On my recent visit to Illinois, my colleage Nathan Dunfield (now blogging!) explained to me the following interesting open question, whose answer is supposed to be “yes”:

**Q1: **Let f be a pseudo-Anosov mapping class on a Riemann surface Sigma of genus at least 2, and M_f the mapping cylinder obtained by gluing the two ends of Sigma x interval together by means of f. Then M_f is a hyperbolic 3-manifold with first Betti number 1. Is there a finite cover M of M_f with b_1(M) > 1?

You might think of this as (a special case of) a sort of “relative virtual positive Betti number conjecture.” The usual vpBnC says that a 3-manifold has a finite cover with positive Betti number; this says that when your manifold *starts* life with Betti number 1, you can get “extra” first homology by passing to a cover.

Of course, when I see “3-manifold fibered over the circle” I whip out a time-worn analogy and think “algebraic curve over a finite field.” So here’s the number theorist’s version of the above question:

**Q2:** Let X/F_q be an algebraic curve of genus at least 2 over a finite field. Does X have a finite etale cover Y/F_{q^d} such that the action of Frobenius on H^1(Y,Z_ell) has an eigenvalue equal to q^{d/2}?

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