## 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.

## 4 thoughts on “Multiple height zeta functions?”

1. Jason Starr says:

We expect lots of rational points only if various obstructions vanish (e.g., the elementary obstruction of Colliot-Th’el`ene and Sansuc, the Brauer-Manin obstruction, etc.). Of course we always expect a Zariski dense set of rational points after a finite field extension.

2. Jason Starr says:

There is one other observation. If the geometric Picard group has rank r > 1, nonetheless, the Picard group over your global field K can still have rank 1 (so usual zeta function instead of multiple zeta function). In the case of a Fano variety of Picard rank r > 1 over a global function field K = F(C) associated to a curve C, this means that sometimes the K-points of X do not come in “packets” over F-points of the Jacobian of C, but instead they come in packets over F-points of more general Abelian varieties. Yi Zhu formulated carefully the replacement of the Jacobian of C in the case of r > 1. (I have been thinking about these Abelian varieties a lot recently.)

3. mixedmath says:

This is interesting. When you say that this is related to GL3 Eisenstein series, do you mean that the Multiple Dirichlet Series you indicated is related to GL3 Eisenstein series (which it is, and which I’m familiar with), or do you mean the multiple height zeta function? I would be very interested in hearing of any connection between the multiple height zeta function and Eisenstein series.

4. JSE says:

I mean this multiple height zeta function has something to do with GL3 Eisenstein series, see e.g. this paper:

https://arxiv.org/abs/1502.04209