How is Taylor-Wiles patching "horizontal Iwasawa theory"?

Question

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?

Answer 1

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.