Bourgain, Gamburd, Sarnak on Markoff triples

Such a great colloquium last week by Peter Sarnak, this year’s Hilldale Lecturer, on his paper with Bourgain and Gamburd.  My only complaint is that he promised to talk about the mapping class group and then barely did!  So I thought I’d jot down what their work has to do with mapping class groups and spaces of branched covers.

Let E be a genus 1 Riemann surface — that is, a torus — and O a point of E.  Then pi_1(E-O) is just a free group on two generators, whose commutator is (the conjugacy class of) a little loop around the puncture.  If G is a group, a G-cover of E branched only at O is thus a map from pi_1(E-O) to G, which is to say a pair (a,b) of elements of G.  Well, such a pair considered up to conjugacy, since we didn’t specify a basepoint for our pi_1.  And actually, we might as well just think about the surjective maps, which is to say the connected G-covers.

Let’s focus on the case G = SL_2(Z).  And more specifically on those maps where the puncture class is sent to a matrix of trace -2.  Here’s an example:  we can take

You can check that in this case the puncture class has trace -2; that is, it is the negative of a unipotent matrix.  Actually, I gotta be honest, these matrices don’t generate SL_2(Z); they generate a finite-index subgroup H of SL_2(Z), its commutator.

Write S for the set of all conjugacy classes of pairs (a,b) of matrices which generate H and have commutator with trace -2.  It turns out that this set is the set of integral points of an affine surface called the Markoff surface:  namely, if we take x = Tr(a)/3, y = Tr(b)/3, and z = Tr(ab)/3, then the three traces obey the relation

and indeed every solution to this equation corresponds to an element of S.

So the integral points on the Markoff surface are acted on by an infinite discrete group.  Which if you just look at the equation seems like kind of a ridiculous miracle.  But in the setting of H-covers is very natural.  Because there’s a natural group acting on S: namely, the mapping class group Γ of type (1,1).  This group’s whole purpose in life is to act on the fundamental group of a once-punctured torus!  (For readers unfamiliar with mapping class groups, I highly recommend Benson Farb and Dan Margalit’s wonderful textbook.)   So you start with a surjection from pi_1(E-O) to H, you compose with the action of  Γ, and you get a new homomorphism.  The action of  Γ on pi_1(E-O) is only outer, but that’s OK, because we’re only keeping track of conjugacy classes of homomorphisms from pi_1(E-O) to H.

So Γ acts on S; and now the lovely theorem is that this action is transitive.

I don’t want to make this mapping class group business sound more abstract than it is.  Γ isn’t a mystery group; it acts on H_1(E-O), a free abelian group of rank 2, which gives a map from Γ to SL_2(Z), which turns out to be an isomorphism.  What’s more, the action of Γ on pairs (a,b) is completely explicit; the standard unipotent generators of SL_2(Z) map to the moves

(a,b) -> (ab,b)

(a,b) -> (a,ab)

(Sanity check:  each of these transformations preserves the conjugacy class of the commutator of a and b.)

Sarnak, being a number theorist, is interested in strong approximation: are the integral solutions of the Markoff equation dense in the adelic solutions?   In particular, if I have a solution to the Markoff equation over F_p — which is to say, a pair (a,b) in SL_2(F_p) with the right commutator — can I lift it to a solution over Z?

Suppose I have a pair (a,b) which lifts to a pair (a,b).  We know (a,b) = g(a_0,b_0) for some g in Γ.  Thus (a,b) = g(a_0,b_0).  In other words, if strong approximation is true, Γ acts transitively on the set S_p of Markoff solutions mod p.  And this is precisely what Bourgain, Gamburd, and Sarnak conjecture.  (In fact, they conjecture more:  that the Cayley-Schreier graph of this action is an expander, which is kind of a quantitative version of an action being transitive.)  One reason to believe this:  if we replace F_p with C, we replace S with the SL_2(C) character variety of pi_1(E-O), and Goldman showed long ago that the action of mapping class groups on complex character varieties of fundamental groups was ergodic; it mixes everything around very nicely.

Again, I emphasize that this is on its face a question of pure combinatorial group theory.  You want to know if you can get from any pair of elements in SL_2(F_p) with negative-unipotent commutator to any other via the two moves above.  You can set this up on your computer and check that it holds for lots and lots of p (they did.)  But it’s not clear how to prove this transitivity for all p!

They’re not quite there yet.  But what they can prove is that the action of Γ on S_p has a very big orbit, and has no very small orbits.

Now that G is the finite group SL_2(F_p), we’re in my favorite situation, that of Hurwitz spaces.  The mapping class group Γ is best seen as the fundamental group of the moduli stack M_{1,1} of elliptic curves.  So an action of Γ on the finite set S_p is just a cover H_p of M_{1,1}.  It is nothing but the Hurwitz space parametrizing maps (f: X -> E) where E is an elliptic curve and f an SL_2(F_p)-cover branched only at the origin.  What Bourgain, Gamburd, and Sarnak conjecture is that H_p is connected.

If you like, this is a moduli space of curves with nonabelian level structure as in deJong and Pikaart.  Or, if you prefer (and if H_p is actually connected) it is a noncongruence modular curve corresponding to the stabilizer of an element of S_p in Γ = SL_2(Z).  This stabilizer is in general going to be a noncongruence subgroup, except it is a congruence subgroup in the more general sense of Thurston.

This seems like an interesting family of algebraic curves!  What, if anything, can we say about it?

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.

 

What I learned from Zhiwei Yun about Hilbert schemes

One knows, of course, that Hilbert schemes of smooth curves and smooth surfaces are nice, and Hilbert schemes of varieties of dimension greater than two are terrifying.

Zhiwei Yun was here giving a talk about his work with Davesh Maulik on Hilbert schemes of curves with planar singularities, and he made a point I’d never appreciated; it’s not the dimension of the variety, but the dimension of its tangent space that really measures the terrifyingness of the  Hilbert space.  Singular curves C with planar singularities are not so bad — you still have a nice Hilbert scheme with an Abel-Jacobi map to the compactified Jacobian.  But let C be the union of the coordinate axes in A^3 and all bets are off.  Hideous extra high-dimensional components aplenty.  If I had time to write a longer blog post today I would think about what the punctual Hilbert scheme at the origin looks like.  But maybe one of you guys will just tell me.

Update:  Jesse Kass explains that I am wrong about C; its Hilbert scheme has a non-smoothable component, but it doesn’t have any components whose dimension is too large.

Bourqui on spaces of rational curves and motivic Batyrev-Manin

David Bourqui just posted a really nice paper, “Asymptotic Behavior of Rational Curves,” notes from a lecture series he gave last summer at the Institut Fourier at Grenoble.  I’ll try to sum up here what it’s about and hopefully entice people to have a look!

Let N_X(B) net of rational points of height at most B on a Fano variety X endowed with an ample line bundle L over a global field K.  By now we are used to the idea (summed up in greatest generality by the Baytrev-Manin conjecture, refined by Peyre) that N_X(B) satisfies an asymptotic regularity — it approaches

for some constants a and b (which are integers) and c (a real number.)  Batyrev and Manin gave predictions for a and b, Peyre pinned down c.  To fix ideas, let’s suppose that the projective embedding of X is given by the anticanonical divisor.  Then a is 1 and b is the Picard rank of X (over K.)  So for instance when X = P^1, you can see immediately that the number of points of height B should be linear in B, and that’s true (remembering that the canonical height is the square of the usual Weil height on P^1.)

Now these conjectures are not exactly right.  There is the problem of accumulating subvarieties, like the lines on a cubic surface, which have way too many rational points; you have to strip these out before you can expect to get down to the expected asymptotic.  And there are more subtle counterexamples, like the one produced by Batyrev and Tschinkel, where the number of rational points is too high by some power of log B.  But the conjecture has been proved for many classes of varieties (toric, homogeneous, very low degree relative to dimension…)

Bourqui’s approach starts from the consideration of these conjectures over the global function field K = k(t), where k = F_q is a finite field.  For simplicity let’s take X to be defined over the constant field k.  Now N_K(B) has two meanings.  You can think of it as a set of K-points of X of bounded height — or you can think it as the number of k-points on the space C_d(X) of degree-d rational curves on X, where q^d = B.  The Batyrev-Manin conjecture, as we phrased it here, is about the first interpretation.  But you can also read it as a statement about the varieties C_d, and it turns out what it says is that

|C_d(k)| / |A^(k)|^d

approaches a limit as d goes to infinity.

Doesn’t this seem a rather astonishing claim at first glance?  These are higher and higher-dimensional varieties over a fixed finite field; the Weil conjectures offer us no useful control.  Why shouldn’t their point-counts fluctuate wildly?

To get some idea of what might be gone, think of a family of varieties that’s easily seen to display this behavior:  the projective spaces P^d.  Evidently,

.

Why is there a limit?  I can think of two reasons.

First reason:  projective spaces have stable cohomology;  the compactly supported cohomology has one dimension in degrees 2d,2d-2, … 0 and is empty in the odd degrees, and all the cohomology is of Tate type.  Note that (apart from the location of the top cohomology group) this description is independent of d, and it follows that the point count provided by the Lefschetz trace formula is (apart from a prepended power of q) independent of d as well.

Second reason:  Forget about finite fields — the expression

remains true motivically (in the suitably completed version of the Grothendieck ring of varieties.)  To be fair, this alone doesn’t quite imply that the point-counting limits hold.  (For instance, the sequence $1 + latex 2^{2^d} [\mathbf{A}^1]^{-d}$ converges motivically to 1, while none of its point-counts converge)  But the motivic convergence is highly suggestive.

My own work in this circle of ideas is mostly concerned with stable cohomology.  What Bourqui is interested in, on the other hand, is whether one has a motivic Batyrev-Manin conjecture; is it the case that

approaches a limit in d, and what is this limit?  (This is question 1.11.2 in Bourqui’s paper — to be precise, Bourqui asks for something more precise where one breaks up C_d according to the numerical equivalence class of the rational curve.)  Bourqui proves this is indeed the case when X is a smooth projective toric variety over a field of characteristic 0.  This is by no means a straightforward imitation of the proof of Batyrev-Manin for toric varieties over global fields:  proving motivic identities is hard!

In the case of toric varieties, by the way, both routes to Batyrev-Manin are available; the fact that spaces of rational curves on toric varieties have stable cohomology was proved by Martin Guest.  Guest shows that the cohomology stabilization holds for all smooth projective toric varieties and some of the singular ones as well — the main tool is a diffeomorphism between this space of rational curves and a certain kind of decorated configuration space on the sphere.  I wonder, is Guest’s configuration space implicitly present in Bourqui’s proof?

“Every curve is a Teichmuller curve,” or “Why SL_2(Z) has the congruence subgroup property.”

A Teichmüller curve in M_g, the moduli space of genus-g curves, is an algebraic curve V in M_g such that the inclusion V -> M_g induces an isometry between the constant-curvature metric on V and the restriction of the Teichmüller metric on M_g.

Alternatively:  the cotangent bundle of M_g, considered as a real manifold, admits a natural action of SL_2(R); the orbits are all copies of SL_2(R) / SO(2), or the upper half-plane.  Most of the time, when you project that hyperbolic plane H down to M_g, you get a dense orbit that wanders all over M_g.  But every once in a while, the fibers of the map H -> M_g are a lattice in H, and the image is actually an algebraic curve; that, again, is a Teichmüller curve.

Teichmüller curves are the subject of lots of recent research; for now, let me just say that they are interestingly canonical curves inside M_g.  Matt Bainbridge proved strong results about their intersection numbers in Hilbert modular surfaces.  McMullen classified Teichmuller curves in M_2, giving a very nice algebraic description of the 1-parameter families of genus-2 curves parametrized by Teichmüller curves.  (As far as I know, there’s no such description in higher genus.)  In a recent note, McMullen proved that they are all defined over number fields.

This leads one to ask:  which curves defined over algebraic number fields are Teichmüller curves?  This is the subject of a paper Ben McReynolds and I just posted to the arXiv, “Every curve is a Teichmüller curve.”  The title should be read birationally; what we prove is that every curve X over an algebraic number field is birational (over C) to a Teichmüller curve in some M_g.  (In the posted version, we prove the slightly weaker statement that X is birational to a Teichmüller curve in M_{g,n}), but we’ve since tweaked the argument to get the closed-surface version.)

So why does SL_2(Z) have the congruence subgroup property?  Especially given that it, y’know, doesn’t?

Here’s what I mean.  Let Gamma_{g,n} be the mapping class group of a genus-g surface with n punctures.  Then Gamma_{g,n} acts as a group of outer automorphisms of the fundamental group pi_{g,n} of the surface; and from this, you get an action of Gamma_{g,n} on the finite set

Hom(pi_{g,n},G)/~

where G is a finite group and ~ is conjugacy.

By a congruence subgroup of Gamma_{g,n} let’s mean a stabilizer in this action.  Why this definition?  Well, when g = 1, n = 0, and G = Z/NZ, the stabilizer is just the standard congruence subgroup Gamma_0(N).  And you can easily check that the class of congruence subgroups of Gamma_{1,0} is cofinal with the usual class of congruence subgroups in SL_2(Z).

Now Gamma_{1,1} is also isomorphic to SL_2(Z), but the notion of “congruence subgroup of SL_2(Z)” afforded by this isomorphism is much more general than the usual one.  So much so that one gets the following, which is really the main point of my paper with Ben:

Every finite-index subgroup of Gamma_{1,1} containing the center and contained in Gamma(2) is a congruence subgroup.

It turns out that the finite covers of the moduli space M_{1,1} corresponding to such finite-index subgroups are always Teichmüller curves; since, by Belyi’s theorem, every curve over a number field can be so expressed, we get the desired result.

The italicized assertion above can be thought of as a very strong kind of “congruence subgroup property.”  Of course, CSP usually refers to the property that every finite-index subgroup contains a principal congruence subgroup.  That finite-index subgroups Gamma_{1,1} (and even Gamma_{1,n}) always contain congruence subgroups as defined above is a theorem of Asada, and it’s conjectured to be true for all g,n.  But the statement that every finite-index subgroup of a mapping class group is a congruence subgroup on the nose is substantially stronger, and I imagine it’s true only for (1,1) and the closely related case (0,4), which was proved, in somewhat different language, in the paper “Every curve is a Hurwitz space,” by Diaz, Donagi, and Harbater.  Our argument is very much inspired by theirs — it was to emphasize this debt that we gave our paper more or less the same title.

Anabelian puzzle 4: What is the probability that a set of n points has no 3 collinear?

OK, this isn’t really an anabelian puzzle, but it was presented to me at the anabelian conference by Alexei Skorobogatov.

Let X_n be the moduli space of n-tuples of points in A^2 such that no three are collinear.  The comment section of this blog computed the number of components of X_n(R) back in January.  Skorobogatov asked what I could say about the cohomology of X_n(C).  Well, not a lot!  But if I were going to make a good guess, I’d start by trying to estimate the number of points on X_n over a finite field F_q.

So here’s a question:  can you estimate the number of degree-n 0-dimensional subschemes S of A^2/F_q which have no three points collinear?  It seems very likely to me that the answer is of the form

for some power series P.

One way to start, based on the strategy in Poonen’s Bertini paper:  given a line L, work out the probability P_L that S doesn’t have three points on L.  Now your first instinct might be to take the product of P_L over all lines in A^2; this will be some version of a special value of the zeta function of the dual P^2.  But it’s not totally clear to me that “having three points on L_1” and “having three points on L_2” are independent.

Turkelli on Hurwitz spaces and Malle’s conjecture

My Ph.D. student Seyfi Turkelli recently posted a really nice paper, “Connected components of Hurwitz schemes and Malle’s conjecture,” to the arXiv. It’s a beautiful example of the “hidden geometry” behind questions about arithmetic distributions, so I thought I’d say a little about it here.

The story begins with the old conjecture, sometimes attributed to Linnik, that the number of degree-n extensions of Q of discriminant at most X grows linearly with X, as X grows with n held constant. When n=2, this is easy; when n = 3, it is a theorem of Davenport and Heilbronn; when n=4 or 5, it is recent work of Bhargava; when n is at least 6, we have no idea.

Having no idea is, of course, no barrier to generalization. Here’s a more refined version of the conjecture, due to Gunter Malle. Let K be a number field, let G be a finite subgroup of S_n, and let N_{K,G}(X) be the number of extensions L/K of degree n whose discriminant has norm at most K, and whose Galois closure has Galois group G. Then there exists a constant c_{K,G} such that

Conjecture: N_{K,G}(X) ~ c_{K,G} X^a(G) (log X)^(b(K,G))

where a and b are constants explicitly described by Malle. (Malle doesn’t make a guess as to the value of c_{K,G} — that’s a more refined statement still, which I hope to blog about later…)

Akshay Venkatesh and I wrote a paper (“Counting extensions of function fields…”) in which we gave a heuristic argument for Malle’s conjecture over K = F_q(t). In that case, N_{K,G}(X) is the number of points on a certain Hurwitz space, a moduli space of finite covers of the projective line. We were able to control the dimensions and the number of irreducible components of these spaces, using in a crucial way an old theorem of Conway, Parker, Fried, and Volklein. The heuristic part arrives when you throw in the 100% shruggy guess that an irreducible variety of dimension d over F_q has about q^d points. When you apply this heuristic to the Hurwitz spaces, you get Malle’s conjecture on the nose.

So we were a little taken aback a couple of years later when Jurgen Kluners produced counterexamples to Malle’s theorem! We quickly figured out what was going on. There wasn’t anything wrong with our theorem; just our analogy. Our Hurwitz spaces were counting geometrically connected covers of the projective line. But a cover Y -> P^1/F_q which is connected, but not geometrically connected, provides a perfectly good field extension of F_q(t). If we’re trying to imitate the number field question, we’d better count those too. It had never occurred to us that they might outnumber the geometrically connected covers — but that’s just what happens in Kluners’ examples.

What Turkelli does in his new paper is to work out the dimensions and components for certain twisted Hurwitz spaces which parametrize the connected but not geometrically connected covers of P^1. This is a really subtle thing to get right — you can’t rely on your geometric intuition, because the phenomenon you’re trying to keep track of doesn’t exist over an algebraically closed field! But Turkelli nails it down — and as a consequence, he gets a new version of Malle’s conjecture, which is compatible with Kluners’ examples, and which I think is really the right statement. Which is not to say I know whether it’s true! But if it’s not, it’s at least the correct false guess given our present state of knowledge.