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From the reviews:"The book...is a thorough and very readable introduction to the arithmetic of function fields of one variable over a finite field, by an author who has made fundamental contributions to the field. It serves as a definitive reference volume, as well as offering graduate students with a solid understanding of algebraic number theory the opportunity to quickly reach the frontiers of knowledge in an important area of mathematics...The arithmetic of function fields is a universe filled with beautiful surprises, in which familiar objects from classical number theory reappear in new guises, and in which entirely new objects play important roles. Goss'clear exposition and lively style make this book an excellent introduction to this fascinating field." MR 97i:11062

Inhaltsverzeichnis

Frontmatter

1. Additive Polynomials

Abstract
Let k be a field of finite characteristic p and let \(\bar k\) be a fixed algebraic closure. Let \(P\left( x \right) \in k\left[ x \right] \) be a polynomial in x with coefficients in k. We say that P(x) is additive if and only if \( P\left( {\alpha + \beta } \right) = P\left( \alpha \right) + P\left( \beta \right)\,\,for\,\,\left\{ {\alpha ,\beta ,\alpha + \beta } \right\} \subseteq k \). We say that P(x) is absolutely additive if and only if P(x) is additive over \(\bar k\).
David Goss

2. Review of Non-Archimedean Analysis

Abstract
In this section we will give a rapid presentation of those properties of non-Archimedean analysis necessary for later sections. For more we refer the reader to [BGR1], [Bru1], [Kob1], [R1,144–159], [L1], and the first two chapters of [DGS1].
David Goss

3. The Carlitz Module

Abstract
We present here the details of the Carlitz module. This is the simplest of all Drinfeld modules and may be given in a concrete, elementary fashion. At the same time, most essential ideas about Drinfeld modules appear in the theory of the Carlitz module. Thus it is an excellent example for the reader to master and keep in mind when reading the more abstract general theory. Our basic reference is [C1], but see also [Go2].
David Goss

4. Drinfeld Modules

Abstract
In this section, we begin discussing Drinfeld modules, our basic objects of study. Drinfeld modules are generalizations of the Carlitz module of our last section. Moreover, most of the salient features of Drinfeld modules are seen on the Carlitz module. Among these are: 1. The “lattice” F r [T]ξ; 2. The exponential of F r [T]ξ (= the Carlitz exponential e C (x)); 3. The multiplication law of e C (x) (= the Carlitz module C); 4. The algebraicity of C and its reductions to fields of “finite” characteristic; and 5. The action of F r [T] on rational points via C.
David Goss

5. T-modules

Abstract
In this section we will present an important generalization of Drinfeld modules called “T-modules.” This theory is due to G. Anderson [A1]. Very roughly, if Drinfeld modules are analogous to elliptic curves, then T-modules are analogous to abelian varieties.
David Goss

6. Shtukas

Abstract
In this section we will introduce “shtukas” which are also called “F-sheaves” or “FH-sheaves.” Let A and k be defined as in Subsection 4.1; so k is a global field over the finite field F r and A is the subring of functions regular away from a fixed place ∞. As in Sections 4 and 5, we have seen that Drinfeld modules and T-modules correspond to representing A as a ring of operators on G a d for some d. The notion of a shtuka, then corresponds to a proper model of this action, i.e., the shtukas will be certain locally free sheaves on the complete curve X corresponding to k (or X base changed to an overfield of F r ). One can then study shtukas through powerful projective methods.
David Goss

7. Sign Normalized Rank 1 Drinfeld Modules

Abstract
In this section we will present the construction of “sign-normalized” rank one Drinfeld modules. These play the role of the Carlitz module for general A. This basic construction is due to David Hayes [Ha3] [Ha2]. Thus we will also call them “Hayes-modules.” We will use Hayes-modules to construct a “cyclotomic theory” of function fields. The reader will find much that is familiar in these extensions from classical theory.
David Goss

8. L-series

Abstract
We retain the notation of Section 4; thus X is our base curve over F r (r = p m 0) with fixed closed point ∞ ∈ X, function field k and where Spec(A) = X−∞. We let C be the completion of an algebraic closure of the completion K of k at ∞. Let \(\bar k \subset {\text{ }}{{\text{C}}_\infty }\) be the algebraic closure and ksep \(\subset \bar k\) the separable closure. Set
$$G: = Gal\left( {{k^{sep}}/k} \right).$$
David Goss

9. Γ-functions

Abstract
In this section we will introduce Γ-functions into the arithmetic of function fields. We do this by building on a basic, and still quite mysterious, construction of L. Carlitz in the A = F r [T]-case. Recall that in Section 3.3 we introduced the Carlitz exponential
$${e_C}\left( z \right) = \sum\limits_{j = 0}^\infty {{z^{{r^j}}}} /{D_j}$$
where \({D_0} = 1,{D_j} = \left[ j \right]{\left[ {j - 1} \right]^r} \cdots {\left[ 1 \right]^{{r^{j - 1}}}},\) for j > 1, and \(\left[ j \right] = {T^{{r^j}}} - T \). In Proposition 3.1.6 we showed that
$${D_j} = {\text{ }}\mathop {\mathop {\mathop \prod \limits_{g \in A} }\limits_{g{\text{ monic}}} }\limits_{\deg {\text{ }}g = j} g.$$
David Goss

10. Additional Topics

Abstract
In this last section, we will briefly describe some current areas of research into the arithmetic described in the previous nine sections. We emphasize the word “some” as this section is not meant to be considered exhaustive.
David Goss

Backmatter

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