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The notion of class fields is usually attributed to Hilbert, but the concept was already in the mind of Kronecker and the term was used by Weber before the appearance of the fundamental papers of Hilbert.

As we have seen, there is a close analogy between algebraic number fields and algebraic functions, and this analogy is even more pronounced if we consider the case of congruence function fields, that is, when the field of constants is finite.

In this chapter we present a brief introduction to Drinfeld modules or, as they were called by Drinfeld himself, elliptic modules . The main goal of V. G. Drinfeld [30] was to generalize three classical results: a) the Kronecker-Weber Theorem; b) the Eichler Shimura Theorem on ζ functions of modular curves and c) the fundamental theorem on complex multiplication.

In this chapter we continue our study of the arithmetic of extensions in function fields. We study the group $$ \varsigma \left( s \right) = \sum {_{n = 1}^\infty \frac{1} {{n^s }}} $$ where K/k is an arbitrary function field. When gK is 0 or 1, the group G is infinite, except in the case that k is a finite field. For gk ≥ 2, G is almost always a finite group. In order to investigate G , we need to consider some special points in K called the Weierstrass points. We also need to know the genus gk of K . It is often difficult to determine precisely the genus of a function field, so we will derive some bounds for the genus in special cases. This result is the Castelnuovo-Severi inequality .