It is widely expected that, if a curve over a global field has no rational points, that there is an obstruction to existence of rational points coming from the Brauer group. One piece of evidence for this is an heuristic due to Poonen. We show that Poonen's argument also applies to p-primary subgroups of the Brauer group (for any prime p) but that there are examples of curves with no rational point but not having an obstruction coming from the p-primary subgroups of the Brauer group. Joint work with B. Creutz and B. Viray.