Roadmap to Iwasawa Theory

Question

I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good amount of iwasawa theory? In what sequence should they study this material?

Answer 1

Iwasawa theory has at least two aspects, algebraic and $p$-adic analytic. Initially, they were developped independently until they merged into one single subject dominated by the so-called Main Conjecture. Since you ask for a map road, here are first some minimal milestones of "classical" Iw. theory :

1) The algebraic part started with the study of $\mathbf Z_p$-extensions, i.e. infinite Galois extensions with Galois group isomorphic to ($\mathbf Z_p$ ,+). An example of a deep problem is the famous Leopoldt conjecture, which asserts that a number field $K$ possesses exactly $(1+ r_2)$ "independent" $\mathbf Z_p$-extensions, where $r_2$ is the number of pairs of conjugate embeddings of $K$ into $\mathbf C$. The conjecture has been proved by Brumer for abelian $K$, using $p$-adic transcendental methods, but it remains unsolved in general.

Another motivation came from ANT : given $K$ and $p$, how do the $p$-class groups (of ideal classes) behave when going up the layers $K_n$ of a $\mathbf Z_p$-extension $K_{\infty}/K$ ? The answer given by Iw. is one of the first emblematic results of the theory : their orders are asymptotically of the form $p^{c(n)}$, where $c(n)=\mu p^n + \lambda n + \nu$, where $\lambda, \mu \in \mathbf N$ and $\nu \in \mathbf Z$ are parameters depending only on $K$. This theorem is an illustration of a basic "going up and down" process which has become a trademark in Iw. theory : when an arithmetic problem appears too complicated at the level of $K$, one goes up along the layers of $K_{\infty}/K$ hoping that the corresponding problems will stabilize, and then one goes back down to attack the initial problem with the new weapons found upstairs. This was exactly the method of proof of the above result : the projective limit of the $p$-class groups along the layers $K_n$ is a finitely generated torsion module over the ring of formal power series $\mathbf Z_p [[T]]$, and Iw. gave a stucture theorem for such modules, analogous to the one for finitely generated modules over a principal domain.

I must add an example showing that this "algebraic" part of the theory actually contains some deep arithmetic. It was initially thought that $\mu=0$ in general, but a counter-example was built by Iw., and the nullity conjecture nowadays is restricted to the so-called cyclotomic $\mathbf Z_p$-extensions, which mysteriously play a prominent role when arithmetic enters the game. As in the case of Leopoldt, the conjecture «$\mu=0$» has been proved for abelian $K$ by many people (Washington, Sinnott, etc.), but none of the proofs is algebraic.

2) Of course, $p$-adic analysis seems to appear with $\mathbf Z_p [[T]]$, but under this form it remains at the surface, since the Weierstrass preparation theorem allows us, roughly speaking, to replace series by polynomials. From this point of view, the main point is the introduction of an innocent looking invariant attached to any module $X$ as above. If $\Gamma=Gal(K_{\infty}/K)$ is written multiplicatively, it admits a topological generator $\gamma$, and if we set $T = \gamma - 1$, then the formal power series ring $\mathbf Z_p [[T]]$ is no other than the complete group ring $\mathbf Z_p [[\Gamma]]$. As in elementary linear algebra, we can attach to the action of $\gamma - 1$ on $X$ a "characteristic polynomial (series)" $f_X (T)$. This is the first side of the mountain.

The other side comes from the theory of $p$-adic L-functions attached to a totally real number field $K$. For commodity, let us look only at the $p$-adic zeta function $\zeta_p (K, s)$, with $p$ odd. Thanks to an analytic theorem of Siegel, the "special values" (= first non zero terms of the Taylor expansion) of the complex function $\zeta(K, s)$ at negative integers are rational, hence live also in $\mathbf Q_p$. The natural idea is then to "interpolate" them by some $p$-adic zeta function. This was done by Kubota-Leopoldt for $K=\mathbf Q$, and for a general totally real $K$ by Barsky, Cassou-Noguès, Deligne-Ribet. A technical step in one of the constructions used the classical equivariant (=living in a group algebra) Stickelberger element which annihilates the class group. This gave Iwasawa the idea of the Main Conjecture : take the proj. lim. $X$ of the "minus" parts (= inverted by complex conjugation) of the $p$-class groups along the cyclotomic extension $K(\mu_{p^{\infty}})$, where $\mu_{p^{\infty}}$ is the group of all $p^n$-th roots of $1$. Then, up to an adequate change of variables, $\zeta_p(K, s)$ should be "the same" as $f_X (T)$ ! This astonishing conjecture was proved, for $K=\mathbf Q$ by Mazur and Wiles, and in general by Wiles, using modular forms (which are at the core of the Deligne-Ribet construction). A more simple non modular proof was later given by Kolyvagin, who introduced the important notion of Euler systems, but this approach works only over $\mathbf Q$.

You can learn almost all the above stuff (outside modular forms) in now classical textbooks such as Washington's "Introduction to Cyclotomic Fields", Lang's "Cyclotomic Fields" (2 volumes), Coates § Sujatha's "Cyclotomic Fields and Zeta Values". Basically you'll need to have a good enough knowledge of ANT and CFT. Other more powerful tools (cohomology, modular forms...) you can acquire later. I refrain from talking of the current developments of Iw. theory (elliptic curves, Euler systems, Tamagawa Number conjectures, $p$-adic representations...) for which the literature is still at the research level ./.