for some valuation v of K, whose kernel contains J(K), and whose restriction to C(K sub v) is represented explicitly as the integral of a differential. This paper is an attempt to make the case, through a detailed examination of the case of Fermat curves, that this method can be fashioned into a quite precise tool for bounding rational points on curves. In particular, we will present an explicit description of the logarithm in question, given the existence of certain elements in a Selmer group attached to J.