## Soumya Sankar: Proportion of ordinarity in families of curves over finite fields

What’s the chance that a random curve has ordinary Jacobian? You might instinctively say “It must be probability 1” because the non-ordinary locus is a proper closed subvariety of M_g. (This is not obvious by pure thought, at least to me, and I don’t know who first proved it! I imagine you can check it by explicitly exhibiting a curve of each genus with ordinary Jacobian, but I’m not sure this is the best way.)

Anyway, the point is, this instinctive response is wrong! At least it’s wrong if you interpret the question the way I have in mind, which is to ask: given a random curve X of genus g over F_q, with g growing as q stays fixed, is there a limiting probability that X has ordinary Jacobian? And this might not be 1, in the same way that the probability that a random polynomial over F_q is squarefree is not 1, but 1-1/q.

Bryden Cais, David Zureick-Brown and I worked out some heuristic guesses for this problem several years ago, based on the idea that the Dieudonne module for a random curve might be a random Dieudonne module, and then working out in some detail what in the Sam Hill one might mean by “random Dieudonne module.” Then we did some numerical experiments which showed that our heuristic looked basically OK for plane curves of high degree, but pretty badly wrong for hyperelliptic curves of high genus. But there was no family of curves for which one could prove either that our heuristic was right or that it was wrong.

Now there is, thanks to my Ph.D. student Soumya Sankar. Unfortunately, there are still no families of curves for which our heuristics are provably right. But there are now several for which it is provably wrong!

15.7% of Artin-Schreier curves over F_2 (that is: Z/2Z-covers of P^1/F_2) are ordinary. (The heuristic proportion given in my paper with Cais and DZB is about 42%, which matches data drawn from plane curves reasonably well.) The reason Sankar can prove this is because, for Artin-Schreier curves, you can test ordinarity (or, more generally, compute the a-number) in terms of the numerical invariants of the ramification points; the a-number doesn’t care where the ramification points are, which would be a more difficult question.

On the other hand, 0% of Artin-Schreier curves over F are ordinary for any finite field of odd characteristic! What’s going on? It turns out that it’s only in characteristic 2 that the Artin-Schreier locus is irreducible; in larger characteristics, it turns out that the locus has irreducible components whose number grows with genus, and the ordinary curves live on only one of these components. This “explains” the rarity of ordinarity (though this fact alone doesn’t prove that the proportion of ordinarity goes to 0; Sankar does that another way.) Natural question: if you just look at the ordinary component, does the proportion of ordinary curves approach a limit? Sankar shows this proportion is bounded away from 0 in characteristic 3, but in larger characteristics the combinatorics get complicated! (All this stuff, you won’t be surprised to hear, relies on Rachel Pries’s work on the interaction of special loci in M_g with the Newton stratification.)

Sankar also treats the case of superelliptic curves y^n = f(x) in characteristic 2, which turns out to be like that of Artin-Schreier in odd characteristics; a lot of components, only one with ordinary points, probability of ordinarity going to zero.

Really nice paper which raises lots of questions! What about more refined invariants, like the shape of the Newton polygon? What about other families of curves? I’d be particularly interested to know what happens with trigonal curves which (at least in characteristic not 2 or 3, and maybe even then) feel more “generic” to me than curves with extra endomorphisms. Is there any hope for our poor suffering heuristics in a family like that?

## Is the Dedekind sum really a function on SL_2?

Here’s an idle thought I wanted to record before I forgot it.

The Dedekind sum comes up in a bunch of disparate places; it’s how you keep track of the way half-integral weight forms like the eta function aka discriminant to the one-twelfth transforms under SL_2, it shows up in the topology of modular knots, the alternating sum of continued fraction coefficients, etc.  It has a weird definition which I find it hard to get a feel for.  The Dedekind sum also satsfies Rademacher reciprocity: $D(a,b;c) + D(b,c;a) + D(c,a;b) = \frac{1}{12abc}(a^2 + b^2 + c^2 - 3abc)$

If that right-hand side looks familiar, it’s because it’s the very same cubic form whose vanishing defines the Markoff numbers!  Here’s a nice way to interpret it.  Suppose A,B,C are matrices with ABC = 1 and

(1/3)Tr A = a

(1/3)Tr B = b

(1/3)Tr C = c

(Why 1/3?  See this post from last month.)

Then $a^2 + b^2 + c^2 - 3abc = (1/9)(2 + \mathbf{Tr}([A,B]))$

(see e.g. this paper of Bowditch.)

The well-known invariance of the Markoff form under moves like (a,b,c) -> (a,b,ab-c) now “lifts” to the fact that (the conjugacy class of) [A,B] is unchanged by the action of the mapping class group Gamma(0,4) on the set of triples (A,B,C) with ABC=1.

The Dedekind sum can be thought of as a function on such triples:

D(A,B,C) = D((1/3)Tr A, (1/3) Tr B; (1/3) Tr C).

Is there an alternate definition or characterization of D(A,B,C) which makes Rademacher reciprocity $D(A,B,C) + D(B,C,A) + D(C,A,B) = (1/9)(2 + \mathbf{Tr}([A,B]))$

more manifest?

## Naser Talebizadeh Sardari, Hecke eigenvalues, and Chabauty in the deformation space

Naser Sardari is finishing a postdoc at Wisconsin this year and just gave a beautiful talk about his new paper.  Now Naser thinks of this as a paper about automorphic forms — and it is — but I want to argue that it is also a paper which develops an unexpected new form of the Chabauty method!  As I will now explain.  Tell me if you buy it.

First of all, what does Naser prove?  As the title might suggest, it’s a statement about the multiplicity of Hecke eigenvalues a_p; in this post, we’re just going to talk about the eigenvalue zero.  The Hecke operator T_p acts on the space of weight-k modular forms on Gamma_0(N); how many zero eigenvectors can it have, as k goes to infinity with N,p fixed?  If you believe conjectures of Maeda type, you might expect that the Hecke algebra acts irreducibly on the space S_k(Gamma_0(N)); of course this doesn’t rule out that one particular Hecke operator might have some zeroes, but it should make it seem pretty unlikely.

And indeed, Naser proves that the number of zero eigenvectors is bounded independently of k, and even gives an explicit upper bound. (When the desired value of a_p is nonzero, T_p has finite slope and we can reduce to a problem about modular forms in a single p-adic family; in that context, a uniform bound is easy, and one can even show that the number of such forms of weight <k grows very very very very slowly with k, where each "very" is a log; this is worked out on Frank Calegari’s blog.. On the other hand, as Naser points out below in comments, if you ask about the “Hecke angle” a_p/p^{(k-1)/2}, we don’t know how to get any really good bound in the nonzero case. I think the conjecture is that you always expect finite multiplicity in either setting even if you range over all k.)

What I find most striking is the method of proof and its similarity to the Chabauty method!  Let me explain.  The basic idea of Naser’s paper is to set this up in the language of deformation theory, with the goal of bounding the number of weight-k p-adic Galois representations rho which could be the representations attached to weight-k forms with T_p = 0.

We can pin down the possible reductions mod p of such a form to a finite number of possibilities, and this number is independent of k, so let’s fix a residual representation rhobar once and for all.

The argument takes place in R_loc, the ring of deformations of rhobar|G_{Q_p}.  And when I say “the ring of deformations” I mean “the ring of deformations subject to whatever conditions are important,” I’m just drawing a cartoon here.  Anyway, R_loc is some big p-adic power series ring; or we can think of the p-adic affine space Spec R_loc, whose Z_p-points we can think of as the space of deformations of rhobar to p-adic local representations.  This turns out to be 5-dimensional in Naser’s case.

Inside Spec R_loc, we have the space of local representations which extend to global ones; let’s call this locus Spec R_glob.  This is still a p-adic manifold but it’s cut out by global arithmetic conditions and its dimension will be given by some computation in Galois cohomology over Q; it turns out to be 3.

But also inside Spec R_loc, we have a submanifold Z cut out by the condition that a_p is not just 0 mod p, it is 0 on the nose, and that the determinant is the kth power of cyclotomic for the particular k-th power you have in mind.  This manifold, which is 2-dimensional, is something you could define without ever knowing there was such a thing as Q; it’s just some closed locus in the deformation space of rhobar|Gal(Q_p).

But the restriction of rho to Gal(Q_p) is a point psi of R_loc which has to lie in both these two spaces, the local one which expresses the condition “psi looks like the representation of Gal(Q_P) attached to a weight-k modular form with a_p = 0” and the global one which expresses the condition “psi is the restriction to Gal(Q_p) of representation of Gal(Q) unramified away from some specified set of primes.”  So psi lies in the intersection of the 3-dimensional locus and the 2-dimensional locus in 5-space, and the miracle is that you can prove this intersection is transverse, which means it consists of a finite set of points, and what’s more, it is a set of points whose cardinality you can explicitly bound!

If this sounds familiar, it’s because it’s just like Chabauty.  There, you have a curve C and its Jacobian J.  The analogue of R_loc is J(Q_p), or rather let’s say a neighborhood of the identity in J(Q_p) which looks like affine space Q_p^g.

The analogue of R_glob is (the p-adic closure of) J(Q), which is a proper subspace of dimension r, where r is the rank of J(Q), something you can compute or at least bound by Galois cohomology over Q.  (Of course it can’t be a proper subspace of dimension r if r >= g, which is why Chabauty doesn’t work in that case!)

The analogue of Z is C(Q_p); this is something defined purely p-adically, a locus you could talk about even if you had no idea your C/Q_p were secretly the local manifestation of a curve over Q.

And any rational point of C(Q), considered as a point in J(Q_p), has to lie in both C(Q_p) and J(Q), whose dimensions 1 and at most g-1, and once again the key technical tool is that this intersection can be shown to be transverse, whence finite, so C(Q) is finite and you have Mordell’s conjecture in the case r < g.  And, as Coleman observed decades after Chabauty, this method even allows you to get an explicit bound on the number of points of C(Q), though not an effective way to compute them.

I think this is a very cool confluence indeed!  In the last ten years we've seen a huge amount of work refining Chabauty; Matt Baker discusses some of it on his blog, and then there’s the whole nonabelian Chabauty direction launched by Minhyong Kim and pushed forward by Jen Balakrishnan and Netan Dogra and many others.  Are there other situations in which we can get meaningful results from “deformation-theoretic Chabauty,” and are the new technical advances in Chabauty methods relevant in this context?

## Wanlin Li, “Vanishing of hyperelliptic L-functions at the central point”

My Ph.D. student Wanlin Li has posted her first paper!  And it’s very cool.  Here’s the idea.  If chi is a real quadratic Dirichlet character, there’s no reason the special value L(1/2,chi) should vanish; the functional equation doesn’t enforce it, there’s no group whose rank is supposed to be the order of vanishing, etc.  And there’s an old conjecture of Chowla which says the special value never vanishes.  On the very useful principle that what needn’t happen doesn’t happen.

Alexandra Florea (last seen on the blog here)  gave a great seminar here last year about quadratic L-functions over function fields, which gave Wanlin the idea of thinking about Chowla’s conjecture in that setting.  And something interesting developed — it turns out that Chowla’s conjecture is totally false!  OK, well, maybe not totally false.  Let’s put it this way.  If you count quadratic extensions of F_q(t) up to conductor N, Wanlin shows that at least c N^a of the corresponding L-functions vanish at the center of the critical strip.  The exponent a is either 1/2,1/3, or 1/5, depending on q.  But it is never 1.  Which is to say that Wanlin’s theorem leaves open the possibility that o(N) of the first N hyperelliptic L-functions vanishes at the critical point.  In other words, a density form of Chowla’s conjecture over function fields might still be true — in fact, I’d guess it probably is.

The main idea is to use some algebraic geometry.  To say an L-function vanishes at 1/2 is to say some Frobenius eigenvalue which has to have absolute value q^{1/2} is actually equal to q^{1/2}.  In turn, this is telling you that the hyperelliptic curve over F_q whose L-function you’re studying has a map to some fixed elliptic curve.  Well, that’s something you can make happen by physically writing down equations!  Of course you also need a lower bound for the number of distinct quadratic extensions of F_q(t) that arise this way; this is the most delicate part.

I think it’s very interesting to wonder what the truth of the matter is.  I hope I’ll be back in a few months to tell you what new things Wanlin has discovered about it!

## Rational points on solvable curves over Q via non-abelian Chabauty (with Daniel Hast)

New paper up!  With my Ph.D. student Daniel Hast (last seen on the blog here.)

We prove that hyperelliptic curves over Q of genus at least 2 have only finitely many rational points.  Actually, we prove this for a more general class of high-genus curves over Q, including all solvable covers of P^1.

But wait, don’t we already know that, by Faltings?  Of course we do.  So the point of the paper is to show that you can get this finiteness in a different way, via the non-abelian Chabauty method pioneered by Kim.  And I think it seems possible in principle to get Faltings for all curves over Q this way; though I don’t know how to do it.

## Multiple height zeta functions?

Idle speculation ensues.

Let X be a projective variety over a global field K, which is Fano — that is, its anticanonical bundle is ample.  Then we expect, and in lots of cases know, that X has lots of rational points over K.  We can put these points together into a height zeta function $\zeta_X(s) = \sum_{x \in X(K)} H(x)^{-s}$

where H(x) is the height of x with respect to the given projective embedding.  The height zeta function organizes information about the distribution of the rational points of X, and which in favorable circumstances (e.g. if X is a homogeneous space) has the handsome analytic properties we have come to expect from something called a zeta function.  (Nice survey by Chambert-Loir.)

What if X is a variety with two (or more) natural ample line bundles, e.g. a variety that sits inside P^m x P^n?  Then there are two natural height functions H_1 and H_2 on X(K), and we can form a “multiple height zeta function” $\zeta_X(s,t) = \sum_{x \in X(K)} H_1(x)^{-s} H_2(x)^{-t}$

There is a whole story of “multiple Dirichlet series” which studies functions like $\sum_{m,n} (\frac{m}{n}) m^{-s} n^{-t}$

where $(\frac{m}{n})$ denotes the Legendre symbol.  These often have interesting analytic properties that you wouldn’t see if you fixed one variable and let the other move; for instance, they sometimes have finite groups of functional equations that commingle the s and the t!

So I just wonder:  are there situations where the multiple height zeta function is an “analytically interesting” multiple Dirichlet series?

Here’s a case to consider:  what if X is the subvariety of P^2 x P^2 cut out by the equation $x_0 y_0 + x_1 y_1 + x_2 y_2 = 0?$

This has something to do with Eisenstein series on GL_3 but I am a bit confused about what exactly to say.

## Fox-Neuwirth-Fuks cells, quantum shuffle algebras, and Malle’s conjecture for function fields

I’ve gotten behind on blogging about preprints!  Let me tell you about a new one I’m really happy with, joint with TriThang Tran and Craig Westerland, which we posted a few months ago.

Malle’s conjecture concerns the number of number fields with fixed Galois group and bounded discriminant, a question I’ve been interested in for many years now.  We recall how it goes.

Let K be a global field — that is, a number field or the function field of a curve over a finite field.  Any degree-n extension L/K (here L could be a field or just an etale algebra over K — hold that thought) gives you a homomorphism from Gal(K) to S_n, whose image we call, in a slight abuse of notation, the Galois group of L/K.

Let G be a transitive subgroup of S_n, and let N(G,K,X) be the number of degree-n extensions of K whose Galois group is G and whose discriminant has norm at most X.  Every permutation g in G has an index, which is just n – the number of orbits of g.  So the permutations of index 1 are the transpositions, those of index 2 are the three-cycles and the double-flips, etc.  We denote by a(G) the reciprocal of the minimal index of any element of G.  In particular, a(G) is at most 1, and is equal to 1 if and only if G contains a transposition.

(Wait, doesn’t a transitive subgroup of S_n with a transposition have to be the whole group?  No, that’s only for primitive permutation groups.  D_4 is a thing!)

Malle’s conjecture says that, for every $\epsilon > 0$, there are constants $c,c_\epsilon$ such that $c X^{a(G)} < N(G,K,X) < c_\epsilon X^{a(G)+\epsilon}$

We don’t know much about this.  It’s easy for G = S_2.  A theorem of Davenport-Heilbronn (K=Q) and Datskovsky-Wright (general case) proves it for G = S_3.  Results of Bhargava handle S_4 and S_5, Wright proved it for abelian G.  I kind of think this new theorem of Alex Smith implies for K=Q and every dihedral G of 2-power order?  Anyway:  we don’t know much.  S_6?  No idea.  The best upper bounds for general n are still the ones I proved with Venkatesh a long time ago, and are very much weaker than what Malle predicts.

Malle’s conjecture fans will point out that this is only the weak form of Malle’s conjecture; the strong form doesn’t settle for an unspecified $X^\epsilon$, but specifies an asymptotic $X^a (log X)^b$.   This conjecture has the slight defect that it’s wrong sometimes; my student Seyfi Turkelli wrote a nice paper which I think resolves this problem, but the revised version of the conjecture is a bit messy to state.

Anyway, here’s the new theorem:

Theorem (E-Tran-Westerland):  Let G be a transitive subgroup of S_n.  Then for all q sufficiently large relative to G, there is a constant $c_\epsilon$ such that $N(G,\mathbf{F}_q(t),X) < c_\epsilon X^{a(G)+\epsilon}$

for all X>0.

In other words:

The upper bound in the weak Malle conjecture is true for rational function fields.

1.  We are still trying to fix the mistake in our 2012 paper about stable cohomology of Hurwitz spaces.  Craig and I discussed what seemed like a promising strategy for this in the summer of 2015.  It didn’t work.  That result is still unproved.  But the strategy developed into this paper, which proves a different and in some respects stronger theorem!  So … keep trying to fix your mistakes, I guess?  There might be payoffs you don’t expect.
2. We can actually bound that $X^\epsilon$ is actually a power of log, but not the one predicted by Malle.
3. Is there any chance of getting the strong Malle conjecture?  No, and I’ll explain why.  Consider the case G=S_4.  Then a(G) = 1, and in this case the strong Malle’s conjecture predicts N(S_4,K,X) is on order X, not just X^{1+eps}.   But our method doesn’t really distinguish between quartic fields and other kinds of quartic etale algebras.  So it’s going to count all algebras L_1 x L_2, where L_1 and L_2 are quadratic fields with discriminants X_1 and X_2 respectively, with X_1 X_2 < X.  We already know there’s approximately one quadratic field per discriminant, on average, so the number of such algebras is about the number of pairs (X_1, X_2) with X_1 X_2 < X, which is about X log X.  So there’s no way around it:  our method is only going to touch weak Malle.  Note, by the way, that for quartic extensions, the strong Malle conjecture was proved by Bhargava, and he observes the same phenomenon:

…inherent in the zeta function is a sum over all etale extensions” of Q, including the “reducible” extensions that correspond to direct sums of quadratic extensions. These reducible quartic extensions far outnumber the irreducible ones; indeed, the number of reducible quartic extensions of absolute discriminant at most X is asymptotic to X log X, while we show that the number of quartic field extensions of absolute discriminant at most X is only O(X).

4.  I think there is, on the other hand, a chance of getting rid of the “q sufficiently large relative to G” condition and proving something for a fixed F_q(t) and all finite groups G.

OK, so how did we prove this?

## Prime subset sums

Efrat Bank‘s interesting number theory seminar here before break was about sums of arithmetic functions on short intervals in function fields.  As I was saying when I blogged about Hast and Matei’s paper, a short interval in F_q[t] means:  the set of monic degree-n polynomials P such that

deg(P-P_0) < h

for some monic degree-n P_0 and some small h.  Bank sets this up even more generally, defining an interval in the space V of global sections of a line bundle on an arbitrary curve over F_q.  In Bank’s case, by contrast with the number field case, an interval is an affine linear subspace of some ambient vector space of forms.  This leads one to wonder:  what’s special about these specific affine spaces?  What about general spaces?

And then one wonders:  well, what classical question over Z does this correspond to?  So here it is:  except I’m not sure this is a classical question, though it sort of seems like it must be.

Question:  Let c > 1 be a constant.  Let A be a set of integers with |A| = n and max(A) < c^n.  Let S be the (multi)set of sums of subsets of A, so |S| = 2^n.  What can we say about the number of primes in S?  (Update:  as Terry points out in comments, I need some kind of coprimality assumption; at the very least we should ask that there’s no prime factor common to everything in A.)

I’d like to say that S is kind of a “generalized interval” — if A is the first n powers of 2, it is literally an interval.  One can also ask about other arithmetic functions:  how big can the average of Mobius be over S, for instance?  Note that the condition on max(S) is important:   if you let S get as big as you want, you can make S have no primes or you can make S be half prime (thanks to Ben Green for pointing this out to me.)  The condition on max(S) can be thought of as analogous to requiring that an interval containing N has size at least some fixed power of N, a good idea if you want to average arithmetic functions.

## “On l-torsion in class groups of number fields” (with L. Pierce, M.M. Wood)

New paper up with Lillian Pierce and Melanie Matchett Wood!

Here’s the deal.  We know a number field K of discriminant D_K has class group of size bounded above by roughly D_K^{1/2}.  On the other hand, if we fix a prime l, the l-torsion in the class group ought to be a lot smaller.  Conjectures of Cohen-Lenstra type predict that the average size of the l-torsion in the class group of D_K, as K ranges over a “reasonable family” of algebraic number fields, should be constant.  Very seldom do we actually know anything like this; we just have sporadic special cases, like the Davenport-Heilbronn theorem, which tells us that the 3-torsion in the class group of a random quadratic field is indeed constant on average.

But even though we don’t know what’s true on average, why shouldn’t we go ahead and speculate on what’s true universally?  It’s too much to ask that Cl(K)[l] literally be bounded as K varies (at least if you believe even the most modest version of Cohen-Lenstra, which predicts that any value of dim Cl(D_K)[l] appears for a positive proportion of quadratic fields K) but people do think it’s small:

Conjecture:  |Cl(K)[l]| < D_K^ε.

Even beating the trivial bound

|Cl(K)[l]| < |Cl(K)| < D_K^{1/2 + ε}

is not easy.  Lillian was the first to do it for 3-torsion in quadratic fields.  Later, Helfgott-Venkatesh and Venkatesh and I sharpened those bounds.  I hear from Frank Thorne that he, Bhargava, Shankar, Tsimerman and Zhao have a nontrivial bound on 2-torsion for the class group of number fields of any degree.

In the new paper with Pierce and Wood, we prove nontrivial bounds for the average size of the l-torsion in the class group of K, where l is any integer, and K is a random number field of degree at most 5.  These bounds match the conditional bounds Akshay and I get on GRH.  The point, briefly, is this.  To make our argument work, Akshay and I needed GRH in order to guarantee the existence of a lot of small rational primes which split in K.  (In a few cases, like 3-torsion of quadratic fields, we used a “Scholz reflection trick” to get around this necessity.)  At the time, there was no way to guarantee small split primes unconditionally, even on average.  But thanks to the developments of the last decade, we now know a lot more about how to count number fields of small degree, even if we want to do something delicate like keep track of local conditions.  So, for instance, not only can one count quartic fields of discriminant < X, we can count fields which have specified decomposition at any specified finite set of rational primes.  This turns out to be enough — as long as you are super-careful with error terms! — to  allow us to show, unconditionally, that most number fields of discriminant < D have enough small split primes to make the bound on l-torsion go.  Hopefully, the care we took here to get counts with explicit error terms for number fields subject to local conditions will be useful for other applications too.

## 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?