Open problems : Descending cohomology , geometrically compiled

I was asked by the organizers of the August 2011 birthday conference in honor of Joe Harris to give a short presentation in the session on “Open Problems” in the conference. Now, a great thing when you work together with Joe is that you find yourself in the midst of loads of inspiring problems, thanks to his deep engagement with, and intense curiosity about, all aspects of his subject. He’s a master of formulating problems on somany levels that it’s already something of an open problem simply to choose just one or two of them in his honor. Sometimes Joe introduces a problem very broadly and somewhat obliquely, as when he once asked “How many curves are there defined over Q?” Of course, this was an invitation to discuss the dimension—as a function of the genus g—of the Zariski closure of the set of Q-rational points in Mg the moduli space of curves of genus g. Hyperelliptic curves already gives you 2/3 of the dimension of that moduli space (mod O(1), and as g→ ∞) but can you get, say, a better fraction than that? This question, of course, immediately connects, via celebrated conjectures of Lang, to questions regarding the algebraic geometric structure of Mg.