An introduction to the categorical p-adic Langlands program

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Matthew Emerton, Toby Gee, and Eugen Hellmann

Abstract. We give an introduction to the “categorical” approach to the p-

adic Langlands program, in both the “Banach” and “analytic” settings.

Contents

1. Introduction 2

1.1. A rapid overview of the Langlands program 2

1.2. The p-adic perspective 4

1.3. Proofs of reciprocity 4

1.4. Unramiﬁed geometric Langlands and the Fargues–Scholze conjecture 6

1.5. Some diﬀerences between the ℓ-adic and p-adic cases 8

1.6. Motivation from Taylor–Wiles patching 10

1.7. The “Galois to automorphic” direction 11

1.8. Topics we omit 11

1.9. A brief guide to the notes 12

1.10. Acknowledgements 12

2. Notation 13

2.1. Fields 13

2.2. p-adic Hodge theory 13

2.3. Hodge–Tate weights vs. highest weights 14

3. Taylor–Wiles patching as a motivation for categorical p-adic local

Langlands 14

3.1. A very brief introduction to patching 14

3.2. p-adic Langlands: an overview and history 18

4. Moduli stacks of (φ, Γ)-modules: the Banach case 32

4.1. An overview of [EG23] 33

4.2. Fixed determinant variant 35

5. Moduli stacks of (φ, Γ)-modules: the analytic case 35

M.E. was supported in part by the NSF grants DMS-1902307, DMS-1952705, and DMS-

2201242. T.G. was supported in part by an ERC Advanced grant. This project has received

funding from the European Research Council (ERC) under the European Union’s Horizon 2020

research and innovation programme (grant agreement No. 884596). M.E. and T.G. were both

supported in part by the Simons Collaboration on Perfection in Algebra, Geometry and Topol-

ogy. E.H. was supported by Germany’s Excellence Strategy EXC 2044-390685587 “Mathematics

Münster: Dynamics–Geometry–Structure" and by the CRC 1442 Geometry: Deformations and

Rigidity of the DFG.

arXiv:2210.01404v3 [math.NT] 1 Apr 2025

2 M. EMERTON, T. GEE, AND E. HELLMANN

5.1. (φ, Γ)-modules over the Robba ring 35

5.2. Stacks of de Rham objects 50

5.3. Drinfeld-style compactiﬁcations 57

6. Categorical p-adic local Langlands conjectures 78

6.1. Expectations: The Banach Case 78

6.2. Expectations: the analytic case 87

7. Some known cases of categorical p-adic Langlands 103

7.1. GL1103

7.2. The Banach case for GL2(Qp)— I. The structure of X108

7.3. The Banach case for GL2(Qp)— II. Proof sketches 118

7.4. The Banach case for GL2(Qp)— III. Examples 127

7.5. Semiorthogonal decompositions 134

7.6. Preliminaries in the Banach case for GL2(Qpf)140

7.7. Partial results and examples in the Banach case for GL2(Qp2)145

7.8. The adjoint functor 164

7.9. Global functions and Hecke algebras 170

8. Categorical local Langlands in the case ℓ̸=p170

8.1. C-groups 171

8.2. Langlands parameters in the local case — ℓ̸=p171

8.3. Categorical local Langlands 172

9. Moduli stacks of global Langlands parameters and the cohomology of

Shimura varieties 174

9.1. Global moduli stacks 174

9.2. The global-to-local map 175

9.3. Computing the cohomology of Shimura varieties via categorical

Langlands 177

9.4. An example in the global case: (PGL2)/Q178

9.5. The structure of Xodd

GL2,Q,S 179

9.6. Eigenvarieties and sheaves of overconvergent p-adic automorphic forms181

Appendix A. Background on category theory 192

Appendix B. Coherent, Ind-Coherent, and Pro-Coherent sheaves on

Ind-algebraic stacks 206

Appendix C. Localization and Morita theory 214

Appendix D. Topological abelian groups, rings and modules 224

Appendix E. Representations of p-adic analytic groups 228

Appendix F. Derived moduli stacks of group representations 235

References 237

1. Introduction

The aim of these notes is to discuss some of the p-adic aspects of the Langlands

program, and especially the emerging “categorical” perspective. We begin with an

overview, ﬁrst of some history and then of our goals; we refer the reader to Section 2

for any unfamiliar notation.

1.1. A rapid overview of the Langlands program. The Langlands pro-

gram began with Langlands’s celebrated letter to Weil (reproduced e.g. in [DS15,

§6]), the contents of which were elaborated on in several subsequent writings of

CATEGORICAL p-ADIC LANGLANDS 3

Langlands, including his article “Problems in the theory of automorphic forms” [Lan70]

and his Yale lectures on Euler products [Lan71]. The article focuses on the functo-

riality conjecture for automorphic representations and its consequences, while the

Yale lectures explain his construction of automorphic L-functions, and discuss the

problem of proving their expected analytic properties (analytic continuation and

functional equation).1

In his Yale lectures Langlands already touches, if only tangentially, on the

idea that the automorphic L-functions may include all motivic L-functions. Sub-

sequently, this idea developed into the reciprocity conjecture. (See [Lan79] for

one articulation of this conjecture.) Roughly speaking, the reciprocity conjecture

articulates a correspondence between certain automorphic representations (those

which are algebraic) of G(AF)(for a connected reductive algebraic group Gover

a number ﬁeld F), and motives (with coeﬃcients in Q, say) over Fwhose motivic

Galois group is closely related to to the C-group2cGof G. Conjectures on Galois

representations (especially the Fontaine–Mazur conjecture [FM95]) suggest that

such motives in turn may be identiﬁed with compatible systems of ℓ-adic Galois

representations GalF→cG(Qℓ).3

There are many subtleties involved in trying to formulate a precise reciprocity

conjecture. For example, in the case that Gis not some GLd, one has to worry

about L-packets; and one should restrict to automorphic representations which

are not anomalous in the sense of [Lan79] (or else replace GalFby GalF×SL2

and work in the framework of Arthur’s conjectures regarding non-tempered en-

doscopy [Art89]). We refer to [BG14] for a more technical discussion of the

conjecture, and to [Eme21] for a more thorough historical overview.

We note that there is a relationship between reciprocity and functoriality:

namely, since functoriality is tautologically true for (compatible systems of) Galois

representations, cases of reciprocity can be used to deduce cases of functoriality.

Also, since L-groups and C-groups involve Galois groups in their deﬁnition, rep-

resentations of Galois groups into algebraic groups can sometimes be related to

L-homomorphisms of L-groups or C-homomorphisms of C-groups, and thus some

cases of reciprocity can be subsumed into functoriality. (This is Langlands’s origi-

nal perspective on the Artin conjecture, as explained in [Lan70].) If one replaces

the Galois group GalFby the hypothetical Langlands group LF, and compatible

systems of GalF-representations by representations of LF, then one can also extend

the reciprocity conjecture to non-algebraic automorphic representations, or (using

the “LF-form” of the L-group or C-group) entirely subsume reciprocity into functo-

riality. (A version of this last-mentioned perspective was already adopted at times

by Langlands, by using the “Weil group form” of the L-group.)

1Since the standard L-functions associated to automorphic representations on GLdcoincide

with the L-functions constructed by Godement–Jacquet [GJ72] and Tamagawa [Tam63], which

are known to admit analytic continuations and functional equations, the expected properties would

follow from functoriality. Langlands’s Yale lectures discuss another approach, via constant terms

of Eisenstein series.

2The C-group, introduced in [BG14], is a reﬁnement of the L-group introduced by Langlands,

which is better adapted to the problem of relating automorphic forms and Galois representations;

we recall the deﬁnition below.

3As far as we know, such an expectation can only be made precise for G= GLn, as it is

unclear what the precise deﬁnition of a compatible system of Galois representations should be for

a general G; see e.g. [BHKT19, §6].

4 M. EMERTON, T. GEE, AND E. HELLMANN

1.2. The p-adic perspective. Our intention in these notes is not so much

to focus on reciprocity in the manner described above (the relationship between

automorphic representations and motives, or compatible systems of Galois repre-

sentations), but rather to ﬁx a prime pand consider the relationship between au-

tomorphic representations and p-adic Galois representations (for this ﬁxed choice

of p). At ﬁrst, this doesn’t much change the problem, since (at least for represen-

tations valued in GLd) a compatible system of semisimple Galois representations is

determined by any one of its members, and the Fontaine–Mazur conjecture gives a

purely Galois-theoretic condition for a p-adic Galois representation to be motivic.

But focusing on a particular prime pbrings to the fore certain aspects of the

theory of automorphic forms and Galois representations which are absent in the

more motivic perspective on reciprocity. For example, since at least the work of

Ramanujan, it has been known that automorphic forms can satisfy interesting con-

gruences modulo powers of p. Since the work of Swinnerton–Dyer [SD73] and

Serre [Ser73], it has been understood that these congruences are related to (or

are manifestations of, if one prefers) analogous congruences between p-adic Galois

representations. Extending the notion of congruences between automorphic forms,

one is led to take various p-adic completions of spaces of automorphic forms to ob-

tain the notion of p-adic automorphic forms, with associated Galois representations

that need not be motivic. Related to this, one has notions of mod pautomorphic

forms, to which one might associate mod pGalois representations; and reciprocity

conjectures have been formulated in this context, famously by Serre [Ser87] in

the context of classical modular forms, and more recently in greater generality by

others (e.g. [BDJ10]).

We do not intend at all to survey these developments; rather, our aim here

is simply to indicate that the p-adic perspective emerged naturally over a long

period of time, and has led to a natural collection of problems and concerns: e.g.

how to arrive at some notion of automorphic form which admits p-adic integral

or mod pcoeﬃcients, and which allows for genuinely p-adic objects? And how to

phrase reciprocity in a manner which allows for p-adic integral or mod pGalois

representations, which will be associated to the p-adic integral or mod pobjects

that one introduces on the automorphic side?

We make one last remark on the p-adic theory for now: once one allows oneself

to p-adically interpolate automorphic forms, one sees that automorphic forms nat-

urally lie in families (e.g. Hida families, Coleman families, . . . ; see [Eme11b] for a

survey). And Galois representations also lie in families (e.g. via Mazur’s deforma-

tion theory [Maz89], which he was at least partly motivated to develop in response

to Hida’s theory of ordinary families of p-adic modular forms). These phenomena

of continuous families of objects — on both the automorphic and Galois side — are

not a feature of reciprocity in its more motivic formulation. (Cuspidal automorphic

representations are rigid objects, and so are motives, if one has ﬁxed a particular

number ﬁeld as the ﬁeld of deﬁnition.) Before passing to our next topic, we note

that families of Galois representations are a feature of the geometric Langlands

program. We will return to this point below.

1.3. Proofs of reciprocity. Some of the very ﬁrst results on reciprocity were

proved in the case of representations with solvable image, by combining class ﬁeld

theory (which implies reciprocity in the abelian context) with techniques more clas-

sically automorphic in nature (see in particular the work of Langlands and Tunnell,

CATEGORICAL p-ADIC LANGLANDS 5

[Lan80, Tun81]). In 1995, a breakthrough in our understanding of reciprocity

was achieved with the proof by Wiles [Wil95] and Taylor–Wiles [TW95] of the

modularity of semistable elliptic curves over Q, a result which was soon improved

to handle all elliptic curves over Q[BCDT01]. Building on these methods, reci-

procity has since been proved in many further interesting contexts; see Calegari’s

recent survey [Cal23] for some of the highlights of the last 25 years.

The crux of the method of [Wil95, TW95] and the many subsequent results

that build on their ideas is to prove the modularity — or, more generally, au-

tomorphy — of a p-adic Galois representation. More precisely, the Taylor–Wiles

method gives a way to deduce from the modularity of a single p-adic Galois rep-

resentation ρ1the modularity of another p-adic Galois representation ρ2which is

congruent to ρ1modulo p; this is typically referred to as modularity lifting. (We

“lift” modularity from the mod preduction of ρ1— which, since it coincides with

the mod preduction of ρ2, is inherited from that of ρ2— to the the modularity

of the p-adic representation ρ1itself.) It turns out that there are many such con-

gruences between Galois representations, and modularity can be propagated from

a single representation to many others in this way (either using a ﬁxed prime p,

or using several diﬀerent primes p). Of course, it is necessary to have an initial

supply of representations whose modularity is already known, such as CM forms,

or 2-dimensional Artin representations with solvable image (the latter being used

in Wiles’ proof of Fermat’s Last Theorem [Wil95]).

Modularity lifting theorems are also known as “R=T” theorems, because they

are proved by identifying a Zp-algebra R(a “Galois deformation ring”) parame-

terizing Galois representations congruent to a ﬁxed representation ρ1with another

Zp-algebra T, a “Hecke algebra”, the endomorphism algebra of an appropriate space

of (p-adic) modular forms. Over the last 25 years it has become apparent that such

theorems should hold in great generality, although ﬁnding appropriate deﬁnitions of

Rand Tso that we literally have R=Tis still something of an art. In particular,

it has become clear (following in particular Skinner–Wiles [SW99], Wake–Wang-

Erickson [WWE17], and Newton–Thorne [NT23]) that in general Rshould be

taken to be a so-called pseudodeformation ring, i.e. a deformation ring for pseu-

dorepresentations, rather than a deformation ring for literal Galois representations.

Roughly speaking, pseudorepresentations capture the information given by

characteristic polynomials of representations; for this reason they are also known as

pseudocharacters. They were originally introduced for GL2by Wiles [Wil88], and

were considerably developed for GLdby many authors (we highlight in particular

Chenevier’s notion of a determinant [Che14], which showed how to deﬁne them

for arbitrary primes p). A theory valid for general reductive groups was introduced

by Vincent Laﬀorgue [Laf18].

From Laﬀorgue’s point of view, as recently made precise (in the setting of local

Galois representations) by Fargues–Scholze [FS21, §VIII], a pseudodeformation

ring Ris the ring of global functions on a moduli stack of L-parameters, so that

the moduli space of pseudorepresentations is a coarse moduli space4for the moduli

stack of L-parameters; and for general reductive groups, the Hecke algebra Tis

4Here and below, we use the expression “coarse moduli space” in its usual informal manner;

since stacks of L-parameters are not Deligne–Mumford stacks, they do not admit coarse moduli

spaces in the technical sense, but rather adequate moduli spaces in the sense of [Alp14].

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Anintroductiontothecategoricalp-adicLanglandsprogramMatthewEmerton,TobyGee,andEugenHellmannAbstract.Wegiveanintroductiontothe“categorical”approachtothep-adicLanglandsprogram,inboththe“Banach”and“analytic”settings.Contents1.Introduction21.1.ArapidoverviewoftheLanglandsprogram21.2.Thep-adicperspective...

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