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
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”
There is a whole story of “multiple Dirichlet series” which studies functions like
where 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
This has something to do with Eisenstein series on GL_3 but I am a bit confused about what exactly to say.