## Y. Zhao and the Roberts conjecture over function fields

Before the developments of the last few years the only thing that was known about the Cohen-Lenstra conjecture was what had already been known before the Cohen-Lenstra conjecture; namely, that the number of cubic fields of discriminant between -X and X could be expressed as

$\frac{1}{3\zeta(3)} X + o(X)$.

It isn’t hard to go back and forth between the count of cubic fields and the average size of the 3-torsion part of the class group of quadratic fields, which gives the connection with Cohen-Lenstra in its usual form.

Anyway, Datskovsky and Wright showed that the asymptotic above holds (for suitable values of 12) over any global field of characteristic at least 5.  That is:  for such a field K, you let N_K(X) be the number of cubic extensions of K whose discriminant has norm at most X; then

$N_K(X) = c_K \zeta_K(3)^{-1} X + o(X)$

for some explicit rational constant $c_K$.

One interesting feature of this theorem is that, if it weren’t a theorem, you might doubt it was true!  Because the agreement with data is pretty poor.  That’s because the convergence to the Davenport-Heilbronn limit is extremely slow; even if you let your discriminant range up to ten million or so, you still see substantially fewer cubic fields than you’re supposed to.

In 2000, David Roberts massively clarified the situation, formulating a conjectural refinement of the Davenport-Heilbronn theorem motivated by the Shintani zeta functions:

$N_{\mathbf{Q}}(X) = (1/3)\zeta(3)^{-1} X + c X^{5/6} + o(X^{5/6})$

with c an explicit (negative) constant.  The secondary term with an exponent very close to 1 explains the slow convergence to the Davenport-Heilbronn estimate.

The Datskovsky-Wright argument works over an arbitrary global field but, like most arguments that work over both number fields and function fields, it is not very geometric.  I asked my Ph.D. student Yongqiang Zhao, who’s finishing this year, to revisit the question of counting cubic extensions of a function field F_q(t) from a more geometric point of view to see if he could get results towards the Roberts conjecture.  And he did!  Which is what I want to tell you about.

But while Zhao was writing his thesis, there was a big development — the Roberts conjecture was proved.  Not only that — it was proved twice!  Once by Bhargava, Shankar, and Tsimerman, and once by Thorne and Taniguchi, independently, simultaneously, and using very different methods.  It is certainly plausible that these methods can give the Roberts conjecture over function fields, but at the moment, they don’t.

Neither does Zhao, yet — but he’s almost there, getting

$N_K(T) = \zeta_K(3)^{-1} X + O(X^{5/6 + \epsilon})$

for all rational function fields K = F_q(t) of characteristic at least 5.  And his approach illuminates the geometry of the situation in a very beautiful way, which I think sheds light on how things work in the number field case.

Geometrically speaking, to count cubic extensions of F_q(t) is to count trigonal curves over F_q.  And the moduli space of trigonal curves has a classical unirational parametrization, which I learned from Mike Roth many years ago:  given a trigonal curve Y, you push forward the structure sheaf along the degree-3 map to P^1, yielding a rank-3 vector bundle on P^1; you mod out by the natural copy of the structure sheaf; and you end up with a rank-2 vector bundle W on P^1, whose projectivization is a rational surface in which Y embeds.  This rational surface is a Hirzebruch surface F_k, where k is an integer determined by the isomorphism class of the vector bundle W.  (This story is the geometric version of the Delone-Fadeev parametrization of cubic rings by binary cubic forms.)

This point of view replaces a problem of counting isomorphism classes of curves (hard!) with a problem of counting divisors in surfaces (not easy, but easier.)  It’s not hard to figure out what linear system on F_k contains Y.  Counting divisors in a linear system is nothing but a dimension count, but you have to be careful — in this problem, you only want to count smooth members.  That’s a substantially more delicate problem.  Counting all the divisors is more or less the problem of counting all cubic rings; that problem, as the number theorists have long known, is much easier than the problem of counting just the maximal orders in cubic fields.

Already, the geometric meaning of the negative secondary term becomes quite clear; it turns out that when k is big enough (i.e. if the Hirzebruch surface is twisty enough) then the corresponding linear system has no smooth, or even irreducible, members!  So what “ought” to be a sum over all k is rudely truncated; and it turns out that the sum over larger k that “should have been there” is on order X^{5/6}.

So how do you count the smooth members of a linear system?  When the linear system is highly ample, this is precisely the subject of Poonen’s well-known “Bertini theorem over finite fields.”  But the trigonal linear systems aren’t like that; they’re only “semi-ample,” because their intersection with the fiber of projection F_k -> P^1 is fixed at 3.  Zhao shows that, just as in Poonen’s case, the probability that a member of such a system is smooth converges to a limit as the linear system gets more complicated; only this limit is computed, not as a product over points P of the probability D is smooth at P, but rather a product over fibers F of the probability that D is smooth along F.  (This same insight, arrived at independently, is central to the paper of Erman and Wood I mentioned last week.)

This alone is enough for Zhao to get a version of Davenport-Heilbronn over F_q(t) with error term O(X^{7/8}), better than anything that was known for number fields prior to last year.  How he gets even closer to Roberts is too involved to go into on the blog, but it’s the best part, and it’s where the algebraic geometry really starts; the main idea is a very careful analysis of what happens when you take a singular curve on a Hirzebruch surface and start carrying out elementary transforms at the singular points, making your curve more smooth but also changing which Hirzebruch surface it’s on!

To what extent is Zhao’s method analogous to the existing proofs of the Roberts conjecture over Q?  I’m not sure; though Zhao, together with the five authors of the two papers I mentioned, spent a week huddling at AIM thinking about this, and they can comment if they want.

I’ll just keep saying what I always say:  if a problem in arithmetic statistics over Q is interesting, there is almost certainly interesting algebraic geometry in the analogous problem over F_q(t), and the algebraic geometry is liable in turn to offer some insights into the original question.

## Bhargava and Satriano on Galois closures of rings

Manjul Bhargava and Matt Satriano (starting a postdoc at Michigan this fall) posted a nice paper on the arXIv, “On a notion of “Galois closure” for extensions of rings.” The motivation for this work (I’m guessing) comes from Bhargava’s work on parametrizations of number fields.  Bhargava needs to generalize many classical objects of algebraic number theory from the setting of fields to rings.  For instance, the sextic resolvent of a degree 5 polynomial f(x) (developed by Lagrange, Malfatti, and Vandermonde in the late 18th century) is a degree 6 polynomial g(x) which has a rational root if and only if the equation f(x)=0 is solvable in radicals.

In modern Galois-theoretic language, we would describe the sextic resolvent as follows.  If K/Q is the Galois closure of the field generated by a root of f, then the Galois group Gal(K/Q) acts on the 5 roots of f, and is thus identified with a subgroup of S_5.  On the other hand, S_5 has an index-6 subgroup H, the normalizer of a 5-cycle.  So the action of Gal(K/Q) on S_5 / H gives rise to an extension of Q of degree 6 (maybe a field, maybe a product of fields) and this is Q[x] / g(x).

So far, so good.  But Bhargava needs to define a sextic resolvent which is a rank-6 free Z-module, not a 6-dimensional Q-vector space.  The sticking point is the notion of “Galois closure.”  What is the Galois closure of a rank-5 algebra over Z?  Or, for that matter, over a general commutative ring?

Bhargava gets around this question in his paper on quintic fields by using a concrete construction particular to the case of quintics.  But in the new paper, he and Satriano propose a very nice (“very nice” means “functorial”) completely general construction of a “Galois closure” G(A/B) for any extension of rings A/B such that A is a locally free B-module of rank n.  G(A/B) is a locally free B-module, endowed with an S_n-action, as you might want, and it agrees with the usual definition when A/B is an extension of fields.

But there are surprises — for instance, the Galois closure of the rank 4 algebra C[x,y,z]/(x,y,z)^2 over C is 32-dimensional!  In fact, the authors show that there is no definition of Galois closure which is functorial and for which G(A/B) always has the “expected” rank n! over B.  This might explain why no one has written down this definition before, and I think it is what gives the paper a sort of offbeat charm.  It illustrates a useful point:  you’ve got to know when it’s time to mulch an axiom.

## Happy birthday, Dick Gross

Just returned from Dick Gross’s 60th birthday conference, which functioned as a sort of gathering of the tribe for every number theorist who’s ever passed through Harvard, and a few more besides.  A few highlights (not to slight any other of the interesting talks):

• Curt McMullen talked about Salem numbers and the topological entropy of automorphisms of algebraic surfaces (essentially the material discussed in his 2007 Arbeitstagung writeup.)  In particular, he discussed the fact that the logarithm of Lehmer’s number — conjecturally the “simplest” algebraic integer — is in fact the smallest possible positive entropy for an automorphism of a compact complex surface.  Here’s a question that occurred to me after his talk.  If f is a Cremona transformation, i.e. a birational automorphism of P^2, then there’s a way to define the “algebraic entropy” of f, as follows:   the nth iterate of f is given by two rational functions (R_n(x,y),S_n(x,y)), you let d_n be the maximal degree of R_n and S_n, and you define the entropy to be the limit of (1/n) log d_n.  Question:  do we know how to classify the Cremona transformations with zero entropy?  The elements of PGL_3 are in here, as are the finite-order Cremona transformations (which are themselves no joke to classify, see e.g. work of Dolgachev.)  Are there others?
• Serre spoke about characters of groups taking few values, or taking the value 0 quite a lot — this comes up when you want, e.g., to be sure that two varieties have the same number of points over F_p for all but finitely many p, supposing that they have the same number of points for 99.99% of all p.  The talk included the amusing fact that a character taking only the values -1,0,1 is either constant or a quadratic character.  (But, Serre said, there are lots of characters taking only the values 0,3 — what are they, I wonder?)
• Bhargava talked about his new results with Arul Shankar on average sizes of 2-Selmer groups.  It’s quite nice — at this point, the machine, once restricted to counting orbits of groups acting on the integral points of prehomogenous vector spaces, is far more general:  it seems that the group of people around Manjul is getting a pretty good grasp on the general problem of counting orbits of bounded height of the action of G(Z) on V(Z), where G is a group over Z (even a non-reductive group!) and V is some affine space on which G acts.  With the general counting machine in place, the question is:  how to interpret these orbits?  Manjul showed a list of 70 representations to which the current version of the orbit-counting machine applies; each one, hopefully, corresponds to some interesting arithmetic enumeration problem.  It must be nice to know what your next 70 Ph.D. students are going to do…

Dick has a lot of friends — the open mike at the banquet lasted an hour and a half!  My own banquet story was from my college years at Harvard, where Dick was my first-year advisor.  One time I asked him, in innocence, whether he and Mazur had been in graduate school together.  He fixed me with a very stern look.

“Jordan,” he said, “as you can see, I am a very old man.  But I am not as old as Barry Mazur.

## Turkelli on Hurwitz spaces and Malle’s conjecture

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

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

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

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

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

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

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

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