## 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.

## 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.

## Alexandra Florea on the average central value of hyperelliptic L-functions

Alexandra Florea, a student of Soundararajan, has a nice new paper up, which I heard about in a talk by Michael Rubinstein.  She computes the average of

$L(1/2, \chi_f)$

as f ranges over squarefree polynomials of large degree.  If this were the value at 1 instead of the value at 1/2, this would be asking for the average number of points on the Jacobian of a hyperelliptic curve, and I could at least have some idea of where to start (probably with this paper of Erman and Wood.)  And I guess you could probably get a good grasp on moments by imitating Granville-Soundararajan?

But I came here to talk about Florea’s result.  What’s cool about it is that it has the a main term that matches existing conjectures in the number field case, but there is a second main term, whose size is about the cube root of the main term, before you get to fluctuations!

The only similar case I know is Roberts’ conjecture, now a theorem of Bhargava-Shankar-Tsimerman and Thorne-Taniguchi, which finds a similar secondary main term in the asymptotic for counting cubic fields.  And when I say similar I really mean similar — e.g. in both cases the coefficient of the secondary term is some messy thing involving zeta functions evaluated at third-integers.

My student Yongqiang Zhao found a lovely geometric interpretation for the secondary term the Roberts conjecture.  Is there some way to see what Florea’s secondary term “means” geometrically?  Of course I’m stymied here by the fact that I don’t really know how to think about her counting problem geometrically in the first place.

## Yitang Zhang, bounded gaps, primes as random numbers

In Slate today, I have a piece about Yitang Zhang’s amazing proof of the bounded gaps conjecture.  Actually, very little of the article is about Zhang himself or his proof; I wanted instead to explain why mathematicians believed that bounded gaps (or twin primes) was true in the first place, via Cramér’s heuristic that primes behave like random numbers.

And a lot of twin primes is exactly what number theorists expect to find no matter how big the numbers get—not because we think there’s a deep, miraculous structure hidden in the primes, but precisely because we don’t think so. We expect the primes to be tossed around at random like dirt. If the twin primes conjecture were false, that would be a miracle, requiring that some hitherto unknown force be pushing the primes apart.

## 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]?

## Non-simple abelian varieties in a family

Here’s a funny question. Let f in C[x] be a squarefree polynomial of degree at least 6. Let S be the set of complex numbers t such that the Jacobian of the hyperelliptic curve

$y^2 = f(x)(x-t)$

is not simple. Is S always finite? Even more, is there a bound on |S| which doesn’t depend on f, or depends only on the degree of f?

This question comes from the introduction to “Non-simple abelian varieties in a family: geometric and analytic approaches” , a new paper by me, Christian Elsholtz, Chris Hall, and Emmanuel Kowalski. In its original form this was a four-author, six-page paper — fortunately we’ve now added enough material to make the ratio a bit more respectable!

The paper isn’t about complex algebraic geometry at all — it explains how to get bounds on S when f has rational coefficients and t ranges over rational numbers, which is quite a different story. The point of the paper is partly to prove some theorems and partly to make a metamathematical point — that problems of this kind can be approached via either arithmetic geometry or analytic number theory, and that the two approaches have complementary strengths and weaknesses. Bounds from arithmetic geometry are stronger but less uniform; bounds from analytic number theory are weaker but have better uniformity.

Here’s my favorite example of this phenomenon. Let X be a smooth plane curve over Q of degree d at least 4. Then by Faltings’ Theorem we know that X has only finitely many rational points.

On the other hand, a beautiful theorem of Heath-Brown tells us that the number of rational points on X with coordinates of height at most B is at most C B^(2/d), for some constant C depending only on d. At first, this seems to give much less than Faltings. After all, as B gets larger and larger, the upper bound given by Heath-Brown gets arbitrarily large — whereas we know by Faltings that there are only finitely many points on the whole curve, no matter how large we allow the coordinates to be.

But note that the constant in Heath-Brown’s result doesn’t depend on the curve X. It is what’s called a uniform bound. Faltings’ theorem, by contrast, gives an upper bound on the number of points which depends very badly on the choice of X. Depending on what you’re trying to accomplish, you might be willing to sacrifice uniformity to get finiteness — or the reverse. But it’s best to have both options at hand.

Is it possible to have uniformity and finiteness simultaneously? Conjecturally, yes. Caporaso, Harris, and Mazur showed that, conditional on Lang’s conjecture, there is a constant B(g) such that every genus-g curve X/Q has at most B(g) rational points. The Caporaso-Harris-Mazur paper came out when I was in graduate school, and the idea of such a uniform bound was considered so wacky that CHM was thought of as evidence against Lang’s conjecture. Joe Harris used to wander around the department, buttonholing graduate students and encouraging us to cook up examples of genus-g curves with arbitrarily many points, thus disproving Lang. We all tried, and we all failed — as did many more experienced people. And nowadays, the idea that there might be a uniform bound for the number of rational points on a genus-g curve is considered fairly reputable, even among people who have their doubts about Lang’s conjecture. As far as I know, the world record for the number of rational points on a genus-2 curve is 588, due to Kulesz. Can you beat it?