Iwasawa’s Theory of ℤ-extensions

Abstract

The theory of ℤ p -extensions has turned out to be one of the most fruitful areas of research in number theory in recent years. The subject receives its motivation from the theory of curves over finite fields, which is known to have a strong analogy with the theory of number fields. In the case of curves, it is convenient to extend the field of constants to its algebraic closure, which amounts to adding on roots of unity. There is a natural generator of the Galois group, namely the Frobenius, and its action on various modules yields zeta functions and L-functions. In the number field case, it turns out to be too unwieldy, at least at present, to use all roots of unity. Instead, it is possible to obtain a satisfactory theory by just adjoining the p-power roots of unity for a fixed prime p. This yields a ℤ p -extension. The action of a generator of the Galois group on a certain module yields, at least conjecturally, the p-adic L-functions.