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.

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8 thoughts on “Y. Zhao and the Roberts conjecture over function fields

  1. Tom Leinster says:

    Sorry to leave a superficial comment on a serious post, but perhaps you’re more likely than most to have thoughts on the following question:

    What’s with mathematicians’ habit of referring people by first initial plus surname, as in “J. Smith” or “Y. Zhao”?

    No one else seems to do it. You seldom see “B. Obama” or “J. Austen”. “Barack Obama” or “Jane Austen”, yes. “Obama” and “Austen” are normal too. But “B. Obama” is just weird. So why do mathematicians do it?

  2. Jason Starr says:

    Tiny typo correction: The sheaf you quotient by is O, not O(1). Best, Jason

  3. JSE says:

    Thanks, fixed!

  4. NDE says:

    @T.Leinster: this convention might come from the common bibliographical style of listing references by author(s) and initial(s); if the citation gives “M. Bhargava, A. Shankar, and J. Tsimerman” then one might tend to refer to the authors this way. It’s also often the minimal information needed to specify a mathematician uniquely — which may be why we still write “Cohen-Lenstra” rather than “H.Cohen-H.Lenstra”: alone “Cohen” and “Lenstra” are ambiguous, but in combination there’s a strongly dominant choice.

  5. Frank says:

    So there’s not only “ample” and “very ample”, but also “highly ample” and “semi-ample”? Geesh. Just when I thought I was beginning to get the hang of algebraic geometry vocabulary.

    Seriously though, at this AIM workshop Yongqiang stole the show. I only know a little bit of algebraic geometry, and Yongqiang’s work is one of the factors motivating me to learn more.

    – To oversimplify (in particular to ignore the smoothness issue), to risk saying something wrong, to not address the more technically challenging parts of Yongqiang’s work, and to explain stuff which Jordan probably already knows:

    – You count cubic fields by counting polynomials f(x) = ax^3 + bx^2 + cx + d, such that the maximal order is Z[x]/(f(x)). This is a lattice point counting problem; one counts quadruples (a, b, c, d) subject to a bound on the discriminant, and also one has to represent each cubic field by only one polynomial. (More precisely, one uses the Delone-Faddeev correspondence, and counts GL(2, Z)-orbits of integral binary cubic forms.)

    If a = 1, then your cubic field is monogenic. It is believed that there are asymptotically X^{5/6} cubic fields of discriminant < X that are monogenic, which is in turn related to the conjecture that there are asymptotically X^{5/6} elliptic curves of conductor 1 this doesn’t mean your cubic field isn’t monogenic. Indeed, if I recall correctly what Arul Shankar explained to me once, finding good criteria for non-monogenicity is quite difficult and involves solving some kind of Thue equation (which we can’t really do right now).

    As Jordan explained in a different blog post a long time ago, BST’s geometric approach to counting cubic fields involves “slicing” the fundamental domain and counting quadruples (a, b, c, d) separately for each a. And the reason for the secondary term, roughly speaking, is that although a can be greater than 1, it can’t be less than 1. In other words, a cubic field cannot live too far into the cusp, and this reduces the volume computation by the secondary term.

    (Note: what is in BST’s paper now is not a straight-out volume computation, there is a little bit of analytic trickery. Arul explained to me that the first version of their paper *did* feature a straight volume computation but that the error terms were not good enough.)

    What I gathered from Yongqiang’s explanation at AIM is that having the Maroni invariant k be too large is like having a less than 1. And that counting cubic function fields by their Maroni invariant is like BST’s counting cubic fields by their first coefficient.

    – One interesting note: the paper of mine and Takashi’s you link to is not the one proving Roberts’ conjecture! But I take this as a Freudian slip: I am grateful to have our followup work advertised, and our paper illustrates another reason that we should all be interested in Yongqiang’s work.

    Takashi and I study the counting function of S_3 sextic fields, i.e., splitting fields of non-Galois cubic fields. This is the same as counting non-Galois cubic fields, but with a different weighting.

    All of this was previously observed by Bhargava and Wood, and separately by Belabas and Fouvry, and we extend their work to obtain a power saving error term. But the *really* interesting part is the numerical data at the end. There is a main term, a secondary term, … and… we still haven’t told the full story. There appears to be at least a third term.

    What is it? Takashi and I tried to explain it, and we got nowhere using zeta functions. I don’t see a good way to apply a geometry of numbers argument. But Yongqiang’s approach seems like it might have some power to explain our weird data. This is just one reason I’m eager to read the final version of his paper.

  6. Ben Wieland says:

    There are two things you refer back to without having first mentioned them. One is the Davenport-Heilbronn, which context suggests is the theorem in the first paragraph. The other is the number 12, as if it appeared in the statement of D-H, but it does not in your formulation. What’s up with 12?

  7. Frank says:

    @Ben Wieland:

    Davenport-Heilbronn is indeed the theorem stated in the first paragraph. Here 3 is one suitable value of 12.

    (If you want to count cubic fields with discriminant in [0, X] rather than [-X, X], replace 3 with 12.)

  8. harrison says:

    I will point out that Y. Zhao isn’t exactly uniquely specifying either, although I guess maybe it is within algebraic geometry.

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