Rational Points on Curves and their Jacobians

In this lecture I will discuss two theorems. One, the Skolem-Mahler-Lech theorem, deals with linear recurrent sequences, where the n-th value of the sequence depends linearly on the previous values. For such a sequence ( a(n) : n = 1, 2, ... ), it describes the shape of the set { n : a(n) = 0 }. The other theorem, by Chabauty, gives a partial result in the direction of the now fully known fact that a general algebraic curve has only finitely many rational points. Coleman derived a quantitative statement from Chabauty's method. These seemingly unrelated theorems share a common method of proof: they are based on p-adic analysis. In this talk I will sketch the proofs and point out the similarities between them. I will emphasise the analogies that can be drawn between the quite elementary Skolem-Mahler-Lech Theorem and Chabauty's construction.