## Do all curves over finite fields have covers with a sqrt(q) eigenvalue?

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}?

## The entropy of Frobenius

Since Thurston, we know that among the diffeomorphisms of surfaces the most interesting ones are the pseudo-Anosov diffeomorphisms; these preserve two transverse folations on the surface, stretching one and contracting the other by the same factor.  The factor, usually denoted $\lambda$, is called the dilatation of the diffeomorphism and its logarithm is called the entropy. It turns out that $\lambda$, which is evidently a real number greater than 1, is in fact an algebraic integer, the largest eigenvalue of a matrix that in some sense keeps combinatorial track of the action of the diffeomorphism on the surface.  You might think of it as a kind of measure of the “complexity” of the diffeomorphism.  A recent preprint by my colleague Jean-Luc Thiffeault says much about how to compute these dilatations in practice, and especially how to hunt for diffeomorphisms whose dilatation is as small as possible.