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?

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