Rational Points on Certain Families of Curves of Genus at Least 2

A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows one to give an upper bound for the number of rational points. (See [19, XI, §2].) Unfortunately, the resulting bound depends in quite a nasty manner on the set of primes at which the curve has bad reduction. Prior to Faltings' proof of Mordell's conjecture, there were two methods which in certain rather restrictive cases could be used to prove finiteness of the number of rational points. The first was due to Chabauty [3], and the second to Dem'janenko [5], generalized by Manin [12]. Recently, Coleman [4] has analysed Chabauty's method and used it to give relatively small upper bounds in those cases where it can be applied. For example, he proves that if C/Q is a smooth curve of genus g 3= 2, and if the Jacobian variety J of C satisfies rank/(Q)<g, then