Tag Archives: batyrev-manin

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.

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Bourqui on spaces of rational curves and motivic Batyrev-Manin

David Bourqui just posted a really nice paper, “Asymptotic Behavior of Rational Curves,” notes from a lecture series he gave last summer at the Institut Fourier at Grenoble.  I’ll try to sum up here what it’s about and hopefully entice people to have a look!

Let N_X(B) net of rational points of height at most B on a Fano variety X endowed with an ample line bundle L over a global field K.  By now we are used to the idea (summed up in greatest generality by the Baytrev-Manin conjecture, refined by Peyre) that N_X(B) satisfies an asymptotic regularity — it approaches

c B^a (log B)^{b-1}

for some constants a and b (which are integers) and c (a real number.)  Batyrev and Manin gave predictions for a and b, Peyre pinned down c.  To fix ideas, let’s suppose that the projective embedding of X is given by the anticanonical divisor.  Then a is 1 and b is the Picard rank of X (over K.)  So for instance when X = P^1, you can see immediately that the number of points of height B should be linear in B, and that’s true (remembering that the canonical height is the square of the usual Weil height on P^1.)

Now these conjectures are not exactly right.  There is the problem of accumulating subvarieties, like the lines on a cubic surface, which have way too many rational points; you have to strip these out before you can expect to get down to the expected asymptotic.  And there are more subtle counterexamples, like the one produced by Batyrev and Tschinkel, where the number of rational points is too high by some power of log B.  But the conjecture has been proved for many classes of varieties (toric, homogeneous, very low degree relative to dimension…)

Bourqui’s approach starts from the consideration of these conjectures over the global function field K = k(t), where k = F_q is a finite field.  For simplicity let’s take X to be defined over the constant field k.  Now N_K(B) has two meanings.  You can think of it as a set of K-points of X of bounded height — or you can think it as the number of k-points on the space C_d(X) of degree-d rational curves on X, where q^d = B.  The Batyrev-Manin conjecture, as we phrased it here, is about the first interpretation.  But you can also read it as a statement about the varieties C_d, and it turns out what it says is that

|C_d(k)| / |A^(k)|^d

approaches a limit as d goes to infinity.

Doesn’t this seem a rather astonishing claim at first glance?  These are higher and higher-dimensional varieties over a fixed finite field; the Weil conjectures offer us no useful control.  Why shouldn’t their point-counts fluctuate wildly?

To get some idea of what might be gone, think of a family of varieties that’s easily seen to display this behavior:  the projective spaces P^d.  Evidently,

\lim_{d \rightarrow \infty} (|\mathbf{P}^d(k)| / |\mathbf{A}^1(k)|^d) = (1-1/q)^{-1}.

Why is there a limit?  I can think of two reasons.

First reason:  projective spaces have stable cohomology;  the compactly supported cohomology has one dimension in degrees 2d,2d-2, … 0 and is empty in the odd degrees, and all the cohomology is of Tate type.  Note that (apart from the location of the top cohomology group) this description is independent of d, and it follows that the point count provided by the Lefschetz trace formula is (apart from a prepended power of q) independent of d as well.

Second reason:  Forget about finite fields — the expression

\lim_{d \rightarrow \infty} ([\mathbf{P}^d] / [\mathbf{A}^1]^d) = (1-1/[A^1])^{-1}

remains true motivically (in the suitably completed version of the Grothendieck ring of varieties.)  To be fair, this alone doesn’t quite imply that the point-counting limits hold.  (For instance, the sequence $1 + latex 2^{2^d} [\mathbf{A}^1]^{-d}$ converges motivically to 1, while none of its point-counts converge)  But the motivic convergence is highly suggestive.

My own work in this circle of ideas is mostly concerned with stable cohomology.  What Bourqui is interested in, on the other hand, is whether one has a motivic Batyrev-Manin conjecture; is it the case that

[C_d] / [A^1]^d

approaches a limit in d, and what is this limit?  (This is question 1.11.2 in Bourqui’s paper — to be precise, Bourqui asks for something more precise where one breaks up C_d according to the numerical equivalence class of the rational curve.)  Bourqui proves this is indeed the case when X is a smooth projective toric variety over a field of characteristic 0.  This is by no means a straightforward imitation of the proof of Batyrev-Manin for toric varieties over global fields:  proving motivic identities is hard!

In the case of toric varieties, by the way, both routes to Batyrev-Manin are available; the fact that spaces of rational curves on toric varieties have stable cohomology was proved by Martin Guest.  Guest shows that the cohomology stabilization holds for all smooth projective toric varieties and some of the singular ones as well — the main tool is a diffeomorphism between this space of rational curves and a certain kind of decorated configuration space on the sphere.  I wonder, is Guest’s configuration space implicitly present in Bourqui’s proof?

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