Chabauty-Coleman method for rational points on varieties
Faltings' theorem states that there are finitely many rational points on curves of genus g > 2. His proof is not effective, in the sense that the finite number of rational points obtained from his proof is too large to be computationally useful. On the other hand, the Chabauty-Coleman method does give an explicit upper bound (which is sometimes exact!) on the number of rational points on certain curves of genus g > 2, although it applies to a smaller class of curves. In this talk, we discuss the generalization of the Chabauty-Coleman method to search for rational points on certain higher-dimensional varieties; namely, we will discuss the case of symmetric powers of curves.