## An incidence conjecture of Bourgain over fields of positive characteristic (with Hablicsek)

Marci Hablicsek (a finishing Ph.D. student at UW) and I recently posted a new preprint, “An incidence conjecture of Bourgain over fields of finite characteristic.”

The theme of the paper is a beautiful theorem of Larry Guth and Nets Katz, one of the early successes of Dvir’s “polynomial method.”  They proved a conjecture of Bourgain:

Given a set S of points in R^3, and a set of N^2 lines such that

• No more than N lines are contained in any plane;
• Each line contains at least N points of S;

then S has at least cN^3 points.

In other words, the only way for a big family of lines to have lots of multiple intersections is for all those lines to be contained in a plane.  (In the worst case where all the lines are in a plane, the incidences between points and lines are governed by the Szemeredi-Trotter theorem.)

I saw Nets speak about this in Wisconsin, and I was puzzled by the fact that the theorem only applied to fields of characteristic 0, when the proof was entirely algebraic.  But you know the proof must fail somehow in characteristic p, because the statement isn’t true in characteristic p.  For example, over the field k with p^2 elements, one can check that the Heisenberg surface

$X: x - x^p + yz^p - zy^p = 0$

has a set of p^4 lines, no more than p lying on any plane, and such that each line contains at least p^2 elements of X(k).  If the Guth-Katz theorem were true over k, we could take N = p^2 and conclude that |X(k)| is at least p^6.  But in fact, it’s around p^5.

It turns out that there is one little nugget in the proof of Guth-Katz which is not purely algebraic.  Namely:  they show that a lot of the lines are contained in some surface S with the following property;  at every smooth point s of S, the tangent plane to S at s intersects S with multiplicity greater than 2.  They express this in the form of an assertion that a certain curvature form vanishes everywhere.  In characteristic 0, this implies that S is a plane.  But not so in characteristic p!  (As always, the fundamental issue is that a function in characteristic p can have zero derivative without being constant — viz., x^p.)  All of us who did the problems in Hartshorne know about the smooth plane curve over F_3 with every point an inflection point.  Well, there are surfaces like that too (the Heisenberg surface is one such) and the point of the new paper is to deal with them.  In fact, we show that the Guth-Katz theorem is true word for word as long as you prevent lines not only from piling up in planes but also from piling up in these “flexy” surfaces.

It turns out that any such surface must have degree at least p, and this enables us to show that the Guth-Katz theorem is actually true, word for word, over the prime field F_p.

If you like, you can think of this as a strengthening of Dvir’s theorem for the case of F_p^3.  Dvir proves that a set of p^2 lines with no two lines in the same direction fills up a positive-density subset of the whole space.  What we prove is that the p^2 lines don’t have to point in distinct directions; it is enough to impose the weaker condition that no more than p of them lie in any plane; this already implies that the union of the lines has positive density.  Again, this strengthening doesn’t hold for larger finite fields, thanks to the Heisenberg surface and its variants.

This is rather satisfying, in that there are other situations in this area (e.g. sum-product problems) where there are qualitatively different bounds depending on whether the field k in question has nontrivial subfields or not.  But it is hard to see how a purely algebraic argument can “see the difference” between F_p and F_{p^2}.  The argument in this paper shows there’s at least one way this can happen.

Satisfying, also, because it represents an unexpected application for some funky characteristic-p algebraic geometry!  I have certainly never needed to remember that particular Hartshorne problem in my life up to now.

## Superstrong approximation for monodromy groups (and Galois groups?)

Hey, I posted a paper to the arXiv and forgot to blog about it!  The paper is called “Superstrong approximation for monodromy groups” and it roughly represents the lectures I gave at the MSRI workshop last February on “Thin Groups and Superstrong Approximation.”  Hey, as I write this I see that MSRI has put video of these lectures online:

But the survey paper has more idle speculation in it than the lectures, and fewer “um”s, so I recommend text over video in this case!  I mean, if you like idle speculation.  But if you don’t, would you be reading this blog?

I’m going to recount one of the idle speculations here, but first:

What is superstrong approximation?

Let’s say you have a graph on N vertices, regular of degree d.  One basic thing you want to know about the graph is what the connected components are, or at least how many there are.  That seems like a combinatorial question, and it is, but in a sense it is also a spectral question:  the random walk on the graph, thought of as an operator T on the space of functions on the graph, is going to have eigenvalues between [1,-1], and the mutiplicity of 1 is precisely the number of components; the eigenspace consists of the locally constant functions which are constant on connected components.

So being connected means that the second-largest eigenvalue of T is strictly less than 1.  And so you might say a graph is superconnected (with respect to some positive constant x) if the second-largest eigenvalue is at most 1-x.  But we don’t say “superconnected” because we already have a word for this notion; we say the graph has a spectral gap of size x.  Now of course any connected graph has a spectral gap!  But the point is always to talk about families of graphs, typically with d fixed and N growing; we say the family has a spectral gap if, for some positive x, each graph in the family has a spectral gap of size at least x.  (Such a family is also called an expander family, because the random walks on those graphs tend to bust out of any fixed-size region very quickly; the relation between this point of view and the spectral one would be a whole nother post.)

When does life hand you a family of graphs?  OK, here’s the situation — let’s say you’ve got d matrices in SL_n(Z), or some other arithmetic group.  For every prime p, your matrices project to d elements in SL_n(Z/pZ), which produce a Cayley graph X_p, and X_p is connected just when those elements generate SL_n(Z/pZ).  If your original matrices generate SL_n(Z), their reductions mod p generate SL_n(Z/pZ); this is just the (not totally obvious!) fact that SL_n(Z) surjects onto SL_n(Z/pZ).  But more is true; it turns out that if the group Gamma generated by your matrices is Zariski-dense in SL_n, this is already enough to guarantee that X_p is connected for almost all p.  This statement is called strong approximation for Gamma.

But why stop there — we can ask not only whether X_p is connected, but whether it is superconnected!  That is:  does the family of graphs X_p have a spectral gap?  If so, we say Gamma has superstrong approximation, which is now seen to be a kind of quantitative strengthening of strong approximation.

We know much more than we did five years ago about which groups have superstrong approximation, and what the applications are when this is so.  Sarnak’s paper  from the same conference provides a good overview.

Idle speculation:  superstrong approximation for Galois groups

Another way to express superstrong approximation is to say that Gamma has property tau with respect to the congruence quotients SL_n(Z/pZ).

In the survey paper, I wonder whether there is some way to talk about superstrong approximation for Galois groups with bounded ramification.  For instance; let G be the Galois group of the maximal extension of Q which is tamely ramified everywhere, and unramified away from 2,3,5, and 7.  OK, that’s some profinite group.  I don’t know much about it.  By Golod-Shafarevich I could prove it was infinite, unless I couldn’t, in which case I would toss in some more ramified primes until I could.

We could ask something like the following.  Given any finite quotient Q of G, and any two elements of G whose images generated Q, we get a connected Cayley graph of degree 4 on the elements of Q, by means of those two elements and their inverses.  Is there a uniform spectral gap for all those graphs?

I have no real reason to think so.  But remark:  this would imply immediately that every finite-index subgroup of G has finite abelianization, and that’s true.  It would also imply that there are only finitely many n such that G surjects onto S_n, and that might be true.  Reader survey for those who’ve read this far:  do you think there’s a finite set S of primes such that there are tamely ramified S_n-extensions of Q, for n arbitrarily large, unramified outside S?

Acknowledgment:  I was much aided in formulating this question by the comments on the MathOverflow question I asked about it.

## 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.

## FI-modules over Noetherian rings

New paper on the arXiv, joint with Tom Church, Benson Farb, and UW grad student Rohit Nagpal.  In our first paper on FI-modules (which I blogged about earlier this year) a crucial tool was the fact that the category of FI-modules over a field of characteristic 0 is Noetherian; that is, a submodule of a finitely generated FI-module is again finitely generated.  But we didn’t know how to prove this over a more general ring, which limited the application of some of our results.

In the new paper, we show that the category of FI-modules is Noetherian over an arbitrary Noetherian ring.  Sample consequence:  if M is a manifold and Conf^n M is the configuration space of ordered n-tuples of distinct points on M, then we show that

dim_k H_i(Conf^n M, k)

is a polynomial function of n, for all n greater than some threshold.  In our previous paper, we could prove this only when k had characteristic 0; now it works for mod p cohomology as well.  We also discuss some of the results of Andy Putman’s paper on stable cohomology of congruence subgroups — it is a bad thing that I somehow haven’t blogged about this awesome paper! — showing how, at the expense of losing stable ranges, you can remove from his results some of the restrictions on the characteristic of the coefficient field.

Philosophically, this paper makes me happy because it brings me closer to what I wanted to do in the first place — talk about the representation theory of symmetric groups “for general n” without giving names to representations.  In characteristic 0, this desire might seem a bit perverse, given the rich and beautiful story of the bijection between irreducible representations and partitions.  But in characteristic p, the representation theory of S_n is much harder to describe.  So it is pleasing to be able to talk, in a principled way, about what one might call “representation stability” in this context; I think that when V is a finitely generated FI-module over a finite field k it makes sense to say that the representations V_n of k[S_n] are “the same” for n large enough, even though I don’t have as nice a description of the isomorphism classes of such representations.

## FI-modules and representation stability, III

So how does this paper work?  The main idea is quite simple.  Let’s come back to the example of V_n = H^i(Conf^n M,Q), with i fixed and n ranging over nonnegative integers.  Then we have a sequence of vector spaces

V_0, V_1, V_2, …

But more than a sequence.  You have a map Conf^{n+1} -> Conf^n which is “forget the n+1 st point” — which functorially hands you a map V_n -> V_{n+1}.  So you have a diagram

V_0 -> V_1 -> V_2 -> …..

But in fact you have even more than this!  There’s no reason you have to forget just the n+1 st point.  You have tons of maps from Conf^n to Conf^m for all m <= n; one for each m-element subset of 1..n.  And there are lots of natural identifications between the compositions of these maps.  When you keep track of all the maps at your disposal, what you find is that the vector spaces V_n have a very rigid structure.

Definition:  FI is the category of finite sets with injections.  An FI-module over a ring R is a functor from FI to R-modules.

So V is an FI-module over Q!  (The vector space V_n is revealed as the image of the finite set [1..n] under the functor V.) And the main work of our paper is the study of the category of FI-modules, which sheds a great deal of light on representation stability.  For instance, we show that an FI-module over Q yields a representation-stable sequence in Church-Farb’s original sense if and only if it is finitely generated in the natural sense.  Moreover, the category of FI-modules over Q is Noetherian, in the sense that subobjects of finitely generated FI-modules are again finitely generation.  (The Noetherianness was proven independently by Snowden in a different form.)  Theorems like this very easy to show that tons of examples in nature (like the ones in the previous post) yield representation-stable sequences.  The work is all in the definitions and basic properties; once you have that, proving stability in particular examples is often a matter of a few lines.  For instance, you get a fairly instant proof of Murnaghan’s theorem on stability of Kronecker products; from this point of view, this becomes a theorem about the finite generation of a single object in an abelian category, rather than a theorem about a list of coefficients eventually setting down to constancy.

Sometimes there is more structure still.  Suppose, for example, that the manifold M above has nonempty boundary.  Then there are not only maps from Conf^{n+1} to Conf^n, but maps going the other way; you can add a new point in a little neighborhood of a boundary component.  (This is familiar from the configuration space of the complex plane, where you add new points at “the west pole” in the infinite negative real direction.)  These maps don’t quite compose on the nose, but they’re OK up to homotopy, and so the cohomology groups acquire a system of maps going both up and down.  It turns out that the right structure to describe such systems is given by the category of finite sets with partial injections; i.e. a map from A to B is an isomorphism from a subset of A to a subset of B.  We call this category FI#, and we call a functor from FI# to R-modules an FI#-module over R.

When your vector spaces carry an FI#-module structure you can really go to town.  It turns out that all the “eventuallies” disappear; when M is an open manifold, the dimension of H^i(Conf^n M) is a polynomial in n on the nose, for all n.  What’s more, if you want to show finite generation for FI#-modules, it suffices to show that dim V_n is bounded by some polynomial in n.  Once it’s less than a polynomial, it is a polynomial!  This stuff, unlike some other results in our paper, works in any characteristic and in fact is even fine with integral coefficients.

## FI-modules and representation stability, II

Here are some sequences of vector spaces.  In each case, the sequence is indexed by n, and all other variables are understood to be constant.  So suppose V_n is the space

• H^i(Conf^n M, Q) for M a connected oriented manifold of dimension at least 2.
• The (j_1, .. j_r)-multidegree piece of the diagonal coinvariant algebra on r sets of n variables.
• H^i(M_{g,n},Q), the cohomology of the moduli space of curves of genus g with n marked points.
• The tautological subring of the above.
• The space of degree-d polynomials on the rank variety parametrizing nxn matrices of rank at most r.

By a character polynomial we mean a polynomial with integral coefficients in variables X_1, X_2, X_3, … .  We interpret these symbols (and thus character polynomials) as class functions on the symmetric group by S_n by taking

X_i(s) = number of i-cycles in s

for each permutation s.

Then we show that, in each of the examples above, there’s a character polynomial P such that the character of the action of S_n on V_n is given by P, for all sufficiently large n.  This is one way in which one can say that a sequence of representations of larger and larger symmetric groups are “all the same.”  In particular, by plugging in the identity we find that dim V_n is a polynomial in n, for n large enough.

For many of these examples, almost nothing is known about dimensions of individual spaces!  So a strong regularity theorem like this is perhaps surprising.  Even more surprising (to us at any rate) is that theorems like this require only very meager input from whatever context generate the vector spaces.  You get this stability (and many others) almost for free.

More about how it all works tomorrow!

## FI-modules and representation stability, I

Tom ChurchBenson Farb and I have just posted a new paper, “FI-modules:  a new approach to representation stability,” on the arXiv.  This paper has occupied a big chunk of our attention for about a year, so I’m very pleased to be able to release it!

Here is the gist.  Sometimes life hands you a sequence of vector spaces.  Sometimes these vector spaces even come with maps from one to the next.  And when you are very lucky, those maps become isomorphisms far enough along in the sequence; because at that point you can describe the entire picture with a finite amount of information, all the vector spaces after a certain point being canonically the same.  In this case we typically say we have found a stability result for the sequence.

But sometimes life is not so nice.  Say for instance we study the cohomology groups of configuration spaces of points of n distinct ordered points on some nice manifold M.  As one does.  In fact, let’s fix an index i and a coefficient field k and let V_n be the vector space H^i(Conf^n M, k.)

(In the imaginary world where there are people who memorize every word posted on this blog, those people would remember that I also sometimes use Conf^n M to refer to the space parametrizing unordered n-tuples of distinct points.  But now we are ordered.  This is important.)

For instance, you can let M be the complex plane, in which case we’re just computing the cohomology of the pure braid group.  Or, to put it another way, the cohomology of the hyperplane complement you get by deleting the hyperplanes (x_i-x_j) from C^n.

This cohomology was worked out in full by my emeritus colleagues Peter Orlik and Louis Solomon.  But let’s stick to something much easier; what about the H^1?  That’s just generated by the classes of the hyperplanes we cut out, which form a basis for the cohomology group.  And now you see a problem.  If V_n is H^1(Conf^n C, k), then the sequence {V_n} can’t be stable, because the dimensions of the spaces grow with n; to be precise,

dim V_n = (1/2)n(n-1).

But all isn’t lost.  As Tom and Benson explained last year in their much-discussed 2010 paper, “Representation stability and homological stability,” the right way to proceed is to think of V_n not as a mere vector space but as a representation of the symmetric group on n letters, which acts on Conf^n by permuting the n points.  And as representations, the V_n are in a very real sense all the same!  Each one is

“the representation of the symmetric group given by the action on unordered pairs of distinct letters.”

Of course one has to make precise what one means when one says “V_m and V_n are the same symmetric group representation”, when they are after all representations of different groups.  Church and Farb do exactly this, and show that in many examples (including the pure braid group) some naturally occuring sequences do satisfy their condition, which they call “representation stability.”

So what’s in the new paper?  In a sense, we start from the beginning, defining representation stability in a new way (or rather, defining a new thing and showing that it agrees with the Church-Farb definition in cases of interest.)  And this new definition makes everything much cleaner and dramatically expands the range of examples where we can prove stability.  This post is already a little long, so I think I’ll start a new one with a list of examples at the top.

## Random Dieudonne modules, random p-divisible groups, and random curves over finite fields

Bryden Cais, David Zureick-Brown and I have just posted a new paper,  “Random Dieudonne modules, random p-divisible groups, and random curves over finite fields.”

What’s the main idea?  It actually arose from a question David Bryden asked during Derek Garton‘s speciality exam.  We know by now that there is some insight to be gained about studying p-parts of class groups of number fields (the Cohen-Lenstra problem) by thinking about the analogous problem of studying class groups of function fields over F_l, where F_l has characteristic prime to p.

The question David asked was:  well, what about the p-part of the class group of a function field whose characteristic is equal to p?

That’s a different matter altogether.  The p-divisible group attached to the Jacobian of a curve C in characteristic l doesn’t contain very much information;  more or less it’s just a generalized symplectic matrix of rank 2g(C), defined up to conjugacy, and the Cohen-Lenstra heuristics ask this matrix to behave like a random matrix with respect to various natural statistics.

But p-divisible groups in characteristic p are where the fun is!  For instance, you can ask:

What is the probability that a random curve (resp. random hyperelliptic curve, resp. random plane curve, resp. random abelian variety) over F_q is ordinary?

In my view it’s sort of weird that nobody has asked this before!  But as far as I’ve been able to tell, this is the first time the question has been considered.

We generate lots of data, some of which is very illustrative and some of which is (to us) mysterious.  But data alone is not that useful — much better to have a heuristic model with which we can compare the data.  Setting up such a model is the main task of the paper.  Just as a p-divisible group in characteristic l is decribed by a matrix, a p-divisible group in characteristic p is described by its Dieudonné module;  this is just another linear-algebraic gadget, albeit a little more complicated than a matrix.  But it turns out there is a natural “uniform distribution” on isomorphism classes of  Dieudonné modules; we define this, work out its properties, and see what it would say about curves if indeed their Dieudonné modules were “random” in the sense of being drawn from this distribution.

To some extent, the resulting heuristics agree with data.  But in other cases, they don’t.  For instance:  the probability that a hyperelliptic curve of large genus over F_3 is ordinary appears in practice to be very close to 2/3.  But the probability that a smooth plane curve of large genus over F_3 is ordinary seems to be converging to the probability that a random Dieudonné module over F_3 is ordinary, which is

(1-1/3)(1-1/3^3)(1-1/3^5)….. = 0.639….

Why?  What makes hyperelliptic curves over F_3 more often ordinary than their plane curve counterparts?

(Note that the probability of ordinarity, which makes good sense for those who already know Dieudonné modules well, is just the probability that two random maximal isotropic subspaces of a symplectic space over F_q are disjoint.  So some of the computations here are in some sense the “symplectic case” of what Poonen and Rains computed in the orthogonal case.

We compute lots more stuff (distribution of a-numbers, distribution of p-coranks, etc.) and decline to compute a lot more (distribution of Newton polygon, final type…)  Many interesting questions remain!

## 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 $\bar{\mathbf{Q}}$ itself doesn’t have (B):  consider the sequence $2^{1/2}, 2^{1/3}, 2^{1/4}, \ldots$  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!

## 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.)