Recent Thoughts on Abelian Points

While at MSRI in early 2006, I got asked a very interesting question by Dimitar Jetchev, a Berkeley grad student. It motivated me to study abelian points on algebraic curves (and, to a lesser extent, higher-dimensional algebraic varieties), and MSRI’s special program on Rational and Integral Points was a convenient setting for this. It did not take me long to find families of curves without abelian points; I wrote these up in a paper which will appear in Math. Research Letters. Since then I have continued to try to put these examples into a larger context. Indeed, I have tried several different larger contexts on for size. When, just a couple of weeks ago, I was completing revisions on the paper, the context of “Kodaira dimension” seemed most worth promoting. Now, after ruminating about my upcoming talk for several days it seems that “Field arithmetic” should also be part of the picture. Needless to say, the final and optimal context (whatever that might mean!) has not yet been found. So, after having mentally rewritten the beginning of my talk many times, it strikes me that the revisionist approach may not be best: rather, I will for the most part present things in their actual chronological order.