## 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.)

## Richard Stanley reveals what’s under the turtle

See his preprint, “An application of set theory to cosmology.”

## Modeling lambda-invariants by p-adic random matrices

The paper “Modeling λ-invariants by p-adic random matrices,” with Akshay Venkatesh and Sonal Jain, just got accepted by Comm. Pure. Appl.  Math. But I forgot to blog about it when we finished it!  (I was a little busy at the time with the change in my personal circumstances.)

Anyway, here’s the idea.  As I’ve already discussed here, one heuristic for the Cohen-Lenstra conjectures about the p-rank of the class group of a random quadratic imaginary field K is to view this p-part as the cokernel of g-1, where g is a random generalized symplectic matrix over Z_p.  In the new paper, we apply the same philosophy to the variation of the Iwasawa p-adic λ-invariant.

The p-adic λ-invariant of a number field K is closely related to the p-rank of the class group of K; in fact, Iwasawa theory more or less gets started from the theorem that the p-rank of the class group of $K(\zeta_{p^n})$ is

$\lambda n + \mu p^n + \nu$

for some constants $\lambda, \mu, \nu$ when n is large enough, with $\mu$ expected to be 0 (and proved to be 0 when K is quadratic.)  On the p-adic L-function side, the λ-invariant is (thanks to the main conjecture) related to the order of vanishing of a p-adic L-function.  On the function field side, the whole story is told by the action of Frobenius on the p-torsion of the Jacobian of a curve, which is specified by some generalized symplectic matrix g over F_p.  The p-torsion in the class group is the dimension of the fixed space of g, while the λ-invariant is the dimension of the generalized 1-eigenspace of g, which might be larger.  It’s also in a sense more natural, depending only on the characteristic polynomial of g (which is exactly what the L-function keeps track of.)

So in the paper we do two things.  On the one hand, we study the dimension of the generalized 1-eigenspace of a random generalized symplectic matrix, and from this we derive the following conjecture: for each p > 2 and r >= 0,  the probability that a random quadratic imaginary field K has p-adic λ-invariant r is

$p^{-r} \prod_{t > r} (1-p^{-t})$.

Note that this decreases like $p^{-r}$ with r, while the p-rank of the class group is supposed to be r with probability more like $p^{-r^2}$.  So large λ-invariants should be substantially more common than large p-ranks.

The second part of the paper tests this conjecture numerically, and finds fairly good agreement with the data. A novelty here is that we compute p-adic  λ-invariants of K for small p and large disc(K); previous computational work has held K fixed and considered large p.  It turns out that you can do these computations reasonably efficiently by interpolation; you can compute special values L(s,chi_K) transcendentally for many s; given a bunch of these values, determined to a certain p-adic precision, you can compute the initial coefficients of the p-adic L-function with some controlled p-adic precision as well, and, in particular, you can provably locate the first coefficient which is nonzero mod p.  The location of this coefficient is precisely the λ-invariant.  This method shows that, indeed, large λ-invariants do pop up!  For instance, the 3-adic λ-invariant of $Q(\sqrt{-956238})$ is 14, which I think is a record.

Some questions still floating around:

• Should one expect an upper bound $\lambda \ll_\epsilon D_K^\epsilon$ for each odd p?  Certainly such a bound is widely expected for the p-rank of the class group.
• In the experiments we did, the convergence to the conjectural asymptotic appears to be from below.  For the 3-ranks of class groups of quadratic imaginary fields, this convergence from below was conjectured by Roberts to be explained by a secondary main term with negative coefficient.  Roberts’ conjecture was proved this year — twice!  Bhargava, Shankar, and Tsimerman gave a proof along the lines of Bhargava’s earlier work (involving thoughful decompositions of fundamental domains into manageable regions, and counting lattice points therein) and Thorne and Taniguchi have a proof along more analytic lines, using the Shintani zeta function.  Anyway, one might ask (prematurely, since I have no idea how to prove the main term correct!) whether the apparent convergence from below for the statistics of the λ-invariant is also explained by some kind of negative secondary term.

## Expander graphs, gonality, and variation of Galois representations

Suppose you have a 1-dimensional family of polarized abelian varieties — or, just to make things concrete, an abelian variety A over Q(t) with no isotrivial factor.

You might have some intuition that abelian varieties over Q don’t usually have rational p-torsion points — to make this precise you might ask that A_t[p](Q) be empty for “most” t.

In fact, we prove (among other results of a similar flavor) the following strong version of this statement.  Let d be an integer, K a number field, and A/K(t) an abelian variety.  Then there is a constant p(A,d) such that, for each prime p > p(A,d), there are only finitely many t such that A_t[p] has a point over a degree-d extension of K.

The idea is to study the geometry of the curve U_p parametrizing pairs (t,S) where S is a p-torsion point of A_t.  This curve is a finite cover of the projective line; if you can show it has genus bigger than 1, then you know U_p has only finitely many K-rational points, by Faltings’ theorem.

But we want more — we want to know that U_p has only finitely many points over degree-d extensions of K.  This can fail even for high-genus curves:  for instance, the curve

C:   y^2 = x^100000 + x + 1

has really massive genus, but choosing any rational value of x yields a point on C defined over a quadratic extension of Q.  The problem is that C is hyperelliptic — it has a degree-2 map to the projective line.  More generally, if U_p has a degree-d map to P^1,  then U_p has lots of points over degree-d extensions of K.  In fact, Faltings’ theorem can be leveraged to show that a kind of converse is true.

So the relevant task is to show that U_p admits no map to P^1 of degree less than d; in other words, its gonality is at least d.

Now how do you show a curve has large gonality?  Unlike genus, gonality isn’t a topological invariant; somehow you really have to use the geometry of the curve.  The technique that works here is one we learned from an paper of Abramovich; via a theorem of Li and Yau, you can show that the gonality of U_p is big if you can show that the Laplacian operator on the Riemann surface U_p(C) has a spectral gap.  (Abramovich uses this technique to prove the g=1 version of our theorem:  the gonality of classical modular curves increases with the level.)

We get a grip on this Laplacian by approximating it with something discrete.  Namely:  if U is the open subvariety of P^1 over which A has good reduction, then U_p(C) is an unramified cover of U(C), and can be identified with a finite-index subgroup H_p of the fundamental group G = pi_1(U(C)), which is just a free group on finitely many generators g_1, … g_n.  From this data you can cook up a Cayley-Schreier graph, whose vertices are cosets of H_p in G, and whose edges connect g H with g_i g H.  Thanks to work of Burger, we know that this graph is a good “combinatorial model” of U_p(C); in particular, the Laplacian of U_p(C) has a spectral gap if and only if the adjacency matrix of this Cayley-Schreier graph does.

At this point, we have reduced to a spectral problem having to do with special subgroups of free groups.  And if it were 2009, we would be completely stuck.  But it’s 2010!  And we have at hand a whole spray of brand-new results thanks to Helfgott, Gill, Pyber, Szabo, Breuillard, Green, Tao, and others, which guarantee precisely that Cayley-Schreier graphs of this kind, (corresponding to finite covers of U(C) whose Galois closure has Galois group a perfect linear group over a finite field) have spectral gap; that is, they are expander graphs. (Actually, a slightly weaker condition than spectral gap, which we call esperantism, is all we need.)

Sometimes you think about a problem at just the right time.  We would never have guessed that the burst of progress in sum-product estimates in linear groups would make this the right time to think about Galois representations in 1-dimensional families of abelian varieties, but so it turned out to be.  Our good luck.

## McMullen on dilatation in finite covers

Last year I blogged about a nice paper of Thomas Koberda, which shows that every pseudo-Anosov diffeomorphism of a Riemann surface X acts nontrivially on the homology of some characteristic cover of X with nilpotent Galois group.  (This statement is false with “nilpotent” replaced by “abelian.”)  The paper contains a question which Koberda ascribes to McMullen:

Is the dilatation λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

That question has now been answered by McMullen himself, in the negative, in a preprint released last month.  In fact, he shows that either λ(f) is detected on the homology of a double cover of Σ, or it is not detected by any finite cover at all!

The supremum of the spectral radius of f on the Σ’ is then an invariant of f, which most of the time is strictly bigger than the spectral radius of f on Σ and strictly smaller than λ(f).  Is this invariant interesting?  Are there any circumstances under which it can be computed?

Tagged , ,

## 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.

## 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.

“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]?

## Random pro-p groups, braid groups, and random tame Galois groups

I’ve posted a new paper with Nigel Boston, “Random pro-p groups, braid groups, and random tame Galois groups.”

The paper proposes a kind of “non-abelian Cohen-Lenstra heuristic.”   A typical prediction:  if S is a randomly chosen pair of primes, each of which is congruent to 5 mod 8, and G_S(p) is the Galois group of the maximal pro-2 extension of Q unramified away from S, then G_S(p) is infinite 1/16 of the time.

The usual Cohen-Lenstra conjectures — well, there are a lot of them, but the simplest one asks:  given an odd prime p and a finite abelian p-group A, what is the probability P(A) that a randomly chosen quadratic imaginary field K has a class group whose p-primary part is isomorphic to A?  (Note that the existence of P(A) — which we take to be a limit in X of the corresponding probability as K ranges over quadratic imaginary fields of discriminant at most X — is not at all obvious, and in fact is not known for any p!)

Cohen and Lenstra offered a beautiful conjectural answer to that question:  they suggested that the p-parts of class groups were uniformly distributed among finite abelian p-groups.  And remember — that means that P(A) should be proportional to 1/|Aut(A)|.  (See the end of this post for more on uniform distribution in this categorical setting.)

Later, Friedman and Washington observed that the Cohen-Lenstra conjectures could be arrived at by another means:  if you take K to be the function field of a random hyperelliptic curve X over a finite field instead of a random quadratic imaginary field, then the finite abelian p-group you’re after is just the cokernel of F-1, where F is the matrix corresponding to the action of Frobenius on T_p Jac(X).  If you take the view that F should be a “random” matrix, then you are led to the following question:

Let F be a random element of GL_N(Z_p) in Haar measure:  what is the probability that coker(F-1) is isomorphic to A?

And this probability, it turns out, is precisely the P(A) conjectured by Cohen-Lenstra.

(But now you cry out:  but Frobenius isn’t just any old matrix!  It’s in the generalized symplectic group!  Yes — and Jeff Achter has shown that, at least as far as the probability distribution on A/pA goes, the “right” random matrix model gives you the same answer as the Friedman-Washington quick and dirty model.  Phew.)

Now, in place of a random quadratic imaginary field, pick a prime p and a random set S of g primes, each of which is 1 mod p.  As above, let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S; this is a pro-p group of rank g. What can we say about the probability distribution on G_S(p)?  That is, if G is some pro-p group, can we compute the probability that G_S(p) is isomorphic to G?

Again, there are two approaches.  We could ask that G_S(p) be a “random pro-p group of rank g.”  But this isn’t quite right; G_S(p) has extra structure, imparted to it by the images in G_S(p) of tame inertia at the primes of S.  We define a notion of “pro-p group with inertia data,” and for each pro-p GWID G we guess that the probability that G_S(p) = G is proportional to 1/Aut(G); where Aut(G) refers to the automorphisms of G as GWID, of course.

On the other hand, you could ask what would happen in the function field case if the action of Frobenius on — well, not the Tate module of the Jacobian anymore, but the full pro-p geometric fundamental group of the curve — is “as random as possible.”  (In this case, the group from which Frobenius is drawn isn’t a p-adic symplectic group but Ihara’s “pro-p braid group.”)

And the happy conclusion is that, just as in the Cohen-Lenstra setting, these two heuristic arguments yield the same prediction.  For the relatively few pro-p groups G such that we can compute Pr(G_S(p) = G), our heuristic gives the right answer.  For several more, it gives an answer that seems consistent with numerical experiments.

Maybe it’s correct!