Classical algebraic number theory concerns the properties satisfied by the rational numbers Q and those numbers a which satisfy a polynomial with rational coefficients. It has always been, and remains, a magical subject with the most wonderful and interesting structures one can imagine. The twists and turns of the theory are so subtle that they read like a masterful mystery story. Who in the nineteenth century, for instance, would ever have thought that a solution to Fermat's Last Theorem would arise from the study of elliptic modular functions? Yet this is precisely the way the well known solution due to G. Frey, J.-P. Serre, K. Ribet, R. Taylor and most importantly, A. Wiles, has proceeded. And it is clear that these fantastic results are by no

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Let ffQ(s)= ~n -s, Re{s}>l, be Riemann's zeta function. ~o(s) has many n_>l remarkable properties. Among them are: 1) A meromorphic continuation to all of C with a pole of order one at 1, and a functional equation.2) The values at negative integers are rational and may be interpolated padically to a meromorphic function with a pole of order one at 1. Further, one obtains these p-adic functions essentially by integrating x * against a p-adic measure on Z*.There are similar results for the Dedekind zeta function of any totally real field. Now let k be a function field of characteristic p with Fr, r=p", the full field of constants. We denote the genus by g and its Dedekind zeta function by ~k(s). Among the properties of ~k(s) are:3) The existence of a PeZ [u] of degree 2g so that P(r-s) ~k(s) = (1 -r-s)(1 -rl-s)"4) The functional equation (1) P(u)=rgu2gP ~u " Iwasawa,[5, Chap. 12], tells us there are strong analogies between the constructions in 2) and 3). These parallel the analogy between cyclotomic fields and constant field extensions of k.Fix a place oo of k and let A be those functions regular outside of oo. One knowns A is a Dedekind domain with finite class group and unit group F*. In [1,4], there is an explicit construction of the integral closure, C, of A in the maximal abelian extension of k split totally at oo. This construction is very similar to the construction of cyclotomic fields.0020-9910/79/0055/0107/$02.60

Introduction. The purpose of this article is to introduce the general mathematical community to some recent developments in algebraic geometry and nonarchimedean analysis. Let r = p n ,p a rational prime. Then these developments center around the beginnings of an "arithmetic" theory of the polynomial ring ¥ r [T] over the finite field of r elements. The goal of this theory is to use nonarchimedean analysis to do for Y r [T] what classical analysis does for Z. The theory allows us to find direct analogues of many of the classical functions of arithmetic interest in a situation that, at first glance, seems as nonclassical as possible. In the process much will be learned about the polynomials. Much also will be learned about the unique properties of Z and the classical functions.One of the exciting aspects of the theory is its great generality. Indeed, we could replace ¥ r [T] with much more general affine rings of curves over finite fields. More precisely, if C is a projective, smooth curve over F r , oo a rational point and A the functions regular away from oo, then we may use A instead of ¥ r [T]. Thus one can, so to speak, get a sense of what analysis might have been forced to if Z were not a unique factorization domain. Such observations can only come in the present setting since Q is the only totally real (i.e., all Galois conjugates contained in R) field with a unique absolute-value. We have chosen to stick to the polynomials in order to keep the exposition as simple as possible. The jump from the polynomials to more general rings is not terribly large and most essential features appear for F r [T].Another exciting aspect is that we begin to see how a given 'arithmetic* situation generates an associated harmonic analysis. As classical harmonic analysis is based on the integers, the one developed here is based on F r [T]. In contrast to classical harmonic analysis which is multiplicative, i.e., based on the exponential function, the one here is based on addition.Throughout the paper we compare the theory here with the classical one. In this fashion we hope the reader may speedily develop a feel for the subject.One of the most surprising (and hotly contested) aspects of classical analysis is its harmonic analysis. This centered around the possibility of expanding an arbitrary singly-periodic function in terms of sines and cosines. Since sines and cosines are easily expressed in terms of the exponential function, e (z \ the central role of this function is apparent. Viewed on the complex plane, e (z) has the following very well-known properties: It is never zero, takes addition to multiplication, is invariant under z H> z + 2777 and, finally, it is its own derivative. As a consequence, e (z) gives

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