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I have recently been reading into the proof of modularity of semistable elliptic curves, in particular (what is now known as) the Taylor-Wiles patching argument used to prove the $R=T$ theorem in the minimal case. I feel like I understand the ideas involved at a technical level, but one thing I have not been able to quite make sense of is how certain informal discussions of this work, including from Wiles himself, actually describe this work. Specifically I have been interested in understanding in what ways this patching argument is an instance of "horizontal Iwasawa theory". I have gathered some parts of the picture, mainly from Wiles's original paper, which I will describe below, but I feel like there is more to it.

Sparing all the technical details, to prove that a surjective morphism $R\to T$ of (local complete Noetherian) algebras is an isomorphism, we construct a series of lifts $R_Q\to T_Q$, indexed by certain sets of primes $Q$, which among other properties have a bounded number of topological generators. We choose $R_Q,T_Q$ to be universal deformation rings and Hecke algebras of suitable level.

The idea of patching is that even though those lifts are a priori unrelated, (by some Mittag-Leffler argument) we can pick a sequence of this data and maps between them which let you take an inverse limit $R_\infty\to T_\infty$ of them. By how the $Q$'s are chosen we get that $T_\infty$ is finite free over a power series ring $O[[x_1,\dots,x_r]]$, and the isomorphism then follows for dimension reasons.

To summarize, the basic idea is that while individual Hecke algebras $T_Q$ may be complicated, taking a certain limit of them gives a much nicer ring. This is an idea which is not unfamiliar from Iwasawa theory - there the individual group rings $\mathbb Z_p[G_n]$ are poorly behaved but their inverse limit is isomorphic to $\mathbb Z_p[[T]]$ which has a nice module structure. However I can't help but feel like this similarity is merely superficial.

I am aware of earlier work which Wiles cites which is based on similar ideas - for instance below Proposition 2.15 he discusses a work of Hida in which certain limits of ordinary Hecke rings are taken, with levels $Np^r$ as $r$ varies. In the introduction he also refers to "the projective limit of the Hecke rings for the varying fields in a cyclotomic tower" which (conjecturally) is isomorphic to a power series - this one seems more in Iwasawa-theoretic style, but I'm not sure what construction this refers to.

This was a lot of context with little actual question so let me ask what I'm interested in more explicitly:

In what ways is the Taylor-Wiles patching a form of Iwasawa theory? Is it analogous to some usual types of arguments in classical Iwasawa theory, at a level deeper than some inverse limits of rings being better behaved than individual factors?

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  • $\begingroup$ Dear @Wojowu, You may have already understood what I'm going to say, but in case you haven't, I'll take the opportunity to explain it to you. Among the several reasons why this comparison holds, I believe one might be the control theorem. As you've already noticed, you deduce the freeness at the finite level Q from the freeness at the infinite level. This is also what happens in Hida theory, where it is shown that the limit space of p-adic automorphic forms is free over the Hida algebra. Furthermore, [...] $\endgroup$ Commented Feb 9 at 2:28
  • $\begingroup$ [...] "horizontal" means (I think) that you consider a projective system of algebras and modules ordered by sets of primes q different from p (where in classical Iwasawa theory, you let the power of p grow and fix the integer N, which is vertical). $\endgroup$ Commented Feb 9 at 3:49
  • $\begingroup$ @MarsaultChabat Thanks for these comments. Which control theorem do you have in mind? I'm not sure I'm familiar with the results of Hida theory you mention; is "Hida algebra" here an inverse limit of (ordinary) Hecke algebras I mentioned in my post? $\endgroup$
    – Wojowu
    Commented Feb 9 at 13:54
  • $\begingroup$ Whoops, I forgot to respond. I can't elaborate much more than David's answer, but I can refine my comment. When I use the terminology "control theorem," it's more of an intuitive sense rather than a formal theory. Specifically concerning TW systems, my intuition is this: "If you aim to prove a property about an algebra dependent on a set of primes, then consider taking the limit with respect to this set of primes. This results in an algebra of formal series, facilitating the proof of the property. Subsequently, you can deduce the property at a finite level". [...] $\endgroup$ Commented Feb 12 at 9:24
  • $\begingroup$ [...] In the context of Hida theory, as David mentioned, one examines the Hecke algebra of the limit of ordinary cusp forms of level $Np^{r}$ (taking the limit over $r$). Consequently, this Hecke algebra becomes $\mathcal{O}[[X]]$-free of finite rank, and quotienting by particular primes $P_{k}$ yields the finite (ordinary) Hecke algebra. This process is what we refer to as the "control theorem." It yields numerous consequences, mainly due to the favorable properties of the "infinite" ordinary Hecke algebra. [...] $\endgroup$ Commented Feb 12 at 9:24

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I think your question already contains its own answer.

In classical, "vertical" Iwasawa theory one studies class groups, or other arithetic widgets like elliptic curve Selmer groups, in a limit over $\mathbb{Q}(\zeta_{Np^r})$ as $r \to \infty$ (for fixed $N$); and what makes the theory work is that the Iwasawa algebra $\varprojlim_r \mathbb{Z}_p[G_r]$ is a much better-behaved ring than the individual finite-level group rings, because each $\mathbb{Z}_p[G_r]$ has the form $\mathbb{Z}_p[[T]] / $ (some messy ideal), and as you take the limit the ideals vanish away to 0 and give you $\mathbb{Z}_p[[T]]$.

Likewise, in Taylor-Wiles patching the limiting rings $R_\infty$ and $T_\infty$ are much nicer than the finite-level rings, because we have the same picture of each $T_n$ being $\mathbb{Z}_p[[X_1, \dots, X_k]] / I_r$ for some ideal $I_r$ (and fixed $k$) and the $I_r$'s vanish in the limit.

Hida theory indeed provides a bridge between the two: Hida constructs "nice" rings and modules (everything module-finite over $\mathbb{Z}_p[[X]]$) by taking inverse limits of Hecke algebras; but his construction is "vertical" (limits over level $Np^r$ for fixed $N$ and $r \to \infty$) rather than "horizontal" (introducing lots of auxiliary primes). Hida's paper "Galois reprentations into GL2(Zp[[X]]) attached to ordinary cusp forms" (Inventiones 1986) is a good reference.

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  • $\begingroup$ Thank you David, Hida's work does indeed make sense as a sort of bridge between the two ideas. Would you happen to know what Wiles meant when he talks about "Hecke rings for the varying fields in a cyclotomic tower"? Perhaps this just refers to the towers of varying level $Np^r$ but I'm not sure in what way they're related to cyclotomic towers. $\endgroup$
    – Wojowu
    Commented Feb 9 at 16:32

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