| In 1959, Iwasawa proved that the size of the p-part of the class groups of a Zp -extension grows as a power of p with exponent mupm + lambdam + nu for m sufficiently large. Broadly, I construct conditions to verify if a given m is indeed sufficiently large.;More precisely, let CGim (class group) be the epsiloni-eigenspace component of the p-Sylow subgroup of the class group of the field at the m-th level in a Zp -extension; and let IACGim (Iwasawa analytic class group) be Zp [[T]]/((1 + T &parr0;pm -- 1, f(T, o1-- i)), where f is the associated Iwasawa power series. It is expected that CGim and IACGim be isomorphic, providing us with a powerful connection between algebraic and analytic techniques; however, as of yet, this isomorphism is unestablished in general.;I consider the existence and the properties of an exact sequence 0→ker→CGim→ IACGim→coker→ 0. In the case of a Zp -extension where the Main Conjecture is established, there exists a pseudo-isomorphism between the respective inverse limits of CGim and IACGim . I consider conditions for when such a pseudo-isomorphism immediately gives the existence of the desired exact sequence, and I also consider work-around methods that preserve cardinality for otherwise. However, I primarily focus on constructing conditions to verify if a given m is sufficiently large that the kernel and cokernel of the above exact sequence have become well-behaved, providing similarity of growth both in the size and in the structure of CGim and IACGim ; as well as conditions to determine if any such m exists.;The primary motivating idea is that if IACGim is relatively easy to work with, and if the relationship between CGim and IACGim is understood; then CGim becomes easier to work with.;Moreover, while the motivating framework is stated concretely in terms of the cyclotomic Zp -extension of p-power roots of unity, all results are generally applicable to arbitrary Zp -extensions as they are developed in terms of Iwasawa-Theory-inspired, yet abstracted, algebraic results on maps between inverse limits. |
