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Iwasawa theory gives a formula for the power of $p$ dividing the class group of the $\mathbb{Q}(\zeta_{p^n})$ (where $\zeta_{p^n}$ is a primitive root of unity of exact order $p^n$) for sufficiently large $n$. (See, e.g., Theorem 2 of these notes.) More generally, one gets a similar result for arbitrary $\mathbb{Z}_p$ extensions of number fields. Mazur and Mazur and Rubin have studied the variation of the $p$ part of the Tate-Shafarevich group of an elliptic curve in $\mathbb{Z}_p$ extensions and the variation of the rank of an elliptic curve in $\mathbb{Z}_p$ extensions.

I find this somehow unsatisfying. The restriction to the study of the $p$ part of the ideal class group & Tate Shafarevich, and the restriction to the study of $\mathbb{Z}_p$ extensions seem quite strong.

Yet I've gotten the impression that Iwasawa theory is considered to be fundamental in number theory. So I feel as though I'm missing perspective on why Iwaswa theory is important.

What are some important applications of Iwasawa theory?

I'd also be happy with high-level philosophical comments.

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up vote 11 down vote accepted

Aha, an excuse to quote chunks of my most recent grant proposal :-)

Iwasawa theory is heavily used in work on the BSD conjecture. For instance, the first positive result to be proved in the direction of BSD -- the Coates--Wiles theorem that analytic rank 0 implies algebraic rank 0 for elliptic curves over $\mathbf{Q}$ with complex multiplication -- was shown by using Iwasawa theory. This is a statement which has nothing obviously to do with Zp-extensions, on the face of it, although to be sure they are lurking in the proof. More generally, pretty much all the results on BSD that we now have (thanks to Kolyvagin, Rubin, Kato, Perrin-Riou, Kobayashi, etc...) use Iwasawa theory heavily.

This fits into a general philosophy which states that if $M$ is a motive over a number field $K$, the $L$-values $L(M, j)$ for integer values of $j$ encode information about the cohomology of $M$. Iwasawa theory provides a tool to attack such conjectures, by interposing a third object -- a p-adic L-function -- which one can (sometimes) relate both to the cohomology and to the L-values. This is bound up with the idea that the behaviour of $M$ over your original ground field $K$ might be quite complex, but making a tower of extensions $K = K_0 \subset K_1 \subset K_2 \subset \dots \subset K_\infty$ and taking a limit of objects defined over the $K_n$'s can serve to "smooth out" the behaviour, and then you prove things over $K_\infty$ and see what you can recover over $K$ by some kind of descent argument, splitting your problem into two hopefully easier chunks. (For more philosophy along these lines, see Colmez's Bourbaki expose on p-adic L-functions, http://www.math.jussieu.fr/~colmez/851bourbaki.pdf)

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Thanks David! As before, I need to learn French... What other reading would you recommend on this topic? The original Coates-Wiles paper? –  Jonah Sinick Dec 28 '12 at 19:56
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