Lipnowski-Tsimerman: How large is A_g(F_p)?

Mike Lipnowski and Jacob Tsimerman have an awesome new preprint up, which dares to ask:  how many principally polarized abelian varieties are there over a finite field?

Well, you say, those are just the rational points of A_g, which has dimension g choose 2, so there should be about p^{(1/2)g^2} points, right?  But if you think a bit more about why you think that, you realize you’re implicitly imagining the cohomology groups in the middle making a negligible contribution to the Grothendieck-Lefchetz trace formula.  But why do you imagine that?  Those Betti numbers in the middle are huge, or at least have a right to be. (The Euler characteristic of A_g is known, and grows superexponentially in dim A_g, so you know at least one Betti number is big, at any rate.)

Well, so I always thought the size of A_g(F_q) really would be around p^{(1/2) g^2}, but that maybe humans couldn’t prove this yet.  But no!  Lipnowski and Tsimerman show there are massively many principally polarized abelian varieties; at least exp(g^2 log g).  This suggests (but doesn’t prove) that there is not a ton of cancellation in the Frobenius eigenvalues.  Which puts a little pressure, I think, on the heuristics about M_g in Achter-Erman-Kedlaya-Wood-Zureick-Brown.

What’s even more interesting is why there are so many principally polarized abelian varieties.  It’s because there are so many principal polarizations!  The number of isomorphism classes of abelian varieties, full stop, they show, is on order exp(g^2).  It’s only once you take the polarizations into account that you get the faster-than-expected-by-me growth.

What’s more, some abelian varieties have more principal polarizations than others.  The reducible ones have a lot.  And that means they dominate the count, especially the ones with a lot of multiplicity in the isogeny factors.  Now get this:  for 99% of all primes, it is the case that, for sufficiently large g:  99% of all points on A_g(F_p) correspond to abelian varieties which are 99% made up of copies of a single elliptic curve!

That is messed up.

 

Pila on a “modular Fermat equation”

I like this paper by Pila that just went up on the arXiv, which shows the way that you can get Diophantine consequences from the rapid progress being made in theorems of Andre-Oort type.  (I also want to blog about Tsimerman + Zhang + Yuan on “average Colmez” and Andre-Oort, maybe later!)

Pila shows that if N and M are sufficiently large primes, you can’t have elliptic curves E_1/Q and E_2/Q such that E_1 has an N-isogenous curve E_1 -> E’_1, E_2 has an M-isogenous curve E_2 -> E’_2, and j(E’_1) + j(E’_2) = 1.  (It seems to me the proof uses little about this particular algebraic relation and would work just as well for any f(j(E’_1),j(E’_2)) whose vanishing didn’t cut out a modular curve in X(1) x X(1).)  (This is “Fermat-like” in that it asserts finiteness of rational points on a natural countable family of high-genus curves; a more precise analogy is explained in the paper.)

How this works, loosely:  suppose you have such an (E_1, E_2).  A theorem of Kühne guarantees that E_1 and E_2 are not both CM (I didn’t know this!) It follows (WLOG assume N > M) that the N-isogenies of E_1 are defined over a field of degree at least N^a for some small a (Pila uses more precise bounds coming from a recent paper of Najman.)  So the Galois conjugates of (E’_1, E’_2) give you a whole bunch of algebraic points (E”_1, E”_2) with j(E”_1) + j(E”_2) = 1.

So what?  Rational curves have lots of low-height algebraic points.  But here’s the thing.  These isogenous choices of (E’_1, E’_2) aren’t just any algebraic points on X(1) x X(1); they represent pairs of elliptic curves drawn from a {\em fixed pair of isogeny classes}.  Let H be the hyperbolic plane as usual, and write (z,w) for a point on H x H corresponding to (E’_1, E’_2).  Then the other choices (E”_1, E”_2) correspond to points (gz,hw) with g,h in GL(Q).  GL(Q), not GL(R)!  That’s what we get from working in a fixed isogeny class.  And these points satisfy

j(gz) + j(hw) = 1.

To sum up:  you have a whole bunch of rational points (g,h) on GL_2 x GL_2.  These points are pretty low height (for this Pila gestures at a paper of his with Habegger.)  And they lie on the surface j(gz) + j(hw) = 1.  But this surface is a totally non-algebraic thing, because remember, j is a transcendental function on H!  So (Pila’s version of) the Ax-Lindemann theorem (correction from comments:  the Pila-Wilkie theorem) generates a contradiction; a transcendental curve can’t have too many low-height rational points.

How many rational distances can there be between N points in the plane?

Terry has a nice post up bout the Erdös-Ulam problem, which was unfamiliar to me.  Here’s the problem:

Let S be a subset of R^2 such that the distance between any two points in S is a rational number.  Can we conclude that S is not topologically dense?

S doesn’t have to be finite; one could have S be the set of rational points on a line, for instance.  But this appears to be almost the only screwy case.  One can ask, more ambitiously:

Is it the case that there exists a curve X of degree <= 2 containing all but 4 points of S?

Terry explains in his post how to show something like this conditional on the Bombieri-Lang conjecture.  The idea:  lay down 4 points in general position.  Then the condition that the 5th point has rational distances from x1,x2,x3, and x4 means that point lifts to a rational point on a certain (Z/2Z)^4-cover Y of P^2 depending on x1,x2,x3,x4.  (It’s the one obtained by adjoining the 4 distances, each of which is a square root of a rational function.)

With some work you can show Y has general type, so under Lang its rational points are supported on a union of curves.  Then you use a result of Solymosi and de Zeeuw to show that each curve can only have finitely many points of S if it’s not a line or a circle.  (Same argument, except that instead of covers of P^2 you have covers of the curve, whose genus goes up and then you use Faltings.)

It already seems hard to turn this approach into a proof.  There are few algebraic surfaces for which we can prove Lang’s conjecture.  But why let that stop us from asking further questions?

Question:  Let S be a set of N points on R^2 such that no M are contained in any line or circle.  What is the maximal number of rational distance among the ~N^2 distances between points of S?

The Erdos-Ulam problem suggests the answer is smaller than N^2.  But surely it’s much smaller, right?  You can get at least NM rational distances just by having S be (N/M) lines, each with M rational points.  Can you do better?

 

How many points does a random curve over F_q have?

So asks a charming preprint by Achter, Erman, Kedlaya, Wood, and Zureick-Brown.  (2/5 Wisconsin, 1/5 ex-Wisconsin!)  The paper, I’m happy to say, is a result of discussions at an AIM workshop on arithmetic statistics I organized with Alina Bucur and Chantal David earlier this year.

Here’s how they think of this.  By a random curve we might mean a curve drawn uniformly from M_g(F_q).  Let X be the number of points on a random curve.  Then the average number of points on a random curve also has a geometric interpretation: it is

What about

?

That’s just the average number of ordered pairs of distinct points on a random curve; the expected value of X(X-1).

If we can compute all these expected values, we have all the moments of X, which should give us a good idea as to its distribution.  Now if life were as easy as possible, the moduli spaces of curves would have no cohomology past degree 0, and by Grothendieck-Lefschetz, the number of points on M_{g,n} would be q^{3g-3+n}.  In that case, we’d have that the expected value of X(X-1)…(X-n) was q^n.  Hey, I know what distribution that is!  It’s Poisson with mean q.

Now M_g does have cohomology past degree 0.  The good news is, thanks to the Madsen-Weiss theorem (née the Mumford conjecture) we know what that cohomology is, at least stably.  Yes, there are a lot of unstable classes, too, but the authors propose that heuristically these shouldn’t contribute anything.  (The point is that the contribution from the unstable range should look like traces of gigantic random unitary matrices, which, I learn from this paper, are bounded with probability 1 — I didn’t know this, actually!)  And you can even make this heuristic into a fact, if you want, by letting q grow pretty quickly relative to g.

So something quite nice happens:  if you apply Grothendieck-Lefschetz (actually, you’d better throw in Kai Behrend’s name, too, because M_g is a Deligne-Mumford stack, not an honest scheme) you find that the moments of X still agree with those of a Poisson distribution!  But the contribution of the tautological cohomology shifts the mean from q to q+1+1/(q-1).

This is cool in many directions!

• It satisfies one’s feeling that a “random set,” if it carries no extra structure, should have cardinality obeying a Poisson distribution — the “uniform distribution” on the groupoid of sets.  (Though actually that uniform distribution is Poisson(1); I wonder what tweak is necessary to be able to tune the mean?)
• I once blogged about an interesting result of Bucur and Kedlaya which showed that a random smooth complete intersection curve in P^3 of fixed degree had slightly fewer than q+1 points; in fact, about q+1 – 1/q + o(q^2).  Here the deviation is negative, rather than positive, as the new paper suggests is the case for general curves; what’s going on?
• I have blogged about the question of average number of points on a random curve before.  I’d be very interested to know whether the new heuristic agrees with the answer to the question proposed at the end of that post; if g is a large random matrix in GSp(Z_ell) with algebraic eigenvalues, and which multiplies the symplectic form by q, and you condition on Tr(g^k) > (-q^k-1) so that the “curve” has nonnegatively many points over each extension of F_q, does this give something like the distribution the five authors predict for Tr(g)?  (Note:  I don’t think this question is exactly well-formed as stated.)

 

Are ranks bounded?

Important update, 23 Jul:  I missed one very important thing about Bjorn’s talk:  it was about joint work with a bunch of other people, including one of my own former Ph.D. students, whom I left out of the original post!  Serious apologies.  I have modified the post to include everyone and make it clear that Bjorn was talking about a multiperson project.  There are also some inaccuracies in my second-hand description of the mathematics, which I will probably deal with by writing a new post later rather than fixing this one.

I was only able to get to two days of the arithmetic statistics workshop in Montreal, but it was really enjoyable!  And a pleasure to see that so many strong students are interested in working on this family of problems.

I arrived to late to hear Bjorn Poonen’s talk, where he made kind of a splash talking about joint work by Derek Garton, Jennifer Park, John Voight, Melanie Matchett Wood, and himself, offering some heuristic evidence that the Mordell-Weil ranks of elliptic curves over Q are bounded above.  I remember Andrew Granville suggesting eight or nine years ago that this might be the case.  At the time, it was an idea so far from conventional wisdom that it came across as a bit cheeky!  (Or maybe that’s just because Andrew often comes across as a bit cheeky…)

Why did we think there were elliptic curves of arbitrarily large rank over Q?  I suppose because we knew of no reason there shouldn’t be.  Is that a good reason?  It might be instructive to compare with the question of bounds for rational points on genus 2 curves.  We know by Faltings that |X(Q)| is finite for any genus 2 curve X, just as we know by Mordell-Weil that the rank of E(Q) is finite for any elliptic curve E.  But is there some absolute upper bound for |X(Q)|?  When I was in grad school, Lucia Caporaso, Joe Harris, and Barry Mazur proved a remarkable theorem:  that if Lang’s conjecture were true, there was some constant B such that |X(Q)| was at most B for every genus 2 curve X.  (And the same for any value of 2…)

Did this make people feel like |X(Q)| was uniformly bounded?  No!  That was considered ridiculous!  The Caporaso-Harris-Mazur theorem was thought of as evidence against Lang’s conjecture.  The three authors went around Harvard telling all the grad students about the theorem, saying — you guys are smart, go construct sequences of genus 2 curves with growing numbers of points, and boom, you’ve disproved Lang’s conjecture!

But none of us could.

Mochizuki on ABC

[Update:  Lots of traffic coming in from Hacker News, much of it presumably from outside the usual pro number theory crowd that reads this blog.  If you’re not already familiar with the ABC conjecture, I recommend Barry Mazur’s beautiful expository paper “Questions about Number.”]

[Re-update:  Minhyong Kim’s discussion on Math Overflow is the most well-informed public discussion of Mochizuki’s strategy.  (Of course, it is still very sketchy indeed, as Minhyong hastens to emphasize.)   Both Kim’s writeup and discussions I’ve had with others suggest that the best place to start may be Mochizuki’s 2000 paper “A Survey of the Hodge-Arakelov Theory of Elliptic Curves I.”]

Shin Mochizuki has released his long-rumored proof of the ABC conjecture, in a paper called “Inter-universal Teichmuller theory IV:  log-volume computations and set-theoretic foundations.”

I just saw this an hour ago and so I have very little to say, beyond what I wrote on Google+ when rumors of this started circulating earlier this summer:

I hope it’s true:  my sense is that there’s a lot of very beautiful, very hard math going on in Shin’s work which almost no one in the community has really engaged with, and the resolution of a major conjecture would obviously create such engagement very quickly.

Well, now the time has come.  I have not even begun to understand Shin’s approach to the conjecture.  But it’s clear that it involves ideas which are completely outside the mainstream of the subject.  Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

Let me highlight one point which is clearly important, which I draw from pp.3–6 of the linked paper.

WARNING LABEL:  Of course my attempt to paraphrase is based on the barest of acquaintance with a very small section of the work and is placed here just to get people to look at Mochizuki’s paper — I may have it all wrong!

Mochizuki argues that it is too limiting to think about “the category of schemes over Spec Z,” as we are accustomed to do.  He makes the inarguable point that when X is a kind of thing, it can happen that the category of Xes, qua category, may not tell us very much about what Xes are like — for instance, if there is only one X and it has only one automorphism. Mochizuki argues that the category of schemes over a base is — if not quite this uninformative — insufficiently rich to handle certain problems in Diophantine geometry.  He wants us instead to think about what he calls the “species” of schemes over Spec Z, where a scheme in this sense is not an abstract object in a category, but something cut out by a formula.  In some sense this view is more classical than the conventional one, in which we tend to feel good about ourselves if we can “remove coordinates” and think about objects and arrows without implicitly applying a forgetful functor and referring to the object as a space with a Zariski topology or — ptui! — a set of points.

But Mochizuki’s point of view is not actually classical at all — because the point he wants to make is that formulas can be intepreted in any model of set theory, and each interpretation gives you a different category.  What is “inter-universal” about inter-universal Teichmuller theory is that it is important to keep track of all these categories, or at least many different ones.  What he is doing, he says, is simply outside the theory of schemes over Spec Z, even though it has consequences within that theory — just as (this part is my gloss) the theory of schemes itself is outside the classical theory of varieties, but provides us information about varieties that the classical theory could not have generated internally.

It’s tremendously exciting.  I very much look forward to commentary from people with a deeper knowledge than mine of Mochizuki’s past and present work.

Gonality, the Bogomolov property, and Habegger’s theorem on Q(E^tors)

I promised to say a little more about why I think the result of Habegger’s recent paper, ” Small Height and Infinite Non-Abelian Extensions,” is so cool.

First of all:  we say an algebraic extension K of Q has the Bogomolov property if there is no infinite sequence of non-torsion elements x in K^* whose absolute logarithmic height tends to 0.  Equivalently, 0 is isolated in the set of absolute heights in K^*.  Finite extensions of Q evidently have the Bogomolov property (henceforth:  (B)) but for infinite extensions the question is much subtler.  Certainly itself doesn’t have (B):  consider the sequence   On the other hand, the maximal abelian extension of Q is known to have (B) (Amoroso-Dvornicich) , as is any extension which is totally split at some fixed place p (Schinzel for the real prime, Bombieri-Zannier for the other primes.)

Habegger has proved that, when E is an elliptic curve over Q, the field Q(E^tors) obtained by adjoining all torsion points of E has the Bogomolov property.

What does this have to do with gonality, and with my paper with Chris Hall and Emmanuel Kowalski from last year?

Suppose we ask about the Bogomolov property for extensions of a more general field F?  Well, F had better admit a notion of absolute Weil height.  This is certainly OK when F is a global field, like the function field of a curve over a finite field k; but in fact it’s fine for the function field of a complex curve as well.  So let’s take that view; in fact, for simplicity, let’s take F to be C(t).

What does it mean for an algebraic extension F’ of F to have the Bogomolov property?  It means that there is a constant c such that, for every finite subextension L of F and every non-constant function x in L^*, the absolute logarithmic height of x is at least c.

Now L is the function field of some complex algebraic curve C, a finite cover of P^1.  And a non-constant function x in L^* can be thought of as a nonzero principal divisor.  The logarithmic height, in this context, is just the number of zeroes of x — or, if you like, the number of poles of x — or, if you like, the degree of x, thought of as a morphism from C to the projective line.  (Not necessarily the projective line of which C is a cover — a new projective line!)  In the number field context, it was pretty easy to see that the log height of non-torsion elements of L^* was bounded away from 0.  That’s true here, too, even more easily — a non-constant map from C to P^1 has degree at least 1!

There’s one convenient difference between the geometric case and the number field case.  The lowest log height of a non-torsion element of L^* — that is, the least degree of a non-constant map from C to P^1 — already has a name.  It’s called the gonality of C.  For the Bogomolov property, the relevant number isn’t the log height, but the absolute log height, which is to say the gonality divided by [L:F].

So the Bogomolov property for F’ — what we might call the geometric Bogomolov property — says the following.  We think of F’ as a family of finite covers C / P^1.  Then

(GB)  There is a constant c such that the gonality of C is at least c deg(C/P^1), for every cover C in the family.

What kinds of families of covers are geometrically Bogomolov?  As in the number field case, you can certainly find some families that fail the test — for instance, gonality is bounded above in terms of genus, so any family of curves C with growing degree over P^1 but bounded genus will do the trick.

On the other hand, the family of modular curves over X(1) is geometrically Bogomolov; this was proved (independently) by Abramovich and Zograf.  This is a gigantic and elegant generalization of Ogg’s old theorem that only finitely many modular curves are hyperelliptic (i.e. only finitely many have gonality 2.)

At this point we have actually more or less proved the geometric version of Habegger’s theorem!  Here’s the idea.  Take F = C(t) and let E/F be an elliptic curve; then to prove that F(E(torsion)) has (GB), we need to give a lower bound for the curve C_N obtained by adjoining an N-torsion point to F.  (I am slightly punting on the issue of being careful about other fields contained in F(E(torsion)), but I don’t think this matters.)  But C_N admits a dominant map to X_1(N); gonality goes down in dominant maps, so the Abramovich-Zograf bound on the gonality of X_1(N) provides a lower bound for the gonality of C_N, and it turns out that this gives exactly the bound required.

What Chris, Emmanuel and I proved is that (GB) is true in much greater generality — in fact (using recent results of Golsefidy and Varju that slightly postdate our paper) it holds for any extension of C(t) whose Galois group is a perfect Lie group with Z_p or Zhat coefficients and which is ramified at finitely many places; not just the extension obtained by adjoining torsion of an elliptic curve, for instance, but the one you get from the torsion of an abelian variety of arbitrary dimension, or for that matter any other motive with sufficiently interesting Mumford-Tate group.

Question:   Is Habegger’s theorem true in this generality?  For instance, if A/Q is an abelian variety, does Q(A(tors)) have the Bogomolov property?

Question:  Is there any invariant of a number field which plays the role in the arithmetic setting that “spectral gap of the Laplacian” plays for a complex algebraic curve?

A word about Habegger’s proof.  We know that number fields are a lot more like F_q(t) than they are like C(t).  And the analogue of the Abramovich-Zograf bound for modular curves over F_q is known as well, by a theorem of Poonen.  The argument is not at all like that of Abramovich and Zograf, which rests on analysis in the end.  Rather, Poonen observes that modular curves in characteristic p have lots of supersingular points, because the square of Frobenius acts as a scalar on the l-torsion in the supersingular case.  But having a lot of points gives you a lower bound on gonality!  A curve with a degree d map to P^1 has at most d(q+1) points, just because the preimage of each of the q+1 points of P^1(q) has size at most d.  (You just never get too old or too sophisticated to whip out the Pigeonhole Principle at an opportune moment….)

Now I haven’t studied Habegger’s argument in detail yet, but look what you find right in the introduction:

The non-Archimedean estimate is done at places above an auxiliary prime number p where E has good supersingular reduction and where some other technical conditions are met…. In this case we will obtain an explicit height lower bound swiftly using the product formula, cf. Lemma 5.1. The crucial point is that supersingularity forces the square of the Frobenius to act as a scalar on the reduction of E modulo p.

Yup!  There’s no mention of Poonen in the paper, so I think Habegger came to this idea independently.  Very satisfying!  The hard case — for Habegger as for Poonen — has to do with the fields obtained by adjoining p-torsion, where p is the characteristic of the supersingular elliptic curve driving the argument.  It would be very interesting to hear from Poonen and/or Habegger whether the arguments are similar in that case too!

Hain-Matsumoto, “Galois actions on fundamental groups of curves…”

I recently had occasion to spend some time with Richard Hain and Makoto Matsumoto’s 2005 paper “Galois actions on fundamental groups and the cycle C – C^-,” which I’d always meant to delve into.  It’s really beautiful!  I cannot say I’ve really delved — maybe something more like scratched — but I wanted to share some very interesting things I learned.

Serre proved long ago that the image of the l-adic Galois representation on an elliptic curve E/Q is open in GL_2(Z_l), so long as E doesn’t have CM.  This is a geometric condition on E, which is to say it only depends on the basechange of E to an algebraic closure of Q, or even to C.

What’s the analogue for higher genus curves X?  You might start by asking about the image of the Galois representation G_Q -> GSp_2g(Z_l) attached to the Tate module of the Jacobian of X.  This image lands in GSp_{2g}(Z_l).  Just as with elliptic curves, any extra endomorphisms of Jac(X) may force the image to be much smaller than GSp_{2g}(Z_l).  But the question of whether the image of rho must be open in GSp_2g(Z_l) whenever no “obvious” geometric obstruction forbids it is difficult, and still not completely understood.  (I believe it’s still unknown when g is a multiple of 4…?)  One thing we do know in general, though, is that when X is the generic curve of genus g (that is, the universal curve over the function field Q(M_g) of M_g) the resulting representation

is surjective.

Hain and Matsumoto generalize in a different direction.  When X is a curve of genus greater than 1 over a field K, the Galois group of K acts on more than just the Tate modules (or l-adic H_1) of X; it acts on the whole pro-l geometric fundamental group of X, which we denote pi.  So we get a morphism

What does it mean to ask this representation to have “big image”?

Local points on quadratic twists of X_0(N)

A new paper on the subject by my Ph.D. student, Ekin Ozman, is up on the arXiv today.

The twists in question are isomorphic to X_0(N) over a quadratic field K = Q(sqrt(d)), but not over Q; the twist is via the Atkin-Lehner involution w_N, which is to say that the rational points on such a twist are in bijection with the points P of X_0(N)(K) satisfying

where is the generator of Gal(K/Q).  Call this twist X^d(N).

Why do we care about these twists?  For one thing, X^d(N) parametrizes certain classes of Q-curves, elliptic curves over Qbar which are isogenous to all their Galois conjugates.  These guys turn out to be more flexible than elliptic curves over Q for constructing “Frey curves” attached to Diophantine equations, but have all the same handsome modularity properties as elliptic curves over Q, allowing the usual Mazur-Frey-Serre-Ribet-Wiles style argument to go through.

More generally, though, the X^d(N) form a very natural class of modular curves whose arithmetic hasn’t been much thought about.  For instance:  is there an analogue of Mazur’s theorem?  That is, do these curves have rational points?  Here one immediately encounters a divergence from the untwisted case — X^d(N) might not even have local points!  The cusps, which provide Q_p-points on X_0(N) for every N and p, are not rational points on X^d(N); without them, there’s no reason for X_0(N)(Q_p) to be nonempty.  In a 2004 survey paper about Q-curves, I asked (Problem A in the linked paper) for which d and N the curve X^d(N) had local points everywhere; this certainly has to be settled before any investigation of the global points can start.  Pete Clark (see the appendix to this paper), Jordi Quer, and Josep Gonzalez got partial results on the problem; now Ekin has almost entirely solved it, getting an exact criterion for the existence of local points whenever K and Q(sqrt(-N)) have no common primes of ramification.

I was never able to see a simple way to do this — and it turns out that’s because the answer, as Ekin works it out, is actually pretty complicated!  So I won’t state her theorem here; I’ll just say that it’s competely explicit and it allows you to compute just about whatever you want.  For example:  the number of squarefree d < X such that X^d(17) has local points everywhere is on order of .  (She could compute the constant, too, if for some terrifying reason you needed it…)

In the end, a lot of the X^d(N) have local points everywhere; and because they are all covers of the quotient X_0(N)/w_N, which has finitely many rational points once N is big enough, many of them don’t have any global rational points.  In other words, you have a healthy population of curves violating the Hasse Principle.  (This observation is due to Clark.)  Are these failures of the Hasse principle always due, as some people expect, to the Brauer-Manin obstruction?  In the last section of the paper,  Ekin works out one case in full — namely, X^{17}(23) — and shows that it is indeed so.

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}?