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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2 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.)

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