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?



2d Weil conjecture
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.


