Combinatorial techniques for point-counting on curves

Given a curve over the rational numbers, it is a natural problem to find all its rational points. Faltings proved in the 80s that when the curve has genus at least 2, the number of rational points is finite. His proof, though, is not constructive, and it does not give any upper bound on the number of rational points for any curve. More techniques are needed to get explicit upper bounds. In this talk, I will briefly discuss how p-adic integration and combinatorics are used to accomplish this, with an emphasis on the combinatorial aspects.