# 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

## 1 Answer

The theory of arithmetic motivic integration (see, for example, the paper by Denef and Loeser in the proceedings of the 2002 ICM) takes into account the numbers you wish to encode in your two variable zeta function.

As Kevin Buzzard rightly points out, if the scheme is smooth, then the counts along powers of primes do not give much additional information. This motivic theory is useful only in so far as it allows for possible singularities.

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Correct me if I'm wrong---they might take these numbers into account, but they do not construct a two-variable zeta function, right? – Kevin Buzzard Apr 28 '10 at 16:24
You are right. They construct a Poincaré series with coefficients in certain Grothendieck rings associated to the theory of pseudofinite fields. The motivic Poincaré series specializes to the p-adic series so that their theorem on rationality of the motivic series yields uniform rationality for the p-adic series. The existence of a motivic proof of Dwork's rationality theorem is still open (especially as its naïve motivic formulation is false by the work of Larsen and Lunts). – Thomas Scanlon Apr 28 '10 at 18:06
Thanks! I will take a look... – Bugs Bunny Apr 29 '10 at 12:32