Tag Archives: hurwitz spaces

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

a_0 = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right]

b_0 = \left[ \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right]

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

x^2 + y^2 + z^2 = 3xyz

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?

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Smectic crystals, fingerprints, Klein bottles

Amazing colloquium this week by Randall Kamien, who talked about this paper with Chen and Alexander, this one with Liarte, Bierbaum, Mosna, and Sethna, and other stuff besides.

I’ve been thinking about his talk all weekend and I’m just going to write down a bit about what I learned.  In a liquid crystal, the molecules are like little rods; they have an orientation and nearby molecules want to have nearby orientations.  In a nematic crystal, that’s all that’s going on — the state of the crystal in some region B is given by a line field on B.   A smectic crystal has a little more to it — here, the rods are aligned into layers

smecticA

(image via this handy guide to liquid crystal phases)

separated by — OK, I’m not totally clear on whether they’re separated by a sheet of something else or whether that’s just empty space.  Doesn’t matter.  The point is, this allows you to tell a really interesting topological story.  Let’s focus on a smectic crystal in a simply connected planar region B.   At every point of B, you have, locally, a structure that looks like a family of parallel lines in the plane, each pair of lines separated by a unit distance.  (The unit is the length of the molecule, I think.)

Alternatively, you can think of such a “local smectic structure” as a line in the plane, where we consider two lines equivalent if they are parallel and the distance between them is an integer.  What’s the moduli space M — the “ground state manifold” — of such structures?    Well, the line family has a direction, so you get a map from M to S^1.  The lines in a given direction are parametrized by a line, and the equivalence relation mods out by the action of a lattice, so the fiber of M -> S^1 is a circle; in fact, it’s not hard to see that this surface M is a Klein bottle.

Of course this map might be pretty simple.  If B is the whole plane, you can just choose a family of parallel lines on B, which corresponds to the constant map.  Or you can cover the plane with concentric circles; the common center doesn’t have a smectic structure, and is a defect, but you can map B = R^2 – 0 to M.  Homotopically, this just gives you a path in M, i.e. an element of pi_1(M), which is a semidirect product of Z by Z, with presentation

\langle S,F: FSF^{-1} = S^{-1} \rangle

The concentric circle smectic corresponds the map which sends the generator of pi_1(B) to F.

So already this gives you a nice topological invariant of a plane smectic with k defects; you get a map from pi_1(B), which is a free group on k generators, to pi_1(M).  Note also that there’s a natural notion of equivalence on these maps; you can “stir” the smectic, which is to say, you can apply a diffeomorphism of the punctured surface, which acts by precomposition on pi_1(B).  The action of (the connected components of) Diff(B) on Hom(pi_1(B), pi_1(M)) is my favorite thing; the Hurwitz action of a mapping class group on the space of covers of a Riemann surface!  In particular I think the goal expressed in Chen et al’s paper of “extending our work to the topology of such patterns on surfaces of nontrivial topology (rather than just the plane)” will certainly involve this story.  I think in this case the Hurwitz orbits are pretty big; i.e. if what you know is the local appearance of the defects (i.e. the image in pi_1(M) of the conjugacy class in pi_1(B) corresponding to the puncture) you should almost be able to reconstruct the homotopy type of the map (up to stirring.)  If I understood Randy correctly, those conjugacy classes are precisely what you can actually measure in an experiment.

There’s more, though — a lot more!  You can’t just choose a map from B to M and make a smectic out of it.  The layers won’t line up!  There’s a differential criterion.  This isn’t quite the way they express it, but I think it amounts to the following:  the tangent bundle of M has a natural line bundle L sitting inside it, consisting of those directions of motion that move a line parallel to itself.  I think you want to consider only those maps from B to M such that the induced map on tangent bundles TB -> TM takes image in L.  More concretely, in coordinates, I think this means the following:  if you think of the local smectic structure at p as the preimage of Z under some real-valued function f in the neighborhood of p, then f should satisfy

(df/dx)^2 + (df/dy)^2 = 1.

This restricts your maps a lot, and it accounts for all kinds of remarkable behavior.  For one thing, it forbids certain conjugacy classes in pi_1(M) from appearing as local monodromy; i.e. the set of possible defect types is strictly smaller than the set of conjugacy classes in pi_1(M).  Moreover, it forbids certain kinds of defects from colliding and coalescing — for algebraic geometers, this naturally makes you feel like there’s a question about boundaries of Hurwitz spaces floating around.

Best of all, the differential equation forces the appearance of families of parallel ellipses, involute spirals, and other plane curves of an 18th century flavor.  The cyclides of Dupin put in an appearance.  Not just in the abstract — in actual liquid crystals!  There are pictures!  This is great stuff.  

Update:  Wait a minute — I forgot to say anything about fingerprints!  Maybe because I don’t have anything to say at the moment.  Except that the lines of a fingerprint are formally a lot like the lines of a smectic crystal, the defects can be analyzed in roughly the same way, etc.  Whether the diffeomorphism type of a fingerprint is an interesting forensic invariant I don’t rightly know.  I’ll bet whoever made my iPhone home button knows, though.

 

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“Homological stability for Hurwitz spaces… II” temporarily withdrawn

Akshay Venkatesh, Craig Westerland and I have temporarily withdrawn our preprint “Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II,” because there is a gap in the paper which we do not, at present, see how to remove.  There is no reason to think any of the theorems stated in the paper aren’t true, but because some of them are not proved at this time, we’ve pulled back the whole paper until we finish preparing a revised version consisting just of the material that does in fact follow from the arguments in their current form, together with some patches we’ve come up with.   We are extremely grateful to Oscar Randall-Williams for alerting us to the problem in the paper.

I’ll explain where the gap is below the fold, and which parts of the paper are still OK, but first a few thoughts about the issue of mistakes in mathematics.  Of course we owe a lot of people apologies.  All three of us have given talks in which we told people we had a proof of (a certain version of) the Cohen-Lenstra conjecture over F_q(t).  But we do not.  I know several people who had work in progress using this theorem, and so of course this development disrupts what they were doing, and I’ve kept those people up-to-date with the situation of the paper.  If there are others planning immediately to use the result, I hope this post will help draw their attention to the fact that they need to go back to treating this assertion as a conjecture.

One thing I found, when I talked to colleagues about this situation, is that it’s more common than I thought.  Lots of people have screwed up and said things in public or written things in papers they later realized were wrong.  One senior colleague told me an amazing story — he was in the shower one day when he suddenly realized that a paper he’d published in the Annals four years previously, a result he hadn’t even thought about in months, was wrong; there was an induction argument starting from a false base case!  Fortunately, after some work, he was able to construct a repaired argument getting to the same results, which he published as a separate paper.

Naturally nobody likes to talk about their mistakes, and so it’s easy to get the impression that this kind of error is very rare.  But I’ve learned that it’s not so rare.  And I’m going to try to talk about my own error more than I would in my heart prefer to, because I think we have to face the fact that mathematicians are human, and it’s not safe to be certain something is true because we found it on the arXiv, or even in the Annals.

In a way, what happened with our paper is exactly what people predicted would happen once we lost our inhibitions about treating unrefereed preprints as papers.  We wrote the paper, we made it public, and people cited it before it was refereed, and it was wrong.

But what would have happened in a pre-arXiv world?  The mistake was pretty subtle, resting crucially on the relation between this paper and our previous one.  Would the referee have caught it, when we didn’t?  I’m not so sure.  And if the paper hadn’t been openly shared before publication, Oscar wouldn’t have seen it.  It might well have been published in its incorrect form.  On balance, I’d guess wide distribution on arXiv makes errors less likely to propagate through mathematics, not more.

Sociology of mathematics ends here; those who want to know more about the mistake, keep reading past the fold.

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Stable cohomology for Hurwitz spaces: Upgoer Five version

People ask how many of a kind of thing there are; the thing might be a kind of number, or something like a number. I, together with others, work out how many of those things there are by understanding the way some kinds of spaces look; these spaces are, in a way, the same as the things about which we ask, “how many,” but in another way they are different.  This allows us to use different ideas when we think about them, and answer some questions about numbers which could not be answered before.

Make your own!

(inspired by xkcd, natch.)

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Homological Stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II

Akshay Venkatesh, Craig Westerland, and I, recently posted a new paper, “Homological Stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II.” The paper is a sequel to our 2009 paper of the same title, except for the “II.”  It’s something we’ve been working on for a long time, and after giving a lot of talks about this material it’s very pleasant to be able to show it to people at last!

The main theorem of the new paper is that a version of the Cohen-Lenstra conjecture over F_q(t) is true.  (See my blog entry about the earlier paper for a short description of Cohen-Lenstra.)

For instance, one can ask: what is the average size of the 5-torsion subgoup of a hyperelliptic curve over F_q? That is, what is the value of

\lim_{n \rightarrow \infty} \frac{\sum_C |J(C)[5](\mathbf{F}_q)|}{\sum_C 1}

where C ranges over hyperelliptic curves of the form y^2 = f(x), f squarefree of degree n?

We show that, for q large enough and not congruent to 1 mod 5, this limit exists and is equal to 2, exactly as Cohen and Lenstra predict. Our previous paper proved that the lim sup and lim inf existed, but didn’t pin down what they were.

In fact, the Cohen-Lenstra conjectures predict more than just the average size of the group J(C)[5](\mathbf{F}_q) as n gets large; they propose a the isomorphism class of the group settles into a limiting distribution, and they say what this distribution is supposed to be! Another way to say this is that the Cohen-Lenstra conjecture predicts that, for each abelian p-group A, the average number of surjections from J(C)(\mathbf{F}_q) to A approaches 1. There are, in a sense, the “moments” of the Cohen-Lenstra distribution on isomorphism classes of finite abelian p-groups.

We prove that this, too, is the case for sufficiently large q not congruent to 1 mod p — but, it must be conceded, the value of “sufficiently large” depends on A. So there is still no q for which all the moments are known to agree with the Cohen-Lenstra predictions. That’s why I call what we prove a “version” of the Cohen-Lenstra conjectures. If you think of the Cohen-Lenstra conjecture as being about moments, we’re almost there — but if you think of it as being about probability distributions, we haven’t started!

Naturally, we prefer the former point of view.

This paper ended up being a little long, so I think I’ll make several blog posts about what’s in there, maybe not all in a row.

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Arithmetic Veech sublattices of SL_2(Z)

Ben McReynolds and I have just arXived a retitled and substantially revised version of our paper “Every curve is a Teichmuller curve,” previously blogged about here.  If you looked at the old version, you probably noticed it was very painful to try to read.  My only defense is that it was even more painful to try to write.

With the benefit of a year’s perspective and some very helpful comments from the anonymous referee at Duke, we more or less completely rewrote the paper, making it much more readable and even a bit shorter.

The paper is related to the question I discussed last week about “4-branched Belyi” — or rather the theorem of Diaz-Donagi-Harbater that inspired our paper is related to that question.  The 4-branched Belyi question essentially asks whether every curve C in M_g is a Hurwitz space of 4-branched covers.  (Surely not!) The DDH theorem shows that if you’re going to prove C is not a Hurwitz curve, you can’t do it by means of the birational isomorphism class of C alone; every 1-dimensional function field appears as the function field of a Hurwitz curve (though probably in very high genus.)

 

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Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Now I’ll say a little bit about the actual problem treated by the new paper with Venkatesh and Westerland.  It’s very satisfying to have an actual theorem of this kind:  for years now we’ve been going around saying “it seems like asymptotic conjectures in analytic number theory should have a geometric reflection as theorems about stable cohomology of moduli spaces,” but for quite a while it was unclear we’d ever be able to prove something on the geometric side.

The new paper starts with the question: what do ideal class groups of number fields tend to look like?

That’s a bit vague, so let’s pin it down:  if you write down the ideal class group of the quadratic imaginary number fields \mathbf{Q}(\sqrt{-d}), as d ranges over squarefree integers in [0..X],  you get a list of about \zeta(2)^{-1} X finite abelian groups.

The ideal class group is the one of the most basic objects of algebraic number theory; but we don’t know much about this list of groups!  Their orders are more or less under control, thanks to the analytic class number formula.  But their structure is really mysterious.

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The braid group, analytic number theory, and Weil’s three columns

This post is about a new paper of mine with Akshay Venkatesh and Craig Westerland; but I’m not going to mention that paper in the post. Instead, I want to explain why topological theorems about the stable homology of moduli spaces are relevant to analytic number theory.  If you’ve seen me give a talk about this stuff, you’ve probably heard this spiel before.

We start with Weil’s famous quote about “the three columns”:

“The mathematician who studies these problems has the impression of deciphering a trilingual inscription. In the first column one finds the classical Riemannian theory of algebraic functions. The third column is the arithmetic theory of algebraic numbers.  The column in the middle is the most recently discovered one; it consists of the theory of algebraic functions over finite fields. These texts are the only source of knowledge about the languages in which they are written; in each column, we understand only fragments.”

Let’s see how a classical question of analytic number theory works in Weil’s three languages.  Start with the integers, and ask:  how many of the integers between X and 2X are squarefree?  This is easy:  we have an asymptotic answer of the form

\frac{6}{\pi^2}X + O(X^{1/2}) = \zeta(2)^{-1} X + O(X^{1/2}).

(In fact, the best known error term is on order X^{17/54}, and the correct error term is conjectured to be X^{1/4}; see Pappalardi’s “Survey on k-freeness” for more on such questions.)

So far so good.  Now let’s apply the popular analogy between number fields and function fields, going over to Weil’s column 3, and ask: what’s the analogous statement when Z is replaced by F_q[T]?

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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.

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