Another question that came up at the Newton Insitute: can two different curves X,Y over F_q have the same geometrically metabelian pro-l fundamental group?
I would think not, and here’s why. First of all, the actions of Frob_q on H_1(X,Z_ell) and on H_1(Y,Z_ell) agree. This already implies that Jac(X) and Jac(Y) are isogenous. Can this actually happen in large genus? Yes: a recent arXiv preprint by Ben Smith gives lots of explicit examples of pairs of hyperelliptic curves with isogenous Jacobians. From Smith’s paper I learned about the recent construction by J. F. Mestre of pairs of hyperelliptic curves in every genus with isogenous Jacobians.
In other words, the geometrically abelian fundamental group need not distinguish X from Y.
But the fact that the geometrically metabelian pro-l fundamental groups agree implies the following much stronger fact. Let X_n be the maximal abelian cover of X/F_qbar whose Galois group has exponent l^n, and define Y_n similarly. Then X_n and Y_n have isogenous Jacobians for all n. I would think this would be impossible if X and Y were not isomorphic; but I don’t have the slightest idea for a proof.
Baby version of this question: do there exist non-isomorphic curves X and Y of large genus (say, for the moment, over C) whose Jacobians are isogenous, and such that each Prym of X is isogenous to a Prym of Y?