# 2d Weil conjecture

Does there exist a two variable analogue of the Weil conjecture?

What I mean is that the usual Weil involves a one-variable zeta-function which you get by using numbers $V_n = V ( GF(p^n))$ of points of a smooth algebraic variety over finite fields of characteristic $p$. Is it possible to have a sensible two-parameter family of finite rings instead? Any references?

For instance, one can consider finite quotients of Witt vectors, and form a two-parameter family of numbers $V_{n,m} = V ( Witt(GF(p^n))/I^m)$ (where $I$ is the maximal ideal of the Witt vectors) from a variety $V$ (smooth, projective over $\mathbb Z$). Is there a sensible two variable zeta-function cooked with these numbers?

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If V is a variety over Z and if p is a prime of good reduction, then by Hensel's Lemma the number of points in V(W/p^n) will determine the number of points in V(W/p^{n+1}), so your extra degree of freedom is bogus. –  Kevin Buzzard Apr 28 '10 at 15:35
But if $V$ does not have good reduction, one gets a more interesting function. Theses zeta functions were studied well before motivic integration came along, and are called Igusa zeta functions. One place they arise is in the computation of orbital integrals, and it was for this reason that people like Hales thought that motivic integration might be applied to the theory of orbital integrals, and in particular to the fundamental lemma. –  Emerton Apr 28 '10 at 20:36
And Hales was right! –  Wanderer Apr 28 '10 at 21:42
Thanks! I can see that this is not good but I can think of other naive ways to get a two variable function. For instance, I can stratify the variety and count the point as $q^d$ where $d$ is the dimension of the stratum with the point rather than 1. I guess I'd better shut up and read Denef-Loeser before saying anything else... –  Bugs Bunny Apr 29 '10 at 12:37
Whyever do you want a 2-variable L-function when a variety over Z has a perfectly good one-variable L-function with a whole host of interesting theorems and conjectures attached to it? Surely the logic should be "I see a 2-variable L-function in this simple setting, hence I wonder if it generalises and every variety has one", not "I see a 1-variable L-function, hence I wonder if there's a 2-variable L-function despite not ever having seen one anywhere"? –  Kevin Buzzard Apr 29 '10 at 16:04