This is an experimental post: I’m going to put up something I’ve thought about only vaguely and partially, with the idea that it might be an interesting thing to discuss with people at the upcoming workshop on anabelian geometry at the Newton Institute. Please forgive (or, better, correct) any and all mistakes.
Here’s an old puzzle of Oort, which I’ve mentioned here previously:
Does there exist, for every g > 3, an abelian variety over Qbar not isogenous to the Jacobian of any smooth genus-g curve?
Here’s one way you could imagine trying to construct an example. Given a g-dimensional abelian variety A over a number field K, the action of the absolute Galois group G_K on the l-adic Tate module of A provides a Galois representation .
Now suppose A is in fact the Jacobian of a smooth genus g curve X/K. Then lifts to a representation , where G is the automorphism group of the pro-l geometric fundamental group of X. (In fact, this is the case even if A is only isogenous to a Jacobian, as long as the degree of the isogeny is prime to l.)
So you can ask: are there abelian varieties A such that doesn’t lift from the symplectic group to G? More: such that the restriction doesn’t lift to G, for any finite extension L/K? This seems quite difficult; it means you have to find an obstruction which isn’t torsion.
My further thoughts on this are even more disorganized and I’ll keep them to myself. Oh, except I should say: what makes this puzzle “anabelian” is that it has something to do with the notion that a section of the map
ought to come from a point of M_g(K).
Yes, there is an anabelian puzzle 2, which I’ll try to post in the next couple of days.