Rational Points on Symmetric Products of a Curve

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Of course, Faltings' theorem [3] says that the left-hand side of (1) is actually bounded, independently of X. However, Mumford's result has certain uniformity properties which, when combined with Faltings' theorem, help one to bound the number of points in C(K). (See [15]. This idea is due to Parshin.) In this paper we will give estimates similar to (1) for certain higher dimensional varieties. For each integer d 1, let Cd) be the d-fold symmetric product of the curve C. Thus C(d) is a smooth projective variety of dimension d. As above, let