| Although many classically studied curves have some "obvious" rational points (e.g. the three points with xyz = 0 on a Fermat curve), this is probably atypical: over Q one expects a "random" algebraic curve to have no points at all. For a pointless curve X over any field K, it is a fascinating challenge to say something about the set of field extensions L/K for which X has an L-rational point. E.g.: for a fixed field K and a fixed genus g, for which integers n does there exist a genus g curve X over K such that the least degree of a closed point on X is n? For K = Q and g = 1 the question was asked in 1958 by Lang and Tate and answered in 2004 by the speaker; for g >> 0 the problem remains open. We discuss here the case when K is a local field, using deformation theory to reduce the problem to the construction of certain finite graphs. We also address a related question: to what extent can the pointlessness of a curve over extension fields be gleaned solely by looking at the special fiber of a minimal regular model? |