# Reciprocity in the Langlands program since Fermat’s Last Theorem

Frank Calegari  The University of Chicago, 5734 S University Ave, Chicago, IL
60637, USA<EMAIL_ADDRESS>
###### Abstract.

The _reciprocity conjecture_ in the Langlands program links motives to
automorphic forms. The proof of Fermat’s Last Theorem by Wiles [180, 169]
introduced new tools to study reciprocity. This survey reports on developments
using these ideas (and their generalizations) in the last three decades.

The author was supported in part by NSF Grant DMS-2001097

##  1\. Introduction

The _reciprocity conjecture_ in the Langlands program predicts a relationship
between pure motives111Here (in light of the standard conjectures [124]) one
may take pure motives up to numerical or homological equivalence.
Conjecturally, one can also substitute (for irreducible motive) the notion of
an irreducible weakly compatible system of Galois representations [167] or an
irreducible geometric Galois representation in the sense of Fontaine–Mazur
[83]. and automorphic representations. The simplest version (as formulated by
Clozel [48, Conj 2.1]) states that there should be a bijection between
irreducible motives $M$ over a number field $F$ with coefficients in
$\overline{\mathbf{Q}}$ and cuspidal algebraic representations $\pi$ of
$\mathrm{GL}_{n}(\mathbf{A}_{F})$ satisfying a number of explicit additional
compatibilities, including the equality of algebraic and analytic
$L$-functions $L(M,s)=L(\pi,s)$. In light of multiplicity one theorems [105],
this pins down the correspondence uniquely. There is also a version of this
conjecture for more general reductive groups, although its formulation
requires some care (as was done by Buzzard and Gee [32]). Beyond the
spectacular application by Wiles to Fermat’s Last Theorem [180, Theorem 0.5],
the Taylor–Wiles method [180, 169] gave a completely new technique — and to
this date the most successful one — for studying the problem of reciprocity.
The ideas in these two papers have sustained progress in the field for
almost222Wiles in [180] dates the completion of the proof to September 19,
1994. 30 years. In this survey, we explain how the Taylor–Wiles method has
evolved over this period and where it stands today. One warning: the intended
audience for this document is entirely complementary to the audience for my
talk — I shall assume more than a passing familiarity with the arguments of
[180, 169]. Moreover, this survey is as much a personal and historical333A
whiggish history, naturally. Even with this caveat, it should be clear that
the narrative arc of progress presented here at best represents my own
interpretation of events. I have added a few quotes from first hand sources
when I felt they conveyed a sense of what the experts were thinking in a
manner not easily obtainable from other sources. For other survey articles on
similar topics, see [30, 24]. discussion as a mathematical one — giving
anything more than hints on even a fraction of what is discussed here would be
close to impossible given the space constraints and the competence of the
author. Even with the absence of any real mathematical details in this paper,
the sheer amount of activity in this field has led me to discard any
discussion of advances not directly related to $R=\mathbf{T}$ theorems, which
necessitates the omission of a lot of closely related beautiful mathematics.

###  1.1. The Fontaine–Mazur Conjecture

Let $F$ be a number field. The Fontaine–Mazur conjecture444Fontaine told me
(over a _salad de gésiers_ in Roscoff in 2009) that he and Mazur formulated
their conjecture in the mid-80s. (Colmez pointed me towards these notes [81]
from a talk given by Fontaine at the 1988 Mathematische Arbeitstagung in
Bonn.) He noted that Serre had originally been skeptical, particularly of the
claim that any everywhere unramified representation inside
$\mathrm{GL}_{n}(\overline{\mathbf{Q}}_{p})$ must have finite image, and set
off to find a counterexample (using the construction of Golod–Shavarevich
[91]). He (Serre) did not succeed! [83] predicts that any continuous
irreducible $p$-adic Galois representation

$\rho:G_{F}\rightarrow\mathrm{GL}_{n}(\overline{\mathbf{Q}}_{p})$

which is both unramified outside finitely many primes and potentially semi-
stable (equivalently, de Rham [56]) at all places $v|p$ should be associated
to a motive $M/F$ with coefficients in $\overline{\mathbf{Q}}$. Any such
$\rho$ is automatically conjugate to a representation in $\mathrm{GL}_{n}(E)$
for some finite field $E/\mathbf{Q}_{p}$ and further stabilizes an
$\mathcal{O}_{E}$-lattice. The corresponding residual representation
$\overline{\rho}:G_{F}\rightarrow\mathrm{GL}_{n}(k)$ where
$k=\mathcal{O}_{E}/\pi_{E}$ is the residue field of $E$ is unique up to semi-
simplification. Let us assume here for expositional convenience that
$\overline{\rho}$ is absolutely irreducible. Following Mazur [129], one may
define a universal deformation ring which parameterizes all deformations of
$\overline{\rho}$ unramified outside a finite set $S$. One can then further
impose local conditions to define deformation rings $R$ whose
$\overline{\mathbf{Q}}_{p}$-valued points are associated to Galois
representations which are de Rham at $v|p$ with fixed Hodge–Tate weights.
Assuming the Fontaine–Mazur conjecture, these
$\overline{\mathbf{Q}}_{p}$-valued points correspond to all pure motives $M$
unramified outside $S$ whose $p$-adic realizations are Galois representations
with the same local conditions at $p$ and the same fixed residual
representation $\overline{\rho}$. Assuming the reciprocity conjecture, these
motives should then be associated to a finite dimensional space of automorphic
forms. This leads to the extremely non-trivial prediction that $R$ has
finitely many $\overline{\mathbf{Q}}_{p}$-valued points. The problem of
reciprocity is now to link these $\overline{\mathbf{Q}}_{p}$-valued points of
$R$ to automorphic forms.

###  1.2. $R=\mathbf{T}$ theorems

Associated to the (conjectural) space of automorphic forms corresponding to
$\overline{\mathbf{Q}}_{p}$-valued points of $R$ is a ring of endomorphisms
generated by Hecke operators. The naïve version of $\mathbf{T}$ is defined to
be the completion of this ring with respect to a maximal ideal $\mathfrak{m}$
defined in terms of $\overline{\rho}$. The mere existence of $\mathfrak{m}$ is
itself conjectural, and amounts — in the special case of odd absolutely
irreducible $2$-dimensional representations $\overline{\rho}$ of
$G_{\mathbf{Q}}$ — to Serre’s conjecture [156]. Hence, in the Taylor–Wiles
method, one usually assumes the existence of a suitable $\mathfrak{m}$ as a
hypothesis. The usual shorthand way of describing what comes out of the
Taylor–Wiles method is then an “$R=\mathbf{T}$ theorem.” Proving an
$R=\mathbf{T}$ theorem can more or less be divided into three different
problems:

  1. (1)

Understanding $\mathbf{T}$. Why does there exist555At the time of Wiles’
result, this was seen as the easier direction (if not easy), although, in
light of the success of the Taylor–Wiles method, it may well be the harder
direction in general. a map $R\rightarrow\mathbf{T}$? This is the problem of
the “existence of Galois representations.” Implicit here is the problem of
showing that those Galois representations not only exist but have the “right
local properties” at the ramified primes, particularly those dividing $p$.

  2. (2)

Understanding $R$. Wiles introduced a mechanism for controlling $R$ via its
tangent space using Galois cohomology (in particular Poitou–Tate duality
[131]), and this idea has proved remarkably versatile. What has changed,
however, is our understanding of local Galois representations and how this
information can be leveraged to understand the structure of $R$.

  3. (3)

Understanding why the map $R\rightarrow\mathbf{T}$ is an isomorphism.

We begin by summarizing the original $R=\mathbf{T}$ theorem from this
viewpoint (or more precisely, the modification by Faltings which appears as an
appendix to [169]). We only discuss for now the so-called “minimal case666The
case when the Galois representations attached to $R$ and $\mathbf{T}$ have
minimal level $N$ as determined by the residual representation.” since this is
most relevant for subsequent generalizations (see §6.2). Our summary is
cursory, but see [68, 67] for excellent expositional sources on early versions
of the Taylor–Wiles method. We start with a representation
$\overline{\rho}:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$
for $p>2$ which (say) comes from a semistable elliptic curve $E$ and which we
assume to be modular. By a theorem of Ribet [149], we may assume it is modular
of level either $N=N(\overline{\rho})$ or $N=N(\overline{\rho})p$ where
$N(\overline{\rho})$ is the Serre weight [156] of $\overline{\rho}$.

  1. (1)

Understanding $\mathbf{T}$: The construction of Galois representations
associated to modular forms has its own interesting history (omitted here),
but (in the form originally needed by Wiles) was more or less complete for
modular forms (and even Hilbert modular forms) by 1990. The required local
properties at primes different from $p$ followed from work of Carayol [47],
and the local properties at $p$ were well understood either by
Fontaine–Laffaille theory [82], or, in the ordinary case, by Mazur–Wiles [130]
(see also work of Hida [101, 102]).

  2. (2)

Understanding $R$: Here $R$ is a deformation ring of $\overline{\rho}$ subject
to precise local deformation conditions at $p$ and the primes dividing
$N(\overline{\rho})$. For the prime $p$, the local conditions amount either to
an “ordinary” or “finite-flat” restriction. One then interprets the dual of
the reduced tangent space $\mathfrak{m}_{R}/(\mathfrak{m}^{2}_{R},p)$ of $R$
in terms of Galois cohomology, in particular as a subgroup (Selmer group) of
classes in $H^{1}(\mathbf{Q},\mathrm{ad}^{0}(\overline{\rho}))$ satisfying
local conditions. This can be thought of as analogous to a class group, and
one does not have any _a priori_ understanding of how large it can be although
it has some finite dimension $d$. Using the Greenberg–Wiles formula, the
obstructions in $H^{2}(\mathbf{Q},\mathrm{ad}^{0}(\overline{\rho}))$ can be
related to the reduced tangent space, and allow one to realize $R$ as a
quotient of $W(k)\llbracket x_{1},\ldots,x_{d}\rrbracket$ by $d$ relations. In
particular, if $R$ was finite and free as a $W(k)$-module (as would be the
case if $R=\mathbf{T}$) then $R$ would be a complete intersection.

  3. (3)

Understanding why the map $R\rightarrow\mathbf{T}$ is an isomorphism. Here
lies the heart of the Taylor–Wiles method. The ring $\mathbf{T}$ acts on a
natural module $M$ of modular forms. One shows — under a mild hypothesis on
$\overline{\rho}$ — the existence of (infinitely many) sets $Q=Q_{N}$ for any
natural number $N$ of cardinality $|Q|=d$ — so-called Taylor–Wiles primes —
with a number of pleasant properties:

    1. (a)

The primes $q\in Q$ are congruent to $1\bmod p^{N}$.

    2. (b)

Let $R_{Q}$ be the deformation ring capturing the same local properties as $R$
but modified so that the representations at primes in $Q$ may now be ramified
of degree $p^{N}$. There is naturally a surjection $R_{Q}\rightarrow R$, but
for Taylor–Wiles primes, this modification does not increase the size of the
tangent space. In particular, for a fixed ring $R_{\infty}=W(k)\llbracket
x_{1},\ldots,x_{d}\rrbracket$ there are surjections $R_{\infty}\rightarrow
R_{Q}\rightarrow R$ for every $Q$.

    3. (c)

The corresponding rings $\mathbf{T}$ and $\mathbf{T}_{Q}$ act naturally on
spaces of modular forms $M$ and $M_{Q}$ respectively. Using multiplicity one
theorems, Wiles proves (see [180, Theorem 2.1]) that $M$ and $M_{Q}$ are free
of rank one over $\mathbf{T}$ and $\mathbf{T}_{Q}$ respectively. The space $M$
can be interpreted as a space of modular forms for a particular modular curve
$X$. The second key property of Taylor–Wiles primes is that there are no new
modular forms associated to $\overline{\rho}$ at level $X_{0}(Q)$, and hence
$M$ can _also_ be interpreted as a space of modular forms for $X_{0}(Q)$.
There is a Galois cover $X_{1}(Q)\rightarrow X_{0}(Q)$ with Galois group
$(\mathbf{Z}/Q\mathbf{Z})^{\times}$, and hence an intermediate cover
$X_{H}(Q)\rightarrow X_{0}(Q)$ with Galois group
$\Delta_{N}=(\mathbf{Z}/p^{N}\mathbf{Z})^{d}$ acting via diamond operators.
The space $M_{Q}$ is essentially a localization of a certain space of modular
forms for $X_{H}(Q)$ (with some care taken at the Hecke operators for primes
dividing $Q$). Since the cohomology of modular curves (localized at the
maximal ideal corresponding to $\mathfrak{m}$) is concentrated in degree one,
the module $M_{Q}$ turns out to be free over an auxiliary ring
$S_{N}=W(k)[\Delta_{N}]$ of diamond operators, and the quotient
$M_{Q}/\mathfrak{a}_{Q}$ for the augmentation ideal $\mathfrak{a}_{Q}$ of
$S_{N}$ is isomorphic to $M$. It follows that
$\mathbf{T}_{Q}/\mathfrak{a}_{Q}=\mathbf{T}$.

    4. (d)

The diamond operators have an interpretation on the Galois deformation side,
and there is a identification $R_{Q}/\mathfrak{a}_{Q}=R$ where $R_{Q}$ and
$\mathbf{T}_{Q}$ can be viewed compatibly as $S_{N}$-modules.

  4. (4)

Finally, one “patches” these constructions together for larger and larger $Q$.
This is somewhat counterintuitive, since for different $Q$ the Galois
representations involved are not compatible. However, one forgets the Galois
representations and only remembers the structures relative to both the diamond
operators $S_{N}$ and $R_{\infty}$, giving the data of a surjection

$R_{\infty}\rightarrow\mathbf{T}_{\infty}$

with a compatible action of $S_{\infty}=\projlim S_{N}\simeq W(k)\llbracket
t_{1},\ldots,t_{d}\rrbracket$. Using the fact that $\mathbf{T}_{\infty}$ is
free of finite rank over $S_{\infty}$, and that $R_{\infty}$ and $S_{\infty}$
are formally smooth of the same dimension, one deduces that
$R_{\infty}=\mathbf{T}_{\infty}$ and then $R=\mathbf{T}$ after quotienting out
by the augmentation ideal of $S_{\infty}$.

##  2\. The early years

###  2.1. The work of Diamond and Fujiwara

Wiles made essential uses of multiplicity one theorems in order to deduce that
$M_{Q}$ was free over $\mathbf{T}_{Q}$. Diamond [72] and Fujiwara [85]
(independently) had the key insight that one could instead patch the modules
$M_{Q}$ directly — and then argue directly with the resulting object
$M_{\infty}$ as a module over $R_{\infty}$ which was also free over
$S_{\infty}$. Using the fact that $R_{\infty}$ is formally smooth, this
allowed one to deduce _a posteriori_ that $M_{\infty}$ was free over
$R_{\infty}$ using the Auslander–Buchsbaum formula [9]. This not only removed
the necessity of proving difficult multiplicity one results but gave new
proofs of these results777There is an intriguing result of Brochard [27] which
weakens the hypotheses of Diamond’s freeness criterion even further, although
this idea has not yet been fully exploited. which could then be generalized to
situations where the known methods (often using the $q$-expansion principle)
were unavailable888The history of the subject involves difficult theorems in
the arithmetic geometry of Shimura varieties being replaced by insights from
commutative algebra, paving the way to generalizations where further insights
from the arithmetic geometry of Shimura varieties are required.. Diamond had
the following to say about how he came up with the idea to patch modules
rather than use multiplicity one theorems:

My vague memory is that I was writing down examples of ring homomorphisms and
modules, subject to some constraints imposed by a Taylor–Wiles setup, and I
couldn’t break “$M$ free over the group ring implies $M$ free over $R$.” (I
still have the notebook with the calculations somewhere, mostly done during a
short trip with some friends to Portugal.) I didn’t know _what_ commutative
algebra statement I needed, but I knew I needed to learn more commutative
algebra and found my way to Bruns and Herzog’s “Cohen-Macaulay Rings” [28]
(back in the library in Cambridge UK by then). When I saw the statement of
Auslander–Buchsbaum, it just clicked.

Diamond made a second improvement [70, 71] dealing with primes away from $p$
in situations where the corresponding minimal local deformation problem was
not controlled by the Serre level $N(\overline{\rho})$ alone.

###  2.2. Integral $p$-adic Hodge Theory, part I: Conrad–Diamond–Taylor

One early goal after Fermat was the resolution of the full Taniyama–Shimura
conjecture, namely, the modularity of all elliptic curves over $\mathbf{Q}$.
After the improvements of Diamond, the key remaining problem was understanding
deformation rings associated to local Galois representations at $p$ coming
from elliptic curves with bad reduction at $p$. Since Wiles’ method (via
Langlands–Tunnell [127, 177]) was ultimately reliant on working with the prime
$p=3$, this meant understanding deformations at $p$ of level $p^{2}$ and level
$p^{3}$, since any elliptic curve over $\mathbf{Q}$ has a twist such that the
largest power of $3$ dividing the conductor is at most $27$. Ramakrishna in
his thesis [145] had studied the local deformation problem for finite flat
representations (the case when $(N,p)=1$) and proved that the corresponding
local deformation rings were formally smooth. The case when $p$ exactly
divides $N$ was subsumed into the ordinary case, also treated by Wiles. In
level $p^{2}$, one can show that the Galois representations associated to the
relevant modular forms999This is not true for all modular forms of level
$p^{2}$ and weight $2$, but only for those whose conductor at $p$ remains
divisible by $p^{2}$ after any quadratic twist. of level $p^{2}$ become finite
flat after passing to a finite extension $L/\mathbf{Q}_{p}$ with ramification
degree $e\leq p-1$. In this range, Conrad [64, 65] was able to adapt ideas of
Fontaine [80] to give an equivalence between the local Galois deformations
(assuming $\overline{\rho}|_{G_{\mathbf{Q}_{p}}}$ was irreducible) and linear
algebra data. In particular, as in the work of Ramakrishna, one can show that
the relevant local deformation rings are formally smooth, and so Conrad,
Diamond, and Taylor were able to adapt the Taylor–Wiles method to this setting
[66].

###  2.3. Integral $p$-adic Hodge Theory, part II:
Breuil–Conrad–Diamond–Taylor

A central technical ingredient in all of the arguments so far has been some
use of integral $p$-adic Hodge Theory, and in particular the theory of finite
flat group schemes and Barsotti–Tate groups as developed by Fontaine and
others. All integral versions of this theory required a hypothesis on either
the weight or the ramification index $e$ relative to the bound $p-1$. However,
around this time, Christophe Breuil made a breakthrough101010Much of the
development of integral $p$-adic Hodge theory over the last $20$ years since
[25] has been inspired by its use in the Taylor–Wiles method. However, the
timing of Breuil’s work was more of a happy coincidence, although Breuil was
certainly aware of the fact that a computable theory of finite flat group
schemes over highly ramified bases could well have implications in the
Langlands program. by finding a new way to understand the integral theory of
finite flat group schemes over arbitrarily ramified bases [19]. This was just
the technical tool required to push the methods of [66] to level $p^{3}$.
Using these results, Breuil, Conrad, Diamond and Taylor [25] were able to show
that enough suitably chosen local deformation rings were formally smooth to
prove the modularity of all elliptic curves.

###  2.4. Higher weights, totally real fields, and base change

Many of the methods which worked for modular forms were directly adaptable to
the case both of general rank $2$ motives over $\mathbf{Q}$ with distinct
Hodge–Tate weights (corresponding to modular forms of weight $k\geq 2$ rather
than $k=2$) and also to such motives over totally real fields (which are
related to Hilbert modular forms), see in particular the work of Fujiwara [85]
(and more recently Frietas–Le Hung–Siksek [84]). Another very useful
innovation was a base change idea of Skinner–Wiles [161] which circumvented
the need to rely on Ribet’s level lowering theorem. The use of cyclic base
change ([127] in this case and [5] in general) subsequently became a standard
tool in the subject. For example, it meant that one could always reduce to a
situation where the ramification at all primes $v\nmid p$ was unipotent. The
paper [161] was related to a more ambitious plan by Wiles to prove modularity
for all totally real fields:

After Fermat I started to work with Taylor and then Diamond on the general
case but decided very soon that I would rather try to do the totally real case
for $\mathrm{GL}(2)$. I think this was while I was getting back into other
kinds of problems but I thought I should still earn my bread and butter. One
lunch time at the IAS in 1996 Florian Pop spoke to me and explained to me
about finding points over fields totally split at some primes (e.g. real
places) as he had written a paper [92] about this with some others. Was this
any use for the Tate–Shavarevich group? I immediately saw that whether or not
it was any use for TS (I doubted it) it should certainly give potential
modularity. This gave some kinds of lifting so I worked on the other half
(i.e. descent) thinking that just needed a similar insight. At some point I
suggested to Chris that we try to do Ribet’s theorem using cyclic base change
as that would be useful in proving modularity and was buying time while I
waited to get the right idea. Unfortunately I completely misjudged the
difficulty of descent and the problem is still there. I think it is both much
harder than I thought and also more important. I hope still to prove it! Of
course Taylor found potential modularity and then, what I had assumed was much
harder, a way to think about $\mathrm{GL}(n)$.

##  3\. Reducible representations: Skinner–Wiles

One of the key hypotheses in the Taylor–Wiles method concerns restrictions on
the representation $\overline{\rho}$, in particular the hypothesis that
$\overline{\rho}|_{G_{\mathbf{Q}(\zeta_{p})}}$ is absolutely irreducible. In
[162, 160, 159], Skinner and Wiles introduced a new argument in which this
hypothesis was relaxed, at least assuming the representations were ordinary at
$p$. In the ordinary setting, one can replace the rings $R$ and $\mathbf{T}$
(which in the original setting are finite over $W(k)$) by rings which are
finite (and typically flat) over Iwasawa algebras
$\Lambda=W(k)\llbracket(\mathbf{Z}_{p})^{d}\rrbracket$ for some $d$ which
arise as weight spaces, the point being that the ordinary deformations of
varying weight admit a good integral theory. The first innovation (in part)
involves making a base change so that the reducible locus is (relatively)
“small,” (measured in terms of the codimension over $\Lambda$). The second
idea is then to apply a variant of the Taylor–Wiles method to representations
$\varrho:G_{F}\rightarrow\mathrm{GL}_{2}(\mathbf{T}/\mathfrak{p})$ for non-
maximal prime ideals $\mathfrak{p}$ of $F$111111Representations $\varrho$ to
infinite quotients $\mathbf{T}/\mathfrak{p}$ had also arisen in Wiles’ paper
on Galois representations associated to ordinary modular forms [179] where the
concept of pseudo-deformation was also first introduced.. Wiles again:

We had worked out a few cases we could do without big Hecke rings in some
other papers and I would say it was more a feat of stamina and technique to
work through it. Of course the use of these primes was much more general and
systematic than anything that went before. There is also an amusing point in
this paper where we use a result from commutative algebra. It seemed crucial
then though I don’t know if it still is. This is proposition A.1 of Raynaud
[148]. I had thought at some point during the work on Fermat that this result
might be needed and had asked Michel Raynaud about it. He said he would think
about it. A week later he came back to me, somewhat embarrassed that he had
not known right away, to say that it was a result in his wife’s thesis. So the
reference to M.Raynaud is actually to his wife, Michèle Raynaud, though he
gave the reference.

Allen [2] was later able to adapt these arguments to the $p=2$ dihedral case,
which (in a certain sense) realized the original desire121212As far as primary
historical sources go, the introduction of Wiles’ paper [180] is certainly
worth reading. of Wiles to work at the prime $p=2$.

##  4\. The Artin Conjecture

While the approach of [180, 169] applied (in principle) to all Galois
representations associated to modular forms of weight $k\geq 2$, the case of
modular forms of weight $k=1$ is qualitatively quite different (see also
§10.1). It was therefore quite surprising when Buzzard–Taylor [33] proved
weight one modularity lifting theorems for odd continuous representations
$\rho:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{Q}}_{p})$
which were unramified at $p$. Using this, Buzzard–Dickinson–Shepherd-
Barron–Taylor [31] proved the Artin conjecture for a positive proportion of
all odd $A_{5}$ representations, which had previously only been known in a
finite number of cases131313In a computational _tour de force_ for the time,
Buhler [29] in his thesis had previously established the modularity of an
explicit odd projective $A_{5}$ representation of conductor $800$. up to
twist. Standard ordinary modularity theorems showed the existence of ordinary
modular forms associated to such representations $\rho$ — however, the
classicality theorems of Hida [101] do not apply (and are not true!) in weight
one. The main idea of [33] was to exploit the fact that $\rho$ is unramified
to construct _two_ ordinary modular forms each corresponding to a choice of
eigenvalue of $\rho(\mathrm{Frob}_{p})$ assuming these eigenvalues are
distinct141414This argument can be modified to deal with the case when the
eigenvalues of $\overline{\rho}(\mathrm{Frob}_{p})$ coincide by modifying $R$
and $\mathbf{T}$ to include operators corresponding (on the Hecke side) to
$U_{p}$. Geraghty and I discovered an integral version of this idea ourselves
(“doubling,” following Wiese’s paper [178]) during the process of writing
[38], although it turned out that, at least in characteristic zero, Taylor
already had the idea in his back pocket in the early 2000s.. One then has to
argue [33] that these two ordinary forms are the oldforms associated to a
classical eigenform of weight one, which one can do by exploiting both the
rigid geometry of modular curves and the $q$-expansion principle.

Although the original version of this argument required a number of
improvements to the usual Taylor–Wiles method (Dickinson overcame some
technical issues when $p=2$ [73] and Shepherd-Barron–Taylor proved some new
cases of Serre’s conjecture for $\mathrm{SL}_{2}(\mathbf{F}_{4})$ and
$\mathrm{SL}_{2}(\mathbf{F}_{5})$-representations in [157]), it was ripe for
generalization to totally real fields151515The proof all that finite odd
$2$-dimensional representations over $\mathbf{Q}$ are modular was completed by
Khare and Wintenberger as a consequence of their proof of Serre’s conjecture,
see §8. After a key early improvement by Kassaei [106], the $n=2$ Artin
conjecture for totally real fields is now completely resolved under the
additional assumption that the representation is odd by a number of authors,
including Kassaei–Sasaki–Tian and Pilloni–Stroh [108, 107, 109, 152, 141,
143]. On the other hand, the reliance on $q$-expansions in this argument has
proved an obstruction to extending this to other groups. (See also §11.2).

##  5\. Potential Modularity

One new idea which emerged in Taylor’s paper [166] was the concept of
_potential modularity_. Starting with a representation
$\rho:G_{F}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{Q}}_{p})$ for a
totally real field $F$, one could sidestep the (difficult) problem of proving
the modularity of $\overline{\rho}$ by proving it was modular over some finite
totally real extension $F^{\prime}/F$. In the original paper [180], Wiles
employed a $3$-$5$ switch to deduce the modularity of certain mod $5$
representations from the modularity of mod $3$ representations. More
generally, one can prove the modularity of a mod $p$ representation
$\overline{\rho}_{p}$ from the modularity of a mod $q$ representation
$\overline{\rho}_{q}$ if one can find both of them occurring as the residual
representation of a compatible family where the Taylor–Wiles hypotheses apply
to $\overline{\rho}_{q}$. For example, if $\overline{\rho}_{p}$ and
$\overline{\rho}_{q}$ are representations valued in
$\mathrm{GL}_{2}(\mathbf{F}_{p})$ and $\mathrm{GL}_{2}(\mathbf{F}_{q})$
respectively, one can try to find the compatible family by finding an elliptic
curve with a given mod $p$ and mod $q$ representation. The obstruction to
doing such a $p$-$q$ switch over $F$ is that the corresponding moduli spaces
(which in this case are twists of the modular curve $X(pq)$) are not in
general rational, and hence have no reason to admit rational points. However,
exploiting an idea due to Moret-Bailly [132], Taylor showed that these moduli
spaces at least had many points over totally real fields where one could
additionally ensure that the Taylor–Wiles hypothesis applies at the prime $q$.
At the cost of proving a weaker result, this gives a huge amount of extra
flexibility that has proved remarkably useful. Taylor’s first application of
this idea was to prove the Fontaine–Mazur conjecture for many $2$-dimensional
representations, since the potential modularity of these representations was
enough to prove (for example) that they come from compatible families of
Galois representations (even over the original field $F$!), and that they
satisfy purity (which is known for Hilbert modular forms of regular weight).
The concept of potential modularity, however, has proved crucial for other
applications, not least of which is the proof of the Sato–Tate conjecture (see
§9.2).

##  6\. The work of Kisin

A key ingredient in the work of Breuil–Conrad–Diamond–Taylor (§2.2,§2.3) (and
subsequent work of Savitt [153, 154]) was the fact that a certain local
deformation ring $R^{\mathrm{fl}}$ defined in terms of integral $p$-adic Hodge
theory was formally smooth. The calculations of [25, 154], however, applied
only to some (very) carefully chosen situations sufficient for elliptic curves
but certainly not for all $2$-dimensional representations. In the 2000s, Kisin
made a number of significant contributions, both to the understanding of local
deformation rings but also to the structure of the Taylor–Wiles argument
itself [117, 118, 121, 122, 123, 119].

###  6.1. Local deformation rings at $v=p$

One difficulty with understanding local deformation rings $R^{\mathrm{fl}}$
associated to finite flat group schemes over highly ramified bases is that the
group schemes themselves are not uniquely defined by their generic fibres.
Kisin [122] had the idea that one could also define the moduli space of the
group schemes themselves, giving a projective resolution
$\mathscr{G}\kern-1.00006pt{\mathscr{R}}\rightarrow\mathrm{Spec}(R^{\mathrm{fl}})$
(this map is an isomorphism after inverting $p$). Kisin further realized that
the geometry of $\mathscr{G}\kern-1.00006pt{\mathscr{R}}$ was related to local
models of Shimura varieties, for which one had other available techniques to
analyze their structure and singularities. Later, Kisin was also able [119] to
construct local deformation rings $R$ capturing deformations of a fixed local
representation $\overline{\rho}$ which become semi-stable over a fixed
extension $L/\mathbf{Q}_{p}$ and had Hodge–Tate weights in any fixed finite
range $[a,b]$, absent a complete integral theory of such representations.
(There are also are constructions where one fixes the inertial type of the
corresponding representation.) Kisin further proved that the generic fibres of
these rings were indeed of the expected dimension and often formally smooth.

###  6.2. Kisin’s modification of Taylor–Wiles

Beyond analyzing the local deformation rings themselves, Kisin crucially found
a way [122] to modify the Taylor–Wiles method to avoid the requirement that
these rings are formally smooth, thus greatly expanding the scope of the
method. First of all, Kisin reimagined the global deformation ring $R$ as an
algebra over a (completed tensor product)

$R^{\mathrm{loc}}=\widehat{\bigotimes}_{v\in S}R_{v}$

of local deformation rings $R_{v}$ for sets of places $v\in S$, in particular
including the prime $p$161616Since the local residual representations are
typically reducible, Kisin also introduced the notion of _framed_ deformation
rings which are always well-defined, and which (properly taking into account
the extra variables) are compatible with the Taylor–Wiles argument.. Now,
after a Taylor–Wiles patching argument, one constructs a big module
$M_{\infty}$ over $R_{\infty}$ (and free over the auxiliary ring of diamond
operators $S_{\infty}$) but where $R_{\infty}$ is no longer a power series
ring over $W(k)$ but a power series ring over $R^{\mathrm{loc}}$. If the
algebras $R_{v}$ for $v\in S$ are themselves power series rings, one is
reduced precisely to the original Taylor–Wiles setting as modified by Diamond.
On the other hand, if the $R_{v}$ are (for example) not power series rings but
are integral domains over $W(k)$ of the expected dimension, then Kisin
explained how one could still deduce that $M[1/p]$ was a faithful
$R[1/p]$-module, which proves that $R[1/p]=\mathbf{T}[1/p]$ and suffices for
applications to modularity. More generally, assuming only that the $R_{v}$ are
flat over $W(k)$ and that the generic fibre $R_{v}[1/p]$ is equidimensional of
the expected dimension, the modularity of any point of $R$ reduces to showing
that there is at least one modular point which lies on the same component of
$R_{v}[1/p]$171717There are some subtleties to understanding $R[1/p]$ for
complete local Noetherian $W(k)$-algebras that are not obvious on first
consideration. The first and most obvious blunder to avoid is to recognize
that $R[1/p]$ is usually far from being a local ring. Similarly, the ring
$R[1/p]$ can be regular and still have multiple components, as can be seen in
an example as simple as $R=\mathbf{Z}_{p}\llbracket X\rrbracket/X(X-p)$.
Perhaps more importantly, however, the ring $R[1/p]$ “behaves” in some
important ways like a finitely generated algebra over a field..

In the original modularity lifting arguments, one treated the minimal case
first and then deduced the non-minimal cases using a subtle commutative
algebra criterion which detected isomorphisms between complete intersections.
From the perspective of Kisin’s modification, all that is required is to show
that there exists a single modular point with the right non-minimal local
properties. In either case, both Wiles and Kisin used Ihara’s Lemma to
establish the existence of congruences between old and new forms, but Kisin’s
argument is much softer and thus more generalizable to other
situations181818In particular, Wiles’ numerical criterion [68, Thm 5.3] relies
on certain rings being complete intersections, and Kisin’s local deformation
rings are not complete intersections (or even Gorenstein) in general — see
[163].. Kisin had the following to say about his thought process:

The idea of thinking of $R$ as an $R^{\mathrm{loc}}$ algebra just popped into
my head, after I’d been thinking about the Wiles–Poitou–Tate formula, and how
it fit into the Taylor–Wiles patching argument. This was in Germany, I think
in 2002. I had the idea about moduli of finite flat group schemes in the Fall
of 2003, after I arrived in Chicago. It was entirely motivated by modularity.
I had been trying to compute these deformation rings, by looking at
deformations of finite flat group schemes. For $e<p-1$, the finite flat model
is unique, so I knew this gave the deformation ring in this case; this already
gave some new cases. However I was stuck about the meaning of these
calculations in general for quite some time. At some point I thought I’d
better write up what I had, but as soon as I started thinking about that —
within a day — I realized what the correct picture was with the families of
finite flat group schemes resolving the deformation ring. I already knew about
Breuil’s unpublished note [18], and quite quickly was able to prove the
picture was correct. It was remarkable that prior to coming to Chicago, I
didn’t even know the definition of the affine Grassmannian, but within a few
months of arriving, it actually showed up in my own work.

To me the whole project was incredibly instructive. If I had known more about
what was (thought to be) essential in the Taylor–Wiles method, I never would
have started the project. Not having fixed ideas gave me time to build up
intuition. I also should have gotten the idea about moduli of finite flat
group schemes much sooner if I’d been more attentive to what the geometry was
trying to tell me.

##  7\. $p$-adic local Langlands

###  7.1. The Breuil–Mézard conjecture

Prior to Kisin’s work, Breuil and Mézard [26] undertook a study of certain low
weight potentially semi-stable deformation rings, motivated by [25]. They
discovered (in part conjecturally) a crucial link between the geometry of
these Galois deformation rings (in particular, the Hilbert–Samuel
multiplicities of their special fibres) with the mod-$p$ reductions (and
corresponding irreducible constituents) of lattices inside locally algebraic
$p$-adic representations of $\mathrm{GL}_{2}(\mathbf{Z}_{p})$. In the
subsequent papers [20, 21], Breuil raised the hope that there could exist a
$p$-adic Langlands correspondence relating certain mod-$p$ (or $p$-adic Banach
space) representations of $\mathrm{GL}_{2}(\mathbf{Q}_{p})$ to geometric
$2$-dimensional $p$-adic representations of $G_{\mathbf{Q}_{p}}$191919The
starting observation [22] is as follows: if $\pi=\bigotimes^{\prime}\pi_{v}$
is the automorphic representation associated to a modular form $f$, then
$\pi_{v}$ determines (and is determined by) $\rho_{f}|_{G_{\mathbf{Q}_{v}}}$
for all $v\neq p$ (at least up to Frobenius semi-simplification). On the other
hand, $\pi_{p}$ does not determine the $p$-adic representation
$\rho_{f}|_{G_{\mathbf{Q}_{p}}}$ (except in the exceptional setting where
$\pi_{p}$ is spherical and $a_{p}$ is not a $p$-adic unit), raising the
question of what extra $\mathrm{GL}_{2}(\mathbf{Q}_{p})$ structure associated
to $f$ should determine (and be determined by)
$\rho_{f}|_{G_{\mathbf{Q}_{p}}}$.. Breuil recounts the origins of these
conjectures as follows:

The precise moment I became 100% sure that there would be a non-trivial
$p$-adic correspondence for $\mathrm{GL}_{2}(\mathbf{Q}_{p})$ was in the
computations of [21]. In these computations, I reduced mod $p$ certain
$\overline{\mathbf{Z}}_{p}$–lattices in certain locally algebraic
representations of $\mathrm{GL}_{2}(\mathbf{Q}_{p})$, and at some point, I
found out that this reduction mod $p$ had a really nice behaviour, so nice
that clearly, it was predicting (via the mod-$p$ correspondence) what the
reduction mod-$p$ would be on the
$\mathrm{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})$-side.

These ideas were further developed by Colmez in [58, 59, 60] amongst other
papers202020In [116], Kisin had shown that the $p$-adic representations $V$
associated to non-classical finite slope overconvergent modular forms with
$U_{p}$-eigenvalue $a_{p}$ satisfied $\dim D_{\mathrm{cris}}(V)=1$, and
moreover that crystalline Frobenius acted on this space by $a_{p}$. (This
paper was itself apparently motived by the goal of _disproving_ the
Fontaine–Mazur conjecture!) On the way to the $2004$ Durham symposia on
$L$-functions and Galois representations, Fontaine raised the question to
Colmez to what extent this determined the corresponding Galois representation.
Colmez worked out the answer the evening before his talk and incorporated it
into his lecture the following day, ultimately leading to the notion of
trianguline representations [58].: Colmez studied various Banach space
completions defined by Breuil and proved they were non-zero using the theory
of $(\varphi,\Gamma)$-modules. Since the theory of $(\varphi,\Gamma)$-modules
applies to all Galois representations and not just potentially semi-stable
ones, this led Colmez to propose a $p$-adic local Langlands correspondence for
_arbitrary_ $2$-dimensional representations
$G_{\mathbf{Q}_{p}}\rightarrow\mathrm{GL}_{2}(E)$, and he was ultimately able
to construct a functor from suitable
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$-representations to Galois representations of
$G_{\mathbf{Q}_{p}}$. Colmez gave a talk on his construction at a conference
in Montreal in September 2005. At the same conference, Kisin gave a talk
presenting a proof of the Breuil–Mézard conjecture by relating it directly to
$R=\mathbf{T}$ theorems and the Fontaine–Mazur conjecture for odd
$2$-dimensional representations of $G_{\mathbf{Q}}$ with distinct Hodge–Tate
weights. While Kisin’s argument exploited results of Berger–Breuil [14] and
Colmez, it was realized by the key participants (perhaps in real time) that
Colmez’ $p$-adic local Langlands correspondence should be viewed as taking
place over the entire local deformation ring. Subsequently Colmez was able to
construct the inverse functor212121To add some further confusion to the
historical chain of events, the published version of [120] incorporates some
of these subsequent developments. Note also that the current state of affairs
is that the proof of the full $p$-adic local Langlands correspondence for
$\mathrm{GL}_{2}(\mathbf{Q}_{p})$ (for example as proved in [63] but see also
[60, Remarque VI.6.51]) still relies on the global methods of [78], which in
turn relies on [60]. These mutual dependencies, however, are not circular! The
difficulty arises in the supercuspidal case. One philosophical reason that
global methods are useful here is that all global representations are yoked
together by an object (the completed cohomology group
$\widetilde{H}^{1}(\mathbf{Z}_{p})$) with good finiteness properties. One can
then exploit the fact that crystabeline representations (for which the
$p$-adic local Langlands correspondence is known by [60]) are Zariski dense
inside unrestricted global deformation rings ([78, Theorem 1.2.3], using
arguments going back to Böckle [15]).. Colmez writes:

I received a paper of Breuil (a former version of [23]) during my stay at the
Tata Institute in December 2003—January 2004. In December, I was spending
Christmas under Goa’s palm trees with my daughter when Breuil’s paper arrived
in my email. That paper contained a conjecture (in the semi-stable case) that
I was sure I could prove using $(\varphi,\Gamma)$-modules (if it was true…). I
spent January 2004 working on it and after 15 days of computations in the
dark, I finally found a meaning to some part of a painful formula (you can
find some shadow of all of this in (iii) of Remark 0.5 of my unpublished
[57]). By the end of the month, I was confident that the conjecture was proved
and I told so to Breuil who adapted the computations to the crystalline case,
and wrote them down with the help of Berger (which developed into [14]). (One
thing that makes computations easier and more conceptual in the crystalline
case is that you end up with the universal completion of the locally algebraic
representation you start with; something that is crucial in Matthew
[Emerton]’s proof of the FM conjecture.) Durham was in August of that year and
Berger–Breuil had notes from a course they had given in China [13]. Those
notes were instrumental in my dealing with trianguline representations at
Durham (actually, I did some small computation and the theory just developed
by itself during the night before my talk which was supposed to be on
something else…I think I came up with the concept of trianguline
representations later, to justify the computations, I don’t remember what
language I used in my talk which had some part on Banach–Colmez spaces as far
as I can remember.

###  7.2. Local–global compatibility for completed cohomology

From a different perspective, Emerton had introduced the completed cohomology
groups [77] as an alternative means for constructing the Coleman–Mazur
eigencurve [54]. Inspired by Breuil’s work, Emerton formulated [76] a
local–global compatibility conjecture for completed cohomology in the language
of the then nascent $p$-adic Langlands correspondence. After the construction
of the correspondence for $\mathrm{GL}_{2}(\mathbf{Q}_{p})$ by Colmez and
Kisin, Emerton was able to prove most of his conjecture, leading to a new
proof of (many cases of) the Fontaine–Mazur conjecture. The results of Kisin
[120] and Emerton fell short of proving the full version of this conjecture
for two reasons. The first was related to some technical issues with the
$p$-adic local Langlands correspondence, both at the primes $p=2$ and $3$ but
also when the residual representation locally had the shape
$1\oplus\overline{\varepsilon}$ for the cyclotomic character $\varepsilon$.
(The local issues have now more or less all been resolved [63]. The most
general global results for $p=2$ are currently due to Tung [176].) A second
restriction was the Taylor–Wiles hypothesis that $\overline{\rho}$ was
irreducible. Over the intervening years, a number of key improvements to the
local story have been found, in particular by Colmez, Dospinescu, Hu, and
Paškūnas [140, 63, 104]. Very recently, Lue Pan [137] found a way to marry
techniques from Skinner–Wiles in the reducible case (§3) to techniques from
$p$-adic local Langlands to completely prove the modularity (up to twist) of
any geometric representation
$\rho:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{Q}}_{p})$ for
$p\geq 5$ only assuming the hypotheses that $\rho$ has distinct Hodge–Tate
weights and that $\rho$ is odd222222The assumption on the Hodge–Tate weights
is almost certainly removable using recent progress on the ideas discussed in
§4 (Sasaki has announced such a result). Moreover, Pan has found a different
approach to this case as well, see [138, Theorem 1.0.5] and the subsequent
comments. The hypothesis that $\rho$ is odd is more troublesome — see §9.7..

##  8\. Serre’s Conjecture

In Wiles’ original lectures in Cambridge in 1993, he introduced his method
with the statement that it was  _orthogonal_ to Serre’s conjecture [150]. In
some senses, this viewpoint turned out to be the opposite of prophetic, in
that the ultimate resolution of Serre’s conjecture used the Taylor–Wiles
method as its central core. The proof of Serre’s conjecture by Khare and
Wintenberger [111, 112, 113, 114] introduced a new technique for lifting
residual Galois representations to characteristic zero (see §8.2) which has
proved very useful for subsequent modularity lifting theorems.

###  8.1. Ramakrishna lifting

Ramakrishna, in a series of papers in the late 90s [147, 146], studied the
question of lifting an odd Galois representation

$\overline{\rho}:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$

to a global potentially semistable representation in characteristic zero
unramified outside finitely many primes. This is a trivial consequence of
Serre’s conjecture232323Trivial only assuming the results of Tsuji [175] and
Saito [151], of course. but is highly non-obvious without such an assumption.
Ramakrishna succeeded in proving the existence of lifts by an ingenious
argument involving adding auxiliary primes and modifying the local deformation
problem to a setting where there all global obstructions vanished. The
resulting lifts had the added property that they were valued in
$\mathrm{GL}_{2}(W(k))$ whenever $\overline{\rho}$ was valued in
$\mathrm{GL}_{2}(k)$. Adaptations of Ramakrishna’s method had a number of
important applications even under the assumption of residual modularity,
including in [50] where it was used to produce characteristic zero lifts with
Steinberg conditions at some auxiliary primes. There is also recent work of
Fakhruddin, Khare, and Patrikis [79] which considerably extends these results
in a number of directions.

###  8.2. The Khare–Wintenberger method

One disadvantage of Ramakrishna’s method was that it required allowing
auxiliary ramification which (assuming Serre’s conjecture) should not be
necessary242424If one insists on finding a lift is valued in
$\mathrm{GL}_{2}(W(k))$ rather than $\mathrm{GL}_{2}(\mathcal{O}_{E})$ for
some ramified extension $E/W(k)[1/p]$, then some auxiliary ramification is
necessary in general, at least in fixed weight.. Khare and Wintenberger found
a new powerful method for avoiding this. The starting point is the idea that,
given an odd representation
$\overline{\rho}:G_{F}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{F}}_{p})$
for a totally real field $F$ satisfying the Taylor–Wiles hypotheses, one could
find a finite extension $H/F$ where $\overline{\rho}$ is modular (exactly as
in §5). Then, using an $R=\mathbf{T}$ theorem over $H$, one proves that the
corresponding deformation ring $R_{H}$ of $\overline{\rho}|_{G_{H}}$ is finite
over $W(k)$. However, for formal reasons, there is a map $R_{H}\rightarrow
R_{F}$ (where $R$ is the deformation ring corresponding to the original
representation $\overline{\rho}$) which is a finite morphism, and hence the
ring $R_{F}/p$ is Artinian. Then, by Galois cohomological arguments, one
proves the ring $R_{F}$ has dimension at least one, from which one deduces
that $R_{F}$ has $\overline{\mathbf{Q}}_{p}$-valued points. Even more can be
extracted from this argument, however — the $\overline{\mathbf{Q}}_{p}$-valued
point of $R_{F}$ certainly comes from a $\overline{\mathbf{Q}}_{p}$-valued
point of $R_{H}$, and hence comes from a compatible family of Galois
representations over $H$. Using the fact that one member of the family extends
to $G_{F}$, it can be argued that the entire family descends to a compatible
family over $F$. This one can then hope to prove is modular by working at a
different (possibly smaller) prime, where (hopefully) one can prove the
associated residual representation is modular. In this way, one can
inductively reduce Serre’s conjecture [156] to the case $p=2$ and
$N(\overline{\rho})=1$, where Tate had previously proved in a letter to Serre
[62, July 2, 1973] (also [164]) that all such absolutely irreducible
representations are modular by showing that no such representations exist. The
entire idea is very clean, although in practice the difficulty reduces to the
step of proving modularity lifting theorems knowing either that
$\overline{\rho}$ is either modular and absolutely irreducible or is
reducible. Khare and Wintenberger’s timing was such that the automorphy
lifting technology was just good enough for the proof to work, although this
required some extra effort at the prime $p=2$ (both in their own work and in a
key assist by Kisin [121]). As with Ramakrishna’s method, the
Khare–Wintenberger lifting method has also been systematically exploited for
modularity lifting applications (for example in [11] (see §9.6) building on
ideas of Gee [87]).

##  9\. Higher dimensions

Parallel to the developments of $p$-adic Langlands for $n=2$, the first steps
were made to generalize the theory to higher dimensional representations.
Unlike in the case of modular forms, substantially less was known about the
existence of Galois representations until the 90s.

###  9.1. Construction of Galois representations, part I: Clozel–Kottwitz

The first general construction252525Clozel’s paper is from 1991 and thus not
strictly “post–Fermat” as is the remit of this survey. However, it can be
considered a natural starting point for the “modern” arithmetic theory of
automorphic forms for $\mathrm{GL}(n)$ and so it seems reasonable to mention
it here. of Galois representations in dimension $n>2$ was made by Clozel [48]
(see also the work of Kottwitz [125]). Clozel’s theorem applies to certain
automorphic forms for $\mathrm{GL}_{n}(\mathbf{A}_{L})$ for CM fields
$L/L^{+}$. The construction requires three important hypotheses on $\pi$: The
first is that $\pi$ is conjugate self-dual, that is $\pi^{\vee}\simeq\pi^{c}$.
If $\pi$ is a base change from an algebraic representation of $L^{+}$ and
$n=2$ then this condition is automatic262626At least after a twist which is
always possible to achieve in practice, see [50, Lemma 4.1.4]. More generally,
one can work with unitary similitude groups and consider $\pi$ with
$\pi^{\vee}\simeq\pi^{c}\otimes\chi$ for suitable characters $\chi$., but it
is far from automatic when $n>2$. The second condition is an assumption on the
infinitesimal character which (in the case of modular forms) is equivalent to
the condition that the weight $k$ is $\geq 2$. Finally, there is a technical
condition that for some finite place $x$ the representation $\pi_{x}$ is
square integrable. A number of improvements (particularly at the bad primes)
were made by Harris–Taylor in [100, Theorem C], and later by Taylor–Yoshida
and Caraiani [170, 42, 43], bringing the theory roughly in line with that of
modular forms at the time of Wiles, and in particular primed for possible
generalizations of the Taylor–Wiles method to higher dimensions.

###  9.2. The Sato–Tate conjecture, part I

Harris, and Taylor (as early as 1996) started the work of generalizing the
Taylor–Wiles machinery to the setting of $n$-dimensional representations. They
quickly understood that the natural generalization of these ideas in
$n$-dimensions required the hypothesis that the Galois representations were
self-dual up to a twist. This meant that one should not consider general
automorphic forms on the group $\mathrm{GL}_{n}(\mathbf{A}_{\mathbf{Q}})$ but
rather groups of symplectic or orthogonal type depending on the parity of $n$.
If one replaced Galois representations over totally real fields by Galois
representations over imaginary CM fields and then further imposed the
condition that the Galois representations are conjugate self-dual, the
relevant automorphic forms should then come from unitary groups. There were
two benefits of working with these hypotheses. First of all, the relevant
automorphic representations for unitary groups were, as with modular forms,
associated to cohomology classes on Shimura varieties. In particular, under
the assumption that there existed an auxiliary prime $x$ such that $\pi_{x}$
was square integrable, they could be seen inside the “simple” Shimura
varieties of type $U(n-1,1)$ considered by Kottwitz [125]. On the other hand,
the same Hecke eigenclasses (if not Galois representations) also came from a
compact form of the group and thus inside the cohomology of zero-dimensional
varieties272727Inside $H^{0}$, of course. The advantage of working in this
setting is that the freeness of $M_{Q}$ over the ring of diamond operators is
immediate282828In more general contexts, the freeness of $M_{Q}$ is closely
related to the vanishing of cohomology localized at $\mathfrak{m}$ in all but
one degree.. In the fundamental paper [50], Clozel, Harris, and Taylor
succeeded in overcoming many of the technical difficulties generalizing the
arguments of [180, 169] to these representations. Although the argument in
spirit was very much the same, there are a number of points for
$\mathrm{GL}_{2}$ where things are much easier. One representative example of
this phenomenon is understanding Taylor–Wiles primes. While the Galois side
generalizes readily, the automorphic side requires many new ideas and some
quite subtle arguments concerning the mod $p$ structure of certain
$\mathrm{GL}_{n}(\mathbf{Q}_{q})$-representations of conductor $1$ and
conductor $q$. In order to prove the Sato–Tate conjecture for a modular form
$f$, it was already observed by Langlands that it sufficed to prove the
modularity of all the symmetric powers of $f$. However, it turns out that the
weaker assumption that each of these symmetric powers is _potentially modular_
suffices, and by some subterfuge only the even powers are required [95]. In
order to prove potential modularity theorems, one needs to be able to carry
out some version of the $p$-$q$ switch (§5). In order to do _this_ , one needs
a source of motives which both generate Galois representations of the right
shape (conjugate self dual and with distinct Hodge–Tate weights) and yet also
come in positive dimensional families. It turned out that there already
existed such motives in the literature, namely, the so-called Dwork family.
However, given the strength of the automorphy lifting theorems in [50],
considerable effort had to be made in studying the geometry of the Dwork
family to ensure that the $p$-$q$ switch would produce geometric Galois
representations with the right local properties. These issues were precisely
addressed in the companion paper by Harris, Shepherd-Barron and Taylor [97].
Taken together, these papers contained all the ingredients to prove the
potential modularity of higher symmetric powers of modular forms (satisfying a
technical square integrable condition at some auxiliary prime) with one
exception. As mentioned earlier, the work of Kisin had simplified the passage
from the minimal case to the non-minimal case — “all” that was required was to
produce congruences between the original form and forms of higher level rather
than to compute a precise congruence number as in [180]. However, even
applying Kisin’s approach seemed to require Ihara’s Lemma, and despite several
years of effort, the authors of [50] were not able to overcome this
obstacle292929The issue remains unresolved to this day.. Here is Michael
Harris’ recollection of the process:

In the spring of 1995, I was at Brandeis, Richard was at MIT, and I wanted to
understand the brand new proof of Fermat’s Last Theorem. So I asked Richard if
he would help me learn by collaborating on modularity for higher-dimensional
groups. The collaboration took off a year later, when Richard wrote to tell me
about the Diamond–Fujiwara argument and suggested that we work out the
Taylor–Wiles method for unitary groups. This developed over the next 18 months
or so into the early version of what eventually became the IHES paper with
Clozel. But it had no punch line. I was hoping to work out some non-trivial
examples of tensor product functoriality for
$\mathrm{GL}(n)\times\mathrm{GL}(m)$, where one of the two representations was
congruent mod $l$ to one induced from a CM Hecke character. This would have
required some numerical verification. In the meantime we got sidetracked into
proving the local Langlands conjecture [100].

The manuscript on automorphy lifting went through several drafts and was
circulated; you can still read it on my home page [99]. Genestier and Tilouine
[88] quoted it when they proved modularity lifting for Siegel modular forms.
When Clozel saw the draft he told me we should try to prove the Sato–Tate
Conjecture. Although this was in line with my hope for examples of tensor
product functoriality, it seemed completely out of reach, because I saw no way
to prove residual modularity of symmetric powers.

When I heard about the Skinner–Wiles paper I came up with a quixotic plan to
prove symmetric power functoriality for Eisenstein representations, using the
main conjecture of Iwasawa theory to control the growth of the deformation
rings. This was in the spring of 2000, at the IHP special semester on the
Langlands program, where I first met Chris Skinner.

One day Chris told me that Richard had invented potential modularity. This led
me to a slightly less hopeless plan to prove potential symmetric power
functoriality by proving it for $2$-dimensional representations congruent to
potentially abelian representations, as in the potential modularity argument.
I told Richard about this idea, probably the day he arrived in Paris. He
asked: why apply potential modularity to the $2$-dimensional representation;
why not instead apply it to the the symmetric power representations directly?
I then replied: that would require a variation of Hodge structures with a
short list of properties: mainly, the correct $h^{p,q}$’s and large monodromy
groups. We checked that potential modularity was sufficient for Sato–Tate. We
then resolved to ask our contacts if they knew of VHS with the required
properties. The whole conversation lasted about 20 minutes.

I asked a well-known algebraic geometer, who said he did not know of any such
VHS. Richard asked Shepherd-Barron, who immediately told him about the Calabi-
Yau hypersurfaces that had played such an important role in the mirror
symmetry program. (And if my algebraic geometer hadn’t wanted to be
dismissive, for whatever reason, he would have realized this as well.) The
$h^{p,q}$’s were fine but we didn’t know about the monodromy. However, Richard
was staying at the IHES, and by a happy accident so was Katz, and when Richard
asked Katz about the monodromy for this family of hypersurfaces Katz told him
they were called the Dwork family and gave him the page numbers in one of his
books.

So within a week or two of our first conversation, we found ourselves needing
only one more result to complete the proof of Sato–Tate. This was Ihara’s
lemma, which occupied our attention over the next five years. In the meantime,
Clozel had written a manuscript on symmetric powers, based on the reducibility
mod ell of symmetric powers. The argument was incomplete but he had several
ideas that led to his joining the project, and he also hoped to use ergodic
theory to prove Ihara’s lemma. In the summer of 2003 Clozel and I joined
Richard in “old” Cambridge to try to work this out. The rest you know. We
finally released a proof conditional on Ihara’s lemma in the fall of 2005. A
few months later Richard found his local deformation argument, and the proof
was complete.

###  9.3. Taylor’s trick: Ihara Avoidance

Shortly after the preprints [50, 157] appeared, Taylor found a way to overcome
the problem of Ihara’s lemma. Inspired by Kisin’s formulation of the
Taylor–Wiles method (§6.2), Taylor had the idea of comparing two global
deformation rings $R^{1}$ and $R^{2}$. Here (for simplicity) the local
deformation problems associated to $R^{1}$ and $R^{2}$ are formally smooth at
all but a single prime $q$. At the prime $q$, however, the local deformation
problem associated to $R^{1}$ consists of tamely ramified representations
where a generator $\sigma$ of tame inertia has characteristic polynomial
$(X-1)^{n}$, and for $R^{2}$ the characteristic polynomial has the shape
$(X-\zeta_{1})\ldots(X-\zeta_{n})$ for some fixed distinct roots of unity
$\zeta_{i}\equiv 1\bmod p$. On the automorphic side, there are two patched
modules $H_{1}$ and $H_{2}$, and there is an equality $H_{1}/p=H_{2}/p$. The
local deformation ring $R^{1}_{q}$ associated to $R^{1}$ at $q$ is reducible
and has multiple components in the generic fibre, although the components in
characteristic zero are in bijection to the components in the special fibre.
On the other hand, the local deformation ring associated to $R^{2}$ at $q$
consists of a single component, and so using Kisin’s argument one deduces that
$H_{2}$ has full support. Now a commutative algebra argument using the
identity $H_{1}/p=H_{2}/p$ and the structure of $R^{1}_{q}$ implies that
$H_{1}$ has sufficiently large support over $R_{1}$, enabling one to deduce
the modularity of every $\overline{\mathbf{Q}}_{p}$-valued point of
$R_{1}$303030Taylor’s argument proves theorems of the form
$R[1/p]^{\mathrm{red}}=\mathbf{T}[1/p]$ rather than $R=\mathbf{T}$. This is
still perfectly sufficient for proving modularity lifting results, but not
always other interesting corollaries associated to $R=\mathbf{T}$ theorems
like finiteness of the corresponding adjoint Selmer groups (though see [3,
133])..

###  9.4. The Sato–Tate conjecture, part II

After Taylor’s trick, one was _almost_ in a position to complete the proof of
Sato–Tate for all classical modular forms. A few more arguments were required.
One was the tensor product trick due to Harris which enabled one to pass from
conjugate self-dual motives with weights in an arithmetic progression to
conjugate self-dual motives with consecutive Hodge–Tate weights by a judicious
twisting argument using CM characters. A second ingredient was the analysis of
the ordinary deformation ring by Geraghty [89]. One of the requirements of the
$p$-$q$ trick was the condition that certain moduli spaces (the Dwork family
in this case) had points over various local extensions $E$ of
$\mathbf{Q}_{p}$, in order to construct a motive $M$ over a number field $F$
with $F_{v}=E$ for $v|p$. For the purposes of modularity lifting, one wants
strong control over the local deformation ring at $p$, and the choice of local
deformation ring is more or less forced by the geometric properties of the
$p$-adic representations associated to $M$ . One way to achieve this would be
to work in the Fontaine–Laffaille range where the local deformation rings were
smooth. But this requires both that $M$ is smooth at $p$ and that the
ramification degree $e$ of $E/\mathbf{Q}_{p}$ is one. It is not so clear,
however, that the Dwork family contains suitable points (for a fixed residual
representation $\overline{\rho}$) which lie in any unramified extension of
$\mathbf{Q}_{p}$. What Geraghty showed, however, was that certain ordinary
deformation rings313131In Geraghty’s setting, the residual representations
$\overline{\rho}$ were locally trivial. Hence the definition of “ordinary” was
not something that could be defined on the level of Artinian rings, and the
construction (as with Kisin’s construction of local deformation rings
associated to certain types) is therefore indirect. were connected over
arbitrarily ramified bases. The final piece, however, was the construction of
Galois representations for all conjugate self-dual regular algebraic cuspidal
$\pi$ without the extra condition that $\pi_{q}$ was square integrable for
some $q$. This story merits it own separate discussion; suffice to say that it
required the combined efforts of many people and the resolution of many
difficult problems, not least of which was the fundamental lemma by Laumon and
Ngô [128, 136] (see also the Paris book project [49, 93], the work of Shin
[158], and many more references which if I attempted to make complete would
weigh down the bibliography and still contain grievous omissions).

###  9.5. Big image conditions

The original arguments in [180, 169] required a “big image” hypothesis, namely
that $\overline{\rho}$ was absolutely irreducible after restriction to the
Galois group of $\mathbf{Q}(\zeta_{p})$. Wiles’ argument also required the
vanishing of certain cohomology groups associated to the adjoint
representation of the image of $\overline{\rho}$. These assumptions had
natural analogues in [50] (so-called “big image” hypotheses) although they
were quite restrictive, and it wasn’t clear that they would even apply to most
residual representations coming from some irreducible compatible family. In
the setting of $2$-dimensional representations, the Taylor–Wiles hypothesis
guarantees the existence of many primes $q$ such that $q\equiv 1\bmod p$ and
such that $\overline{\rho}(\mathrm{Frob}_{q})$ has distinct eigenvalues. This
ensures, for example, that there cannot be any Steinberg deformations at $q$
because the ratio of the eigenvalues of any Steinberg deformation must be $q$.
In dimension $n$, one natural way to generalize this might be to say that
$\overline{\rho}(\mathrm{Frob}_{q})$ has distinct eigenvalues, although this
is not always possible to achieve for many irreducible representations
$\overline{\rho}$. A weaker condition is that
$\overline{\rho}(\mathrm{Frob}_{q})$ has an eigenvalue $\alpha$ with
multiplicity one. For such $q$, there will be no deformations which are
unipotent on inertia at $q$ for which the generalized $\alpha$ eigenspace is
not associated to a $1$-dimensional block. The translation of this into an
automorphic condition on $U_{q}$-eigenvalues is precisely what is done in [50]
(there are additional technical conditions on $\mathrm{Frob}_{q}$ with respect
to the adjoint representation $\mathrm{ad}(\overline{\rho})$ which we omit
here). In [172], however, Thorne finds a way to allow
$\overline{\rho}(\mathrm{Frob}_{q})$ to have an eigenvalue $\alpha$ with
higher multiplicity, and yet still cut out (integrally) the space of
automorphic forms whose Galois representations decompose at $q$ as an
unramified representation plus a one dimensional representation which is
tamely ramified of $p$-power order. This technical improvement is very
important because (as proved in the appendix by Guralnick, Herzig, Taylor, and
Thorne [172]) it imposes no restrictions on $\overline{\rho}$ when $p\geq
2n+1$ beyond the condition that $\overline{\rho}$ is absolutely irreducible
after restriction to $G_{\mathbf{Q}(\zeta_{p})}$. This improvement is very
useful for applications.

###  9.6. Potentially diagonalizable representations

After the proof of Sato–Tate for modular forms, Barnet-Lamb, Gee, and Geraghty
turned their attention to proving the analogous theorem for Hilbert modular
forms of regular weight. The methods developed so far were well-suited both to
representations $\rho$ which were either ordinary or when $\rho$ was not
ordinary but still Fontaine–Laffaille. (The latter implies that
$\overline{\rho}|_{G_{\mathbf{Q}_{p}}}$ is absolutely irreducible of some
particular shape.) For a modular form over $\mathbf{Q}$, one easily sees that
$\rho$ takes one of one of these forms for any sufficiently large $p$. For
Hilbert modular forms, one certainly expects that the ordinary hypothesis
should hold for all $v|p$ and infinitely many $p$, but this remains open. The
difficulty arises when, for some prime $p$ (that splits completely, say) the
$p$-adic representation is ordinary at some $v|p$ but non-ordinary other
$v|p$. The reason that this causes issues is that, when applying the Moret-
Bailly argument in the $p$-$q$ switch, one wants to avoid any ramification at
$p$ for the non-ordinary case, and yet have large ramification at the ordinary
case to make $\overline{\rho}$ locally trivial, and these desires are not
compatible. The resolution in [10] involved a clever refinement of the Harris
tensor product trick. These ideas were further refined in [11] and led to the
concept of a _potentially diagonalizable_ representation
$\rho:G_{E}\rightarrow\mathrm{GL}_{n}(\overline{\mathbf{Q}}_{p})$ for some
finite extension of $E/\mathbf{Q}_{p}$. Recall from §6.2 that, in the modified
form of the Taylor–Wiles method, proving modularity of some lift of
$\overline{\rho}$ often comes down to showing the existence of a modular lift
lying on a smooth point of the corresponding component of the generic fibre of
$R^{\mathrm{loc}}$. In light of Taylor’s Ihara avoidance trick (§9.3), the
difficulty in this problem is mostly at the prime $p$, and in particular the
fact that one knows very little about the components of general Kisin
potentially crystalline deformation rings. A potentially diagonalizable
representation is one for which, after some finite (necessarily solvable!)
extension $E^{\prime}/E$, the representation $\rho|_{G_{E}^{\prime}}$ is
crystalline and lies on the same generic irreducible component as a diagonal
representation. This notion has a number of felicitous properties. First, it
includes Fontaine–Laffaille representations and ordinary potentially
crystalline representations. Second, it is clearly invariant under base
change. Third, it is compatible with the tensor product trick of Harris. These
features make it supremely well-adapted to the current forms of the
Taylor–Wiles method. By combining this notion with methods of [10, 12], as
well as extensive use of Khare–Wintenberger lifting (§8.2), Barnet–Lamb, Gee,
Geraghty, and Taylor in [11] proved the potential automorphy of _all_
conjugate self-dual irreducible323232One variant proved shortly thereafter by
Patrikis–Taylor [139] replaced the irreducibility condition by a purity
condition (which is automatically satisfied by representations coming from
pure motives). odd333333Although there is no longer a non-trivial complex
conjugation in the Galois group of a CM field, there is still an oddness
condition related to the conjugate self-duality of the representation and the
fact that there are two ways for an irreducible representation to be self-dual
(orthogonal and symplectic). compatible systems of Galois representations over
a totally real field.

###  9.7. Even Galois Representations

The Fontaine–Mazur conjecture for geometric Galois representations
$\rho:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\overline{\mathbf{Q}}_{p})$
predicts that, up to twist, either $\rho$ is modular _or_ $\rho$ is even with
finite image. The methods of [180, 169] required the assumption that
$\overline{\rho}$ was modular and so _a priori_ the assumption that $\rho$ was
odd (at least when $p>2$). Nothing at all was known about the even case before
the papers [34, 35] in which a very simple trick made the problem accessible
to modularity lifting machinery under the assumption that the Hodge–Tate
weights are distinct. The punch line is that, for any CM field $F/F^{+}$, the
restriction
$\mathrm{Sym}^{2}(\rho):G_{F}\rightarrow\mathrm{GL}_{3}(\overline{\mathbf{Q}}_{p})$
is conjugate self-dual and no longer sees the “evenness” of $\rho$343434The
representation $\rho$ itself restricted to $G_{F}$ will not be odd in the
required sense — one exploits the fact here that $3$ is odd whereas symplectic
representations are always even dimensional.. Hence one can hope to prove it
is potentially modular for some CM extension $L/L^{+}$, and then by cyclic
base change [5] potentially modular for the totally real field $L^{+}$. But
Galois representations coming from regular algebraic automorphic forms for
totally real fields will not be even353535I managed to twist Taylor’s arm into
writing the paper [168] which proved this for odd $n$, which sufficed for my
purposes where $n$ was either $3$ or $9$. This is now also known for general
$n$, see Caraiani–Le Hung [44]., and thus one obtains a contradiction. These
ideas are already enough to deduce the main result of [34] directly from [11],
although in contrast [35] uses (indirectly) the full strength of the $p$-adic
local Langlands correspondence via theorems of Kisin [120]. The papers [34,
35] still fall short of completely resolving the Fontaine–Mazur in this case
even for $p>7$, since there remain big image hypotheses on $\overline{\rho}$.
On the other hand, this trick has nothing to say about the case when the
Hodge–Tate weights are equal (see §12).

###  9.8. Modularity of higher symmetric powers

Another parallel development in higher dimensions was the extension of
Skinner–Wiles (§3) to higher dimensions. Many of the arguments of
Skinner–Wiles relied heavily on the fact that any proper submodule of a
$2$-dimensional representation must have dimension $1$, and one-dimensional
representations are very well understood by class field theory. Nonetheless,
in [173], Thorne proved a residually reducible modularity theorem for higher
dimensional representations. In order to overcome the difficulty of
controlling reducible deformations, he imposed a Steinberg condition at some
auxiliary prime. Although this is a definite restriction, it does apply (for
example) to the Galois representation coming from the symmetric power of a
modular form which also satisfies this condition. In a sequence of papers [51,
52, 53], Clozel and Thorne applied this modularity lifting theorem to prove
new cases of symmetric power functoriality (see also the paper of Dieulefait
[74]). A key difficulty here is again the absence of Ihara’s lemma in order to
find automorphic forms with the correct local properties. Very recently (using
a number of new ideas), Newton and Thorne [134, 135] were able to
(spectacularly!) complete this program and prove the full modularity of all
symmetric powers of all modular forms.

##  10\. Beyond self-duality and Shimura varieties

All the results discussed so far — with the exception of those discussed in §4
—- apply only to Galois representations which are both regular and satisfy
some form of self-duality. Moreover, they all correspond to automorphic forms
which can be detected by the (étale) cohomology of Shimura varieties. Once one
goes beyond these representations, many of the established methods begin to
break immediately363636I should warn the reader that this section and the next
(even more than the rest of this paper) is filtered through the lens of my own
personal research journey — _caveat lector_!.

An instructive case to consider is the case of $2$-dimensional geometric
Galois representations of an imaginary quadratic field $F$ with distinct
Hodge–Tate weights. The corresponding automorphic forms for
$\mathrm{GL}_{2}(\mathbf{A}_{F})$ contribute to the cohomology of locally
symmetric spaces $X$ which are arithmetic hyperbolic
$3$-manifolds373737Already by 1970, Serre (following ideas of Langlands) was
trying to link Mennicke’s computation that
$\mathrm{GL}_{2}(\mathbf{Z}[\sqrt{-109}])^{\mathrm{ab}}$ is infinite to the
possible existence of elliptic curves over $\mathbf{Q}(\sqrt{-109})$ with good
reduction everywhere [61, Jan 14, 1970].. These spaces are certainly not
algebraic varieties and their cohomology is hard to access via algebraic
methods. One of the first new questions to arise in this context is the
relationship between torsion classes and Galois representations. Some
speculations about this matter were made by Elstrodt, Grunewald, and Mennicke
at least as far back as 1981 [75], but the most influential conjecture was due
to Ash [6], who conjectured that eigenclasses in the cohomology of congruence
subgroups of $\mathrm{GL}_{n}(\mathbf{Z})$ over $\overline{\mathbf{F}}_{p}$
(which need not lift to characteristic zero) should give rise to
$n$-dimensional Galois representations over finite fields. Later, conjectures
were made [8, 7] in the converse conjecture in the spirit of Serre [156]
linking Galois representations to classes in cohomology modulo $p$. Certainly
around 2004, however, it was not at all clear what exactly one should expect
the landscape to be383838I recall conversations with a number of experts at
the 2004 Durham conference, where nobody seemed quite sure even what the
dimension of the ordinary deformation ring $R$ of a $3$-dimensional
representation
$\rho:G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{3}(\overline{\mathbf{F}}_{p})$
should be. Ash, Pollack, and Stevens had computed numerical examples where a
regular algebraic ordinary cuspidal form for
$\mathrm{GL}_{3}(\mathbf{A}_{\mathbf{Q}})$ not twist-equivalent to a symmetric
square did not appear to admit classical deformations. (I learnt about this
example from Stevens at a talk at Banff in December $2003$.) This would be
easily explained if $R$ had (relative) dimension $0$ over $\mathbf{Z}_{p}$ but
be more mysterious otherwise., and so it was around this time I decided to
start thinking about this question393939One great benefit to me at the time of
thinking about Galois representations over imaginary quadratic fields was that
it didn’t require me to understand the geometry of Shimura varieties which I
have always found too complicated to understand. The irony of course is that
the results of [1, 17] ultimately rely on extremely intricate properties of
Shimura varieties. in earnest. I became convinced very soon (for aesthetic
reasons if not anything else) that if one modified $\mathbf{T}$ to be the ring
of endomorphisms acting on integral cohomology (so that it would see not only
the relevant automorphic forms but also the torsion classes) then there should
still be an isomorphism $R=\mathbf{T}$. Moreover, this equality would not only
be a form of reciprocity which moved beyond the conjecture linking motives to
automorphic forms, but it suggested that the integral cohomology of arithmetic
groups (including the torsion classes) were themselves the fundamental object
of interest. Various developments served only to confirm this point of view.
In my paper with Mazur [40], we gave some theoretical evidence for why
ordinary families of Galois representations of imaginary quadratic fields
might on the one hand be positive dimensional and explained completely by
torsion classes and yet not contain any classical automorphic points at all.
During the process of writing [36], Dunfield (numerically) compared the
torsion classes in the cohomology of inner forms of $\mathrm{GL}_{2}$ and the
data was in perfect agreement with a conjectural Jacquet–Langlands
correspondence for torsion (later taken up in joint work with Venkatesh [41]).
Emerton and I had the idea of working with completed cohomology groups both to
construct Galois representations and even possibly to approach questions of
modularity. The first idea was to exploit the well-known relationship between
the cohomology of these manifolds and the cohomology of the boundary of
certain Shimura varieties. We realized that if we could control the co-
dimension of the completed cohomology groups over the non-commutative Iwasawa
algebra, the Hecke eigenclasses would be forced to be seen by eigenclasses
coming from the middle degree of these Shimura varieties where one had access
to Galois representations404040Unfortunately, these conjectures [37, Conj 1.5]
remain all open in more or less all cases except for Scholze’s results in the
case of certain Shimura varieties [155, Cor 4.2.3].. On the automorphy lifting
side, we had even vaguer ideas [37, §1.8]414141Pan’s remarkable paper [137]
turned some of these pipe dreams into reality. on how to proceed. A different
(and similarly unsuccessful) approach424242This is taken from my 2006 NSF
proposal, and I believe influenced by my conversations with Taylor at Harvard
around that time. was to work with ordinary deformations over a partial weight
space for a split prime $p=vw$ in an imaginary quadratic field $F$. That is,
deformations of $\overline{\rho}$ which had an unramified quotient at $v$ and
$w$ but with varying weight at $v$ and fixed weight at $w$. Here the yoga of
Galois deformations suggested that $R$ should be finite flat over $W(k)$ in
this case (and even a complete intersection). Moreover, one had access to
$\mathbf{T}$ using an overlooked434343One should never overlook results of
Hida. I only learnt about this paper when Hida pointed it out to me (with a
characteristic smile on his face) after my talk in Montreal in 2005. I was
pleased at least that the idea that these families were genuinely non-
classical was not anticipated either in [103] or in §4 of Taylor’s thesis
[171]. result of Hida [103], and in particular one could deduce that
$\mathbf{T}$ has dimension at least one. If one could show that $\mathbf{T}$
was flat over $W(k)$, then one could plausibly apply (assuming the existence
of Galois representations) the original argument of [180, 169]. The flatness
of $\mathbf{T}$, however, remains an open problem444444One might even argue
that there is no compelling argument to believe it is true — the problem is
analogous to the vanishing of the $\mu$-invariant in Iwasawa theoretic
settings..

###  10.1. The Taylor–Wiles method when $l_{0}>0$, part I: Calegari–Geraghty

Shortly before (and then during) the special year on Galois representations at
the IAS in 2010-2011, I started to work with Geraghty in earnest on the
problem of proving $R=\mathbf{T}$ in the case of imaginary quadratic fields,
assuming the existence of a surjection $R\rightarrow\mathbf{T}$. A computation
in Galois cohomology shows that the expected “virtual” dimension of $R$ over
$W(k)$ should be $-1$, and hence the patched module $M_{\infty}$ should have
codimension $1$ over the ring of diamond operators $S_{\infty}$. We realized
this was a consequence of the fact that, after localizing the cohomology at a
non-Eisenstein maximal ideal, the cohomology should be non-zero in exactly two
degrees. More precisely, patching the presentations of these $S_{N}$-modules
would result in a balanced presentation of $M_{\infty}$ as an
$S_{\infty}$-module with the same (finite) number of generators as relations.
We then realized that the same principle held more generally for
$n$-dimensional representations over any number field. In characteristic zero,
the localized cohomology groups were non-zero exactly in a range
$[q_{0},q_{0}+l_{0}]$ (with $q_{0}$ and $l_{0}$ as defined in [26]) where
$-l_{0}$ coincided with the expected virtual dimension of $R$ over $W(k)$
coming from Galois cohomology. We could thus show — assuming the localized
torsion cohomology also vanished in this range — that by patching _complexes_
$P_{Q}$ (rather than modules $M_{Q}$), one arrives at a complex $P_{\infty}$
of free $S_{\infty}$ modules in degrees $[q_{0},q_{0}+l_{0}]$. Because the
ring $R_{\infty}$ of dimension $\dim R_{\infty}=(\dim S_{\infty})-l_{0}$ acts
by patching on $H^{*}(P_{\infty})$, a simple commutative algebra lemma then
shows that $M_{\infty}=H^{*}(P_{\infty})$ has codimension $l_{0}$ over
$S_{\infty}$ and must be concentrated in the final degree. In particular, the
Taylor–Wiles method (as modified by Diamond) could be happily adapted to this
general setting454545David Hansen came up with a number of these ideas
independently [94].. Moreover, the arguments were compatible with all the
other improvements, including Taylor’s Ihara avoidance argument amongst other
things464646These methods only prove
$R[1/p]^{\mathrm{red}}=\mathbf{T}[1/p]^{\mathrm{red}}$, of course. In
situations where $\mathbf{T}\otimes\mathbf{Q}=0$, the methods of [38] in the
minimal case prove not only that $R=\mathbf{T}$ but also that (both) rings are
complete intersections. Moreover, one also has access to level raising (on the
level of complexes) and Ihara’s lemma [41, §4], and I tried for some time
(unsuccessfully) to adapt the original minimal $\Rightarrow$ non-minimal
arguments of [180] to this setting. There certainly seems to be some rich
ideas in commutative algebra in these situations to explore, see for example
recent work of Tilouine–Urban [174].. We also realized that the same idea
applied to Galois representations coming from the coherent cohomology of
Shimura varieties even when the corresponding automorphic forms were not
discrete series. While our general formulation involved a number of
conjectures we considered hopeless, the coherent case had at least one setting
in which many more results were available, namely the case of modular forms of
weight one, where the required vanishing conjecture was obvious, and where we
were able to establish the existence of the required map
$R\rightarrow\mathbf{T}$ with all the required local properties by direct
arguments. Although the state of knowledge concerning Galois representations
increased tremendously between the original conception of [38] and its final
publication, by early 2016 it still seemed out of reach to make any of the
results in [38] unconditional.

###  10.2. Construction of Galois representations, part II

Before one can hope to prove $R=\mathbf{T}$ theorems, one needs to be able to
associate Galois representations to the corresponding automorphic forms. There
are two contexts in which one might hope to make progress. The first is in
situations where the automorphic forms contribute to the Betti cohomology of
some locally symmetric space — for example, tempered algebraic cuspidal
automorphic representations for $\mathrm{GL}_{n}(\mathbf{A}_{F})$ and any $F$.
The second is in situations where the automorphic forms contribute to the
coherent cohomology of some Shimura variety. Here the first and easiest case
corresponds to weight one modular forms, where the Galois representations were
first constructed by Deligne and Serre [69].

In work of Harris–Soudry–Taylor [98, 165], Galois representations were
constructed for regular algebraic forms for $\mathrm{GL}_{2}(\mathbf{A}_{F})$
for an imaginary quadratic field $F$ and satisfying a further restriction on
the central character. Harris, Soudry, and Taylor exploited (more or less) the
fact that the automorphic induction of such forms are self-dual (although not
regular) and still contribute to coherent cohomology, so one can construct
Galois representations using a congruence argument as in the paper of Deligne
and Serre [69]. On the other hand, this does not prove the expected local
properties of the Galois representation at $v|p$.

It was well-known for many years that the Hecke eigenclasses associated to
regular algebraic cuspidal automorphic forms for
$\mathrm{GL}_{n}(\mathbf{A}_{F})$ for a CM field $F$ could be realized as
eigenclasses coming from the boundary of certain unitary Shimura varieties of
type $U(n,n)$. It was, however, also well-known that the corresponding étale
cohomology classes did not realize the desired Galois representations474747For
a more basic example of what can go wrong, note that the Hecke eigenvalues of
$T_{l}$ on $H^{0}(X,\mathbf{Q}_{p})$ of a modular curve are $1+l$, which
corresponds to the Galois representation
$\mathbf{Q}_{p}\oplus\mathbf{Q}_{p}(1)$. However, only the piece
$\mathbf{Q}_{p}$ occurs inside $H^{0}$.. Remarkably, this problem was
completely and unexpectedly resolved in 2011 in [96] by
Harris–Lan–Taylor–Thorne. Richard Taylor writes:

For [96] I knew that the Hecke eigenvalues we were interested in contributed
to Betti cohomology of $U(n,n)$. The problem was to show that they contributed
to overconvergent $p$-adic cusp forms. I was convinced on the basis of
Coleman’s paper “classical and overconvergent modular forms” [55] that this
must be so. I can’t now reconstruct exactly why Coleman’s paper convinced me
of this, and it is possible, even probable, that my reasoning didn’t really
make any sense. However it was definitely this that kept me working at the
problem, when we weren’t really getting anywhere.

Amazingly, this breakthrough immediately inspired the next development:

###  10.3. Construction of Galois representations, part III: Scholze

In [155], Scholze succeeded in constructing Galois representations associated
to torsion classes in the setting of $\mathrm{GL}_{n}(\mathbf{A}_{F})$ for a
CM field $F$. Scholze had the idea after seeing some lectures on [96]:

During a HIM trimester at Bonn, Harris and Lan gave some talks about their
construction of Galois representations. At the time, I had some ideas in my
head that I didn’t have any use for: That Shimura varieties became perfectoid
at infinite level, and that there is a Hodge–Tate period map defined on them.
The only consequence I could draw from this were certain vanishing results for
completed cohomology as conjectured in your work with Emerton; so at least I
knew that the methods were able to say something nontrivial about torsion
classes in the cohomology. After hearing Harris’ and Lan’s talks, I was trying
to see whether these ideas could help in extending their results to torsion
classes. After a little bit of trying, I found the fake-Hasse invariants, and
then it was clear how the argument would go.

Even after this breakthrough, Scholze’s construction still fell short of the
conjectures in [38] in two ways. The first was that the Galois representation
(ignoring here issues of pseudo-representations) was valued not in
$\mathbf{T}$ but in $\mathbf{T}/I$ for some ideal $I$ of fixed nilpotence.
This is not a crucial obstruction to the methods of [38]. The second issue,
however, was that the Galois representations were constructed (in the end) via
$p$-adic congruences, and thus one did not have control over their local
properties at $p$ which _are_ crucial for modularity applications.

###  10.4. The Taylor–Wiles method when $l_{0}>0$, part II: DAG

Although not directly related to new $R=\mathbf{T}$ theorems, one new recent
idea in the subject has been the work of Galatius–Venkatesh [86] on derived
deformation rings in the context of Venkatesh’s conjectures over $\mathbf{Z}$.
This work (in part) reinterprets the arguments of [38] in terms of a derived
Hecke action. The authors define a derived version $\mathcal{R}$ of $R$ with
$\pi_{0}(\mathcal{R})=R$. Under similar hypotheses to [38], the higher
homotopy groups of $\mathcal{R}$ are shown to exist precisely in degrees $0$
to $l_{0}$. One viewpoint of the minimal case of [38] is that one constructs a
(highly non-canonical) formally smooth ring $R_{\infty}$ of dimension
$n-l_{0}$ with an action of a formally smooth ring $S_{\infty}$ of dimension
$n$ such that the minimal deformation ring $R$ is
$R_{\infty}\otimes_{S_{\infty}}S_{\infty}/\mathfrak{a}$ for the augmentation
ideal $\mathfrak{a}$. Moreover, the ring $R$ is identified both with the
action of $\mathbf{T}$ on the entire cohomology and simultaneously on the
cohomology in degree $q_{0}+l_{0}$. On the other hand, when $l_{0}>1$, the
intersection of $R_{\infty}$ and $S_{\infty}/\mathfrak{a}$ over $S_{\infty}$
is never transverse484848When $l_{0}=1$, the intersection _can_ be transverse
when $R$ is a finite ring. In this setting, the relevant cohomology is also
non-zero and finite in exactly one degree. On the other hand, as soon as
$\mathrm{Hom}(R,\overline{\mathbf{Q}}_{p})$ is non-zero (for example, when
there exists an associated motive) and $l_{0}>0$, the intersection will always
be non-transverse., and homotopy groups of the derived intersection recover
the cohomology in all degrees (under the running assumptio2n one also knows
that the patched cohomology is free). On the other hand, there is a more
canonical way to define $\mathcal{R}$, namely to take the unrestricted global
deformation ring $R^{\mathrm{glob}}$ (which has no derived structure) and
intersect it with a suitable local crystalline deformation ring as algebras
over the unrestricted local deformation ring. The expected dimension of this
intersection is also $-l_{0}$ over $W(k)$, although this is not so clear from
this construction. Hence [86] can be viewed as giving an intrinsic definition
of $\mathcal{R}$ independent of any choices of Taylor–Wiles primes and showing
that its homotopy groups are related (as with
$R_{\infty}\otimes^{\mathbf{L}}_{S_{\infty}}S_{\infty}/\mathfrak{a}$) to the
cohomology494949There are some subtleties as to what the precise statement
should be in the presence of global congruences, but already this author gets
confused at the best of times between homology and cohomology, so I will not
try to unentangle these issues here.. These ideas have hinted at a closer
connection between the Langlands program in the arithmetic case and the
function field case than was previously anticipated505050Not anticipated by
many people, at least; Michael Harris has been proselytizing the existence of
a connection for quite some time., see for example work of Zhu [181].

##  11\. Recent Progress

###  11.1. Avoiding conjectures involving torsion I: the 10-author paper

As mentioned in §10.3, even after the results of [155] there remained a
significant gap to make the results of [38] unconditional, namely, the
conjecture that these Galois representations had the right local properties at
$p$ and a second conjecture predicting the vanishing of (integral) cohomology
localized at a non-maximal ideal $\mathfrak{m}$ outside a certain precise
range (corresponding to known results in characteristic zero). It should be
noted that the second conjecture was still open (in all but the easiest cases)
in the simpler setting of Shimura varieties. The first hints that one could
possibly make progress on this second conjecture (at least for Shimura
varieties) was given in an informal talk by Scholze in Bellairs515151I was
invited to give the lecture series in Bellairs after Matthew Emerton didn’t
respond to his emails. Through some combination of the appeal of my own work
and the fact that the lectures were given on a beach in Barbados, I managed to
persuade Patrick Allen, George Boxer, Ana Caraiani, Toby Gee, Vincent Pilloni,
Peter Scholze, and Jack Thorne to come, all of whom are now my co-authors, and
all of whom (if they weren’t already at the time) are now more of an expert in
this subject than I am. The thought that I managed to teach any of them
something about the subject is pleasing indeed. in 2014. This very quickly led
to a long term collaboration between Scholze and Caraiani [45, 46], which
Caraiani describes as follows:

At the Barbados conference in May 2014, Peter gave a lecture on how one might
compute the cohomology of compact unitary Shimura varieties with torsion
coefficients. The key was to have some control for
$R\pi_{HT*}\mathbb{F}_{\ell}$ restricted to any given Newton stratum. He was
expressing this in terms of a conjecture that had grown out of his work on
local Langlands using the Langlands–Kottwitz method. After his talk, I went to
ask him some questions about this conjecture and it sounded like there were
some things that still needed to be made precise. He asked if I wanted to help
him make his strategy work. After some hesitation (because I didn’t think I
knew enough or was strong enough to work with him), I accepted. Later that
evening, I suggested switching from the Langlands–Kottwitz approach to
understanding $R\pi_{HT*}\mathbb{F}_{\ell}$ to an approach more in the style
of Harris–Taylor. This relies on the beautiful Mantovan product formula that
describes Newton strata in terms of Rapoport–Zink spaces and Igusa varieties.
Maybe something like this could help illuminate the geometry of the Hodge–Tate
period morphism? Peter immediately saw that this should work and we made plans
for me to visit Bonn that summer to continue the collaboration.

As Peter and I were finishing writing up the compact case, it became clear to
us that the vanishing theorem would give a way to construct Galois
representations associated to generic mod $p$ classes that preserves the
desired information at $p$. Peter started thinking about the non-compact case
and how that might apply to the local-global compatibility needed for
Calegari–Geraghty. I remember discussing this with him at the Clay Research
conference in Oxford in September 2015. By spring 2016, Richard started
floating the idea of a working group on Calegari–Geraghty and found out that
Peter and I had an approach to local-global compatibility. Around June 2016,
Richard suggested to me to organize the working group with him. Peter was very
excited about the idea, but wasn’t sure he would be able to attend for family
reasons. In the end, we found a date in late October 2016 that worked for
everyone.

The working group met under the auspices of the first “emerging topics”
workshop525252Although later described as a “secret” workshop, it was an
“invitation-only working group.” at the IAS to determine the extent to which
the expected consequences could be applied to modularity lifting: A clear
stumbling point was the vanishing of integral cohomology after localization
outside the range of degrees $[q_{0},q_{0}+l_{0}]$. On the other hand, Khare
and Thorne had already observed in [115] by a localization argument that this
could sometimes be avoided in certain minimal cases. It was this argument we
were able to modify for the general case, thus avoiding the need to prove the
(still open) vanishing conjectures for torsion classes535353I regard my main
contribution to [1] as explaining how the arguments in [38] using Taylor’s
Ihara avoidance (§9.3) were incompatible with any characteristic zero
localization argument in the absence of (unknown) integral vanishing results
in cohomology. The objection (even in the case $l_{0}=0$) was that it was easy
to construct complexes $P^{1}$ and $P^{2}$ of free $S_{\infty}$ modules so
that the support of $H^{*}(P^{1}/p)$ and $H^{*}(P^{2}/p)$ coincided (as they
must) but that (for example) $H^{*}(P^{1})[1/p]$ was zero even though
$H^{*}(P^{2})[1/p]$ was not. The objection to this objection, however, which
was resolved during the workshop (and which to be clear I played no part in
resolving!) is to not merely to compare the support of the complexes $P^{i}/p$
but to consider the entire complex in the derived category. In particular,
even (say) for a finite $\mathbf{Z}_{p}$-module $M$, the module $M[1/p]$ is
non-zero exactly when $M\otimes^{\mathbf{L}}\mathbf{F}_{p}$ has non-zero Euler
characteristic.. The result of the workshop was a success beyond what we could
have reasonably anticipated — we ended up with more or less545454It is worth
emphasizing that an incredible amount of work was required to turn these ideas
into reality, and that this intellectual effort was by and large carried out
by the younger members of the collaboration. the outline of a plan to prove
all the main modularity lifting theorems which finally appeared in [1], namely
the Ramanujan conjecture for regular algebraic automorphic forms for
$\mathrm{GL}_{2}(\mathbf{A}_{F})$ of weight zero for any CM field $F$, and
potential modularity (and the Sato–Tate conjecture) for elliptic curves over
CM fields.

There have already been a number of advancements beyond [1] including in
particular by Allen, Khare, and Thorne [4] proving the modularity of many
elliptic curves over CM fields and a potential automorphy theorem for ordinary
representations by Qian [144]. It does not seem completely implausible that
results of the strength of [11] for $n$-dimensional regular Galois
representations of $G_{\mathbf{Q}}$ are within reach.

###  11.2. Avoiding conjectures involving torsion II: abelian surfaces

A second example that Geraghty and I had considered during the 2010–2011 IAS
special year was the case of abelian surfaces, corresponding to low
(irregular) weight Siegel modular forms of genus $g=2$. It was clear that a
key difficulty was proving the vanishing of
$H^{2}(X,\omega^{2})_{\mathfrak{m}}$ where $X$ was a (compactified) Siegel
$3$-fold with good reduction at $p$, where $\mathfrak{m}$ is maximal ideal of
the Hecke algebra corresponding to an absolutely irreducible representation,
and where $\omega|_{Y}=\det\pi_{*}\Omega^{1}_{\mathcal{A}/Y}$ on the open
moduli space $Y\subset X$ admitting a corresponding universal abelian surface
$\mathcal{A}/Y$. In other irregular weights (corresponding to motives with
Hodge–Tate weights $[0,0,k-1,k-1]$ for $k\geq 4$) the vanishing of the
corresponding cohomology groups was known by Lan and Suh [126]. The vanishing
of $H^{2}(X,\omega^{2})_{\mathfrak{m}}$ was more subtle, however, because the
corresponding group does _not_ vanish in general before localization in
contrast to the previous cases. In [39], we proved a minimal modularity
theorem for these higher weight representations and a minimal modularity
theorem in the abelian case contingent on the vanishing conjecture above which
we did not manage to resolve (and which remains unresolved). I finished and
then submitted the paper after I had moved to Chicago and Geraghty had moved
to Facebook in 2015. I then started working with Boxer and Gee555555George
Boxer had also arrived at Chicago in 2015, and was collaborating with Gee on
companion form results for Siegel modular forms, with the hope (in part) of
deducing the modularity of abelian surfaces from Serre’s conjecture for
$\mathrm{GSp}_{4}$ in a manner analogous to the deduction by of the Artin
conjecture from Serre’s conjecture for $\mathrm{GL}_{2}$ in [110, 113]. They
usually worked together at Plein Air cafe. Since I had thought about similar
questions with Geraghty and frequently went to Plein Air for 6oz cappuccinos,
it was not entirely surprising for us to start working together. on this
vanishing question under certain supplementary local hypotheses. (By this
point, Galois representations associated to torsion classes in coherent
cohomology had been constructed by Boxer [16] and Goldring–Koskivirta [90].)
But then in November of 2016 (one week after the IAS workshop!), Pilloni’s
paper on higher Hida theory [142] was first posted. It was apparent to us that
Pilloni’s ideas would be extremely useful, and the four of us began a
collaboration almost immediately. Just as in [1], we were ultimately able to
avoid proving any vanishing conjectures. However, unlike [1], the way around
this problem was not purely by commutative algebra, but instead by working
with ideas from [142]. Namely, instead of working with the cohomology of the
full Siegel modular variety $X$, one could work with the coherent cohomology
of a certain open variety of $X$ with cohomological dimension one whose
(infinite dimensional) cohomology could still be tamed using the methods of
higher Hida theory [142] in a way analogous to how Hida theory controls the
(infinite dimensional) cohomology of the affine variety (with cohomological
dimension zero) corresponding to the ordinary locus. Generalizing this to a
totally real field, one could then _combine_ these ideas with the Taylor–Wiles
method as modified in [38] to prove the potential modularity of abelian
surfaces over totally real fields [17]. This coincidentally gives a second
proof of the potential modularity of elliptic curves over CM fields proven in
[1]. (The papers [1] and [17] both were conceived of and completed within a
week or so of each other.)

##  12\. The depths of our ignorance

Despite what can reasonably be considered significant progress in proving many
cases of modularity since 1993, it remains the case that many problems appear
just as hopeless as they did then565656Or perhaps harder, since there has been
almost 30 years without any progress whatsoever.. Perhaps most embarrassing is
the case of even Galois representations
$G_{\mathbf{Q}}\rightarrow\mathrm{GL}_{2}(\mathbf{C})$ with non-solvable image
(equivalently, projective image $A_{5}$). For example, we cannot establish the
Artin conjecture for a _single_ Galois representation whose image is the
binary icosahedral group $\mathrm{SL}_{2}(\mathbf{F}_{5})$ of order $120$. The
key problem is that the automorphic forms (Maass forms with eigenvalue
$\lambda=1/4$ in this case) are very hard to access — given an even
(projective) $A_{5}$ Galois representation, we don’t even know how to prove
that there exists a corresponding Maass form with the right Laplacian
eigenvalue, let alone one whose Hecke eigenvalues correspond to the Galois
representation575757Motives can be divided according to a tetrachotomy. The
first form are the Tate (and potentially Tate) motives, whose automorphy was
known to Riemann and Hecke. The second form are the motives (conjecturally)
associated to automorphic representations which are discrete series at
infinity and thus amenable to the Taylor–Wiles method. The third form are the
motives (conjecturally) associated to automorphic representations which are at
least seen by some flavour of cohomology, either by the Betti cohomology of
locally symmetric spaces or the coherent cohomology of Shimura varieties
(possibly in degrees greater than zero) which are amenable in principle to the
modified Taylor–Wiles method. The fourth form consist of the rest, which
(besides a few that can be accessed by cyclic base change) are a complete
mystery.. In many ways, we have made no real progress on this question. The
case of curves of genus $g>2$ whose Jacobians have no extra endomorphisms
seems equally hopeless. One can only take solace in the fact that the
Shimura–Taniyama conjecture seemed equally out of reach before Wiles’
announcement in Cambridge in 1993.

## References

  * [1] P. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor, and J. Thorne, Potential automorphy over CM fields, 2018, preprint, arXiv:1812.09999.
  * [2] P. B. Allen, Modularity of nearly ordinary 2-adic residually dihedral Galois representations. _Compos. Math._ 150 (2014), no. 8, 1235–1346.
  * [3] P. B. Allen, Deformations of polarized automorphic Galois representations and adjoint Selmer groups. _Duke Math. J._ 165 (2016), no. 13, 2407–2460.
  * [4] P. B. Allen, C. Khare, and J. A. Thorne, Modularity of $\operatorname{GL}_{2}(\mathbb{F}_{p})$-representations over CM fields. 2019, %****␣Survey.bbl␣Line␣25␣****arXiv:1910.12986.
  * [5] J. Arthur and L. Clozel, _Simple algebras, base change, and the advanced theory of the trace formula_. Annals of Mathematics Studies 120, Princeton University Press, Princeton, NJ, 1989.
  * [6] A. Ash, Galois representations attached to mod $p$ cohomology of ${\mathrm{GL}}(n,{\mathbf{Z}})$. _Duke Math. J._ 65 (1992), no. 2, 235–255.
  * [7] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology. _Duke Math. J._ 112 (2002), no. 3, 521–579.
  * [8] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of ${\mathrm{GL}}(n,{\mathbf{Z}})$. _Duke Math. J._ 105 (2000), no. 1, 1–24.
  * [9] M. Auslander and D. A. Buchsbaum, Homological dimension in local rings. _Trans. Amer. Math. Soc._ 85 (1957), 390–405.
  * [10] T. Barnet-Lamb, T. Gee, and D. Geraghty, The Sato-Tate conjecture for Hilbert modular forms. _J. Amer. Math. Soc._ 24 (2011), no. 2, 411–469.
  * [11] T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, Potential automorphy and change of weight. _Ann. of Math. (2)_ 179 (2014), no. 2, 501–609.
  * [12] T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy II. _Publ. Res. Inst. Math. Sci._ 47 (2011), no. 1, 29–98.
  * [13] L. Berger and C. Breuil, Towards a $p$-adic Langlands programme, 2004, unpublished, http://perso.ens-lyon.fr/laurent.berger/autrestextes/hangzhou.pdf.
  * [14] L. Berger and C. Breuil, Sur quelques représentations potentiellement cristallines de ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$. _Astérisque_ (2010), no. 330, 155–211.
  * [15] G. Böckle, On the density of modular points in universal deformation spaces. _Amer. J. Math._ 123 (2001), no. 5, 985–1007.
  * [16] G. Boxer, Torsion in the coherent cohomology of Shimura varieties and Galois representations. 2015, arXiv:1507.05922.
  * [17] G. Boxer, F. Calegari, T. Gee, and V. Pilloni, Abelian surfaces over totally real fields are potentially modular. 2018, arXiv:1812.09269.
  * [18] C. Breuil, Schémas en groupe et corps des normes, 1998, URL https://www.imo.universite-paris-saclay.fr/~breuil/PUBLICATIONS/groupesnormes.pdf, unpublished.
  * [19] C. Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés. _Ann. of Math. (2)_ 152 (2000), no. 2, 489–549.
  * [20] C. Breuil, Sur quelques représentations modulaires et $p$-adiques de ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$. I. _Compositio Math._ 138 (2003), no. 2, 165–188.
  * [21] C. Breuil, Sur quelques représentations modulaires et $p$-adiques de ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$. II. _J. Inst. Math. Jussieu_ 2 (2003), no. 1, 23–58.
  * [22] C. Breuil, Introduction générale. _Astérisque_ (2008), no. 319, 1–12.
  * [23] C. Breuil, Série spéciale $p$-adique et cohomologie étale complétée. _Astérisque_ (2010), no. 331, 65–115.
  * [24] C. Breuil, Correspondance de Langlands $p$-adique, compatibilité local-global et applications [d’après Colmez, Emerton, Kisin, $\ldots$]. _Astérisque_ (2012), no. 348, Exp. No. 1031, viii, 119–147.
  * [25] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over $\mathbb{Q}$: wild 3-adic exercises. _J. Amer. Math. Soc._ 14 (2001), no. 4, 843–939.
  * [26] C. Breuil and A. Mézard, Multiplicités modulaires et représentations de ${\mathrm{GL}}_{2}({\mathbf{Z}}_{p})$ et de ${\mathrm{Gal}}(\overline{\mathbf{Q}}_{p}/{\mathbf{Q}}_{p})$ en $l=p$. _Duke Math. J._ 115 (2002), no. 2, 205–310.
  * [27] S. Brochard, Proof of de Smit’s conjecture: a freeness criterion. _Compos. Math._ 153 (2017), no. 11, 2310–2317.
  * [28] W. Bruns and J. Herzog, _Cohen-Macaulay rings_. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.
  * [29] J. P. Buhler, _Icosahedral Galois representations_. Lecture Notes in Mathematics, Vol. 654, Springer-Verlag, Berlin-New York, 1978.
  * [30] K. Buzzard, Potential modularity—a survey. In _Non-abelian fundamental groups and Iwasawa theory_ , pp. 188–211, London Math. Soc. Lecture Note Ser. 393, Cambridge Univ. Press, Cambridge, 2012.
  * [31] K. Buzzard, M. Dickinson, N. Shepherd-Barron, and R. Taylor, On icosahedral Artin representations. _Duke Math. J._ 109 (2001), no. 2, 283–318.
  * [32] K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations. In _Automorphic forms and Galois representations. Vol. 1_ , pp. 135–187, London Math. Soc. Lecture Note Ser. 414, Cambridge Univ. Press, Cambridge, 2014.
  * [33] K. Buzzard and R. Taylor, Companion forms and weight one forms. _Ann. of Math. (2)_ 149 (1999), no. 3, 905–919.
  * [34] F. Calegari, Even Galois representations and the Fontaine-Mazur conjecture. _Invent. Math._ 185 (2011), no. 1, 1–16.
  * [35] F. Calegari, Even Galois Representations and the Fontaine-Mazur conjecture II. _J. Amer. Math. Soc._ 25 (2012), no. 2, 533–554.
  * [36] F. Calegari and N. M. Dunfield, Automorphic forms and rational homology 3-spheres. _Geom. Topol._ 10 (2006), 295–329.
  * [37] F. Calegari and M. Emerton, Completed cohomology—a survey. In _Non-abelian fundamental groups and Iwasawa theory_ , pp. 239–257, London Math. Soc. Lecture Note Ser. 393, Cambridge Univ. Press, Cambridge, 2012\.
  * [38] F. Calegari and D. Geraghty, Modularity lifting beyond the Taylor–Wiles method. _Inventiones mathematicae_ 211 (2018), no. 1, 297–433.
  * [39] F. Calegari and D. Geraghty, Minimal modularity lifting for nonregular symplectic representations. _Duke Math. J._ 169 (2020), no. 5, 801–896.
  * [40] F. Calegari and B. Mazur, Nearly ordinary Galois deformations over arbitrary number fields. _J. Inst. Math. Jussieu_ 8 (2009), no. 1, 99–177.
  * [41] F. Calegari and A. Venkatesh, A torsion Jacquet-Langlands correspondence. _Astérisque_ (2019), no. 409, x+226.
  * [42] A. Caraiani, Local-global compatibility and the action of monodromy on nearby cycles. _Duke Math. J._ 161 (2012), no. 12, 2311–2413.
  * [43] A. Caraiani, Monodromy and local-global compatibility for $l=p$. _Algebra Number Theory_ 8 (2014), no. 7, 1597–1646.
  * [44] A. Caraiani and B. V. Le Hung, On the image of complex conjugation in certain Galois representations. _Compos. Math._ 152 (2016), no. 7, 1476–1488.
  * [45] A. Caraiani and P. Scholze, On the generic part of the cohomology of compact unitary Shimura varieties. _Ann. of Math. (2)_ 186 (2017), no. 3, 649–766.
  * [46] A. Caraiani and P. Scholze, On the generic part of the cohomology of non-compact unitary Shimura varieties. 2019, arXiv:1909.01898.
  * [47] H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert. _Ann. Sci. École Norm. Sup. (4)_ 19 (1986), no. 3, 409–468.
  * [48] L. Clozel, Représentations galoisiennes associées aux représentations automorphes autoduales de ${\mathrm{GL}}(n)$. _Inst. Hautes Études Sci. Publ. Math._ (1991), no. 73, 97–145.
  * [49] L. Clozel, M. Harris, J.-P. Labesse, and B.-C. Ngô (eds.), _On the stabilization of the trace formula_. Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, Volume I: Stabilization of the trace formula, International Press, Somerville, MA, 2011.
  * [50] L. Clozel, M. Harris, and R. Taylor, Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations. _Inst. Hautes Études Sci. Publ. Math._ 108 (2008), 1–181.
  * [51] L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality, I. _Compos. Math._ 150 (2014), no. 5, 729–748.
  * [52] L. Clozel and J. A. Thorne, Level raising and symmetric power functoriality, II. _Ann. of Math. (2)_ 181 (2015), no. 1, 303–359.
  * [53] L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality, III. _Duke Math. J._ 166 (2017), no. 2, 325–402.
  * [54] R. Coleman and B. Mazur, The eigencurve. In _Galois representations in arithmetic algebraic geometry (Durham, 1996)_ , pp. 1–113, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, Cambridge, 1998.
  * [55] R. F. Coleman, Classical and overconvergent modular forms. _Invent. Math._ 124 (1996), no. 1-3, 215–241.
  * [56] P. Colmez, Les conjectures de monodromie $p$-adiques. 2003, séminaire Bourbaki. Vol. 2001/2002.
  * [57] P. Colmez, Une correspondance de langlands locale $p$-adique pour les représentations semi-stables de dimension $2$, 2004, URL https://webusers.imj-prg.fr/~pierre.colmez/sst.pdf, unpublished.
  * [58] P. Colmez, Représentations triangulines de dimension 2. _Astérisque_ (2008), no. 319, 213–258.
  * [59] P. Colmez, La série principale unitaire de ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$. _Astérisque_ (2010), no. 330, 213–262.
  * [60] P. Colmez, Représentations de ${\mathrm{GL}}_{2}(\mathbb{Q}_{p})$ et $(\phi,\Gamma)$-modules. _Astérisque_ (2010), no. 330, 281–509.
  * [61] P. Colmez (ed.), _Correspondance Serre-Tate. Vol. I_. Documents Mathématiques (Paris) [Mathematical Documents (Paris)] 13, Société Mathématique de France, Paris, 2015.
  * [62] P. Colmez (ed.), _Correspondance Serre-Tate. Vol. II_. Documents Mathématiques (Paris) [Mathematical Documents (Paris)] 14, Société Mathématique de France, Paris, 2015.
  * [63] P. Colmez, G. Dospinescu, and V. Paškūnas, The $p$-adic local Langlands correspondence for ${\mathrm{G}L}_{2}(\mathbb{Q}_{p})$. _Camb. J. Math._ 2 (2014), no. 1, 1–47.
  * [64] B. Conrad, Finite group schemes over bases with low ramification. _Compositio Math._ 119 (1999), no. 3, 239–320.
  * [65] B. Conrad, Ramified deformation problems. _Duke Math. J._ 97 (1999), no. 3, 439–513.
  * [66] B. Conrad, F. Diamond, and R. Taylor, Modularity of certain potentially Barsotti-Tate Galois representations. _J. Amer. Math. Soc._ 12 (1999), no. 2, 521–567.
  * [67] G. Cornell, J. H. Silverman, and G. Stevens (eds.), _Modular forms and Fermat’s last theorem_. Springer-Verlag, New York, 1997.
  * [68] H. Darmon, F. Diamond, and R. Taylor, Fermat’s last theorem. In _Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993)_, pp. 2–140, Int. Press, Cambridge, MA, 1997.
  * [69] P. Deligne and J.-P. Serre, Formes modulaires de poids $1$. _Ann. Sci. École Norm. Sup. (4)_ 7 (1974), 507–530 (1975).
  * [70] F. Diamond, On deformation rings and Hecke rings. _Ann. of Math. (2)_ 144 (1996), no. 1, 137–166.
  * [71] F. Diamond, An extension of Wiles’ results. In _Modular forms and Fermat’s last theorem (Boston, MA, 1995)_ , pp. 475–489, Springer, New York, 1997.
  * [72] F. Diamond, The Taylor-Wiles construction and multiplicity one. _Invent. Math._ 128 (1997), no. 2, 379–391.
  * [73] M. Dickinson, On the modularity of certain 2-adic Galois representations. _Duke Math. J._ 109 (2001), no. 2, 319–382.
  * [74] L. Dieulefait, Automorphy of ${\mathrm{Symm}}^{5}(\mathrm{GL}(2))$ and base change. _J. Math. Pures Appl. (9)_ 104 (2015), no. 4, 619–656.
  * [75] J. Elstrodt, F. Grunewald, and J. Mennicke, ${\mathrm{PSL}}(2)$ over imaginary quadratic integers. In _Arithmetic Conference (Metz, 1981)_ , pp. 43–60, Astérisque 94, Soc. Math. France, Paris, 1982.
  * [76] M. Emerton, A local-global compatibility conjecture in the $p$-adic Langlands programme for ${\mathrm{GL}}_{2/{\mathbb{Q}}}$. _Pure Appl. Math. Q._ 2 (2006), no. 2, Special Issue: In honor of John H. Coates. Part 2, 279–393.
  * [77] M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. _Invent. Math._ 164 (2006), no. 1, 1–84.
  * [78] M. Emerton, Local-global compatibility in the $p$-adic Langlands programme for $\operatorname{GL}_{2}/\mathbb{Q}$, 2010, preprint.
  * [79] N. Fakhruddin, C. Khare, and S. Patrikis, Lifting and automorphy of reducible mod p galois representations over global fields. 2020, arXiv:2008.12593.
  * [80] J.-M. Fontaine, Groupes finis commutatifs sur les vecteurs de Witt. _C. R. Acad. Sci. Paris Sér. A-B_ 280 (1975), Ai, A1423–A1425.
  * [81] J.-M. Fontaine, Semi-stable Galois representations, 1988, URL https://webusers.imj-prg.fr/~pierre.colmez/arbeitstagung-1988.pdf, notes from the 1988 Mathematische Arbeitstagung.
  * [82] J.-M. Fontaine and G. Laffaille, Construction de représentations $p$-adiques. _Ann. Sci. École Norm. Sup. (4)_ 15 (1982), no. 4, 547–608 (1983).
  * [83] J.-M. Fontaine and B. Mazur, Geometric Galois representations. In _Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993)_, pp. 41–78, Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995\.
  * [84] N. Freitas, B. V. Le Hung, and S. Siksek, Elliptic curves over real quadratic fields are modular. _Invent. Math._ 201 (2015), no. 1, 159–206.
  * [85] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case. 2006, arXiv:math/0602606.
  * [86] S. Galatius and A. Venkatesh, Derived Galois deformation rings. _Adv. Math._ 327 (2018), 470–623.
  * [87] T. Gee, Automorphic lifts of prescribed types. _Math. Ann._ 350 (2011), no. 1, 107–144.
  * [88] A. Genestier and J. Tilouine, Systèmes de Taylor-Wiles pour ${\mathrm{GSp}}(4)$. _Astérisque_ (2005), no. 302, 177–290.
  * [89] D. Geraghty, Modularity lifting theorems for ordinary Galois representations. _Math. Ann._ 373 (2019), no. 3-4, 1341–1427.
  * [90] W. Goldring and J.-S. Koskivirta, Strata Hasse invariants, Hecke algebras and Galois representations. _Invent. Math._ 217 (2019), no. 3, 887–984.
  * [91] E. S. Golod and I. R. Šafarevič, On the class field tower. _Izv. Akad. Nauk SSSR Ser. Mat._ 28 (1964), 261–272.
  * [92] B. Green, F. Pop, and P. Roquette, On Rumely’s local-global principle. _Jahresber. Deutsch. Math.-Verein._ 97 (1995), no. 2, 43–74.
  * [93] T. Haines and M. Harris (eds.), _Stabilization of the trace formula, Shimura varieties, and arithmetic applications, volume II: Shimura varieties._ London Mathematical Society Lecture Note Series, Cambridge University Press, 2020.
  * [94] D. Hansen, Minimal modularity lifting for $\mathrm{GL}_{2}$ over an arbitrary number field. 2012, arXiv:1209.5309.
  * [95] M. Harris, Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications. In _Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II_ , pp. 1–21, Progr. Math. 270, Birkhäuser Boston, Boston, MA, 2009.
  * [96] M. Harris, K.-W. Lan, R. Taylor, and J. Thorne, On the rigid cohomology of certain Shimura varieties. _Res. Math. Sci._ 3 (2016), 3:37.
  * [97] M. Harris, N. Shepherd-Barron, and R. Taylor, A family of Calabi-Yau varieties and potential automorphy. _Ann. of Math. (2)_ 171 (2010), no. 2, 779–813.
  * [98] M. Harris, D. Soudry, and R. Taylor, $l$-adic representations associated to modular forms over imaginary quadratic fields. I. Lifting to ${\mathrm{G}Sp}_{4}({\mathbf{Q}})$. _Invent. Math._ 112 (1993), no. 2, 377–411.
  * [99] M. Harris and R. Taylor, Deformations of automorphic Galois representations, 1998–2003, http://www.math.columbia.edu/ harris/website/publications.
  * [100] M. Harris and R. Taylor, _The geometry and cohomology of some simple Shimura varieties_. Annals of Mathematics Studies 151, Princeton University Press, Princeton, NJ, 2001.
  * [101] H. Hida, Galois representations into ${\mathrm{GL}}_{2}({\mathbf{Z}}_{p}[[X]])$ attached to ordinary cusp forms. _Invent. Math._ 85 (1986), no. 3, 545–613.
  * [102] H. Hida, Nearly ordinary Hecke algebras and Galois representations of several variables. In _Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988)_ , pp. 115–134, Johns Hopkins Univ. Press, Baltimore, MD, 1989.
  * [103] H. Hida, $p$-adic ordinary Hecke algebras for ${\mathrm{GL}}(2)$. _Ann. Inst. Fourier (Grenoble)_ 44 (1994), no. 5, 1289–1322.
  * [104] Y. Hu and V. Paškūnas, On crystabelline deformation rings of ${\mathrm{G}al}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$. _Math. Ann._ 373 (2019), no. 1-2, 421–487.
  * [105] H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II. _Amer. J. Math._ 103 (1981), no. 4, 777–815.
  * [106] P. L. Kassaei, A gluing lemma and overconvergent modular forms. _Duke Math. J._ 132 (2006), no. 3, 509–529.
  * [107] P. L. Kassaei, Modularity lifting in parallel weight one. _J. Amer. Math. Soc._ 26 (2013), no. 1, 199–225.
  * [108] P. L. Kassaei, Analytic continuation of overconvergent Hilbert modular forms. _Astérisque_ (2016), no. 382, 1–48.
  * [109] P. L. Kassaei, S. Sasaki, and Y. Tian, Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case. _Forum Math. Sigma_ 2 (2014), e18, 58.
  * [110] C. Khare, Remarks on mod $p$ forms of weight one. _Internat. Math. Res. Notices_ (1997), no. 3, 127–133.
  * [111] C. Khare, Serre’s modularity conjecture: the level one case. _Duke Math. J._ 134 (2006), no. 3, 557–589.
  * [112] C. Khare and J.-P. Wintenberger, On Serre’s conjecture for 2-dimensional mod $p$ representations of ${\mathrm{G}al}(\overline{\mathbb{Q}}/\mathbb{Q})$. _Ann. of Math. (2)_ 169 (2009), no. 1, 229–253.
  * [113] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. I. _Invent. Math._ 178 (2009), no. 3, 485–504.
  * [114] C. Khare and J.-P. Wintenberger, Serre’s modularity conjecture. II. _Invent. Math._ 178 (2009), no. 3, 505–586.
  * [115] C. B. Khare and J. A. Thorne, Potential automorphy and the Leopoldt conjecture. _Amer. J. Math._ 139 (2017), no. 5, 1205–1273.
  * [116] M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture. _Invent. Math._ 153 (2003), no. 2, 373–454.
  * [117] M. Kisin, Modularity for some geometric Galois representations. In _$L$ -functions and Galois representations_, pp. 438–470, London Math. Soc. Lecture Note Ser. 320, Cambridge Univ. Press, Cambridge, 2007.
  * [118] M. Kisin, Modularity of 2-dimensional Galois representations. In _Current developments in mathematics, 2005_ , pp. 191–230, Int. Press, Somerville, MA, 2007.
  * [119] M. Kisin, Potentially semi-stable deformation rings. _J. Amer. Math. Soc._ 21 (2008), no. 2, 513–546.
  * [120] M. Kisin, The Fontaine-Mazur conjecture for ${\mathrm{GL}}_{2}$. _J. Amer. Math. Soc._ 22 (2009), no. 3, 641–690.
  * [121] M. Kisin, Modularity of 2-adic Barsotti-Tate representations. _Invent. Math._ 178 (2009), no. 3, 587–634.
  * [122] M. Kisin, Moduli of finite flat group schemes, and modularity. _Annals of Math.(2)_ 170 (2009), no. 3, 1085–1180.
  * [123] M. Kisin, The structure of potentially semi-stable deformation rings. In _Proceedings of the International Congress of Mathematicians. Volume II_ , pp. 294–311, Hindustan Book Agency, New Delhi, 2010.
  * [124] S. L. Kleiman, The standard conjectures. In _Motives (Seattle, WA, 1991)_ , pp. 3–20, Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994.
  * [125] R. E. Kottwitz, On the $\lambda$-adic representations associated to some simple Shimura varieties. _Invent. Math._ 108 (1992), no. 3, 653–665.
  * [126] K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties. _Adv. Math._ 242 (2013), 228–286.
  * [127] R. P. Langlands, _Base change for ${\mathrm{GL}}(2)$_. Annals of Mathematics Studies 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.