Wenhan Dai

Lectures on the Langlands Program

IHES 2022 Summer School

• Time: July 11 to 29, 2022.
• Organizers: Pierre-Henri Chaudouard (IMJ-PRG), Wee Teck Gan (National Univ. of Singapore), Tasho Kaletha (Univ. of Michigan), and Yiannis Sakellaridis (Johns Hopkins Univ.).
• Scientific Committee: Gérard Laumon (Univ. Paris-Sud), Colette Mœglin (IMJ-PRG), Bảo Châu NGÔ (Chicago Univ.) and Jean-Loup Waldspurger (IMJ-PRG).

It has been almost 45 years since the influential summer school held in Corvallis, Oregon in 1977 brought together the leading experts of the Langlands program and defined the research agenda in this area for subsequent decades, at the same time inspiring and enabling several generations of young researchers to join in this exciting journey. This 3-week IHES summer school aims to do the same for the next phase of development in the Langlands program.

Recent decades have brought tremendous progress on the project of endoscopy, the extension of the Langlands program to the “relative” setting of spherical varieties and other related spaces, numerous successful “explicit” methods (such as the theta correspondence) to construct functoriality.

Ideas from the geometric Langlands program have begun impacting and enriching the classical Langlands program in significant ways. In particular, the idea that the “space of Langlands parameters” is not just a set, but a geometric space, can be used to organize a lot of developments around reciprocity.

The Summer School will attempt to bring these exciting new directions together and explore their interactions.

Lecture Notes (in chronological order)

Please use with caution and do not disseminate.

The video links are uploaded by IHES onto YouTube. And a transcription set of videos on Bilibili is at provisional work as well.

July 11

• Organizers – Warm-up Introduction. (video)
  Organizers’ answers to: What is the Langlands Program? Why a summer school on the Langlands Program in 2022? What do you hope to achieve during these three weeks?

• Olivier Taïbi (ENS Lyon) – The Local Langlands Conjecture (1/3). (video, live-TeXed notes, notes)
  We formulate the local Langlands conjecture for connected reductive groups over local fields, including the internal parametrization of L-packets.

• Erez Lapid (Weizmann Institute) – Some Perspective on Eisenstein Series (1/2). (video, live-TeXed notes)
  This is a review of some developments in the theory of Eisenstein series since Corvallis. It seems that some unusual beast appeared in his talk. This expository writing on Eisenstein series contains a relatively mild introduction to the background.

• Olivier Taïbi (ENS Lyon) – The Local Langlands Conjecture (2/3). (video, live-TeXed notes, notes)
  This is a follow-up to the previous session.

• Sophie Morel (ENS Lyon) – Shimura Varieties (1/3). (video, live-TeXed notes, notes)
  Depending on your point of view, Shimura varieties are a special kind of locally symmetric spaces, a generalization of moduli spaces of abelian schemes with extra structures, or the imperfect characteristic 0 version of moduli spaces of shtuka. They play an important role in the Langlands program because they have many symmetries (the Hecke correspondences) allowing us to link their cohomology to the theory of automorphic representations, and on the other hand they are explicit enough for this cohomology to be computable. The goal of these lectures is to give an introduction to Shimura varieties, to present some examples, and to explain the conjectures on their cohomology (at least in the simplest case).

July 12

• Pierre-Henri Chaudouard (IMJ-PRG) – Introduction to the (Relative) Trace Formula (1/2). (video, notes)
  The relative trace formula as envisioned by Jacquet and others is a possible generalization of the Arthur-Selberg trace formula. It is expected to be a useful tool in the relative Langlands program. We will try to present the general principle and give some examples and applications.

• Erez Lapid (Weizmann Institute) – Some Perspectives on Eisenstein Series (2/2). (video, notes)
  This is a follow-up to the previous session.

• Sophie Morel (ENS Lyon) – Shimura Varieties (2/3). (video, live-TeXed notes, notes)
  To continue the previous lecture. Kai-wen Lan’s expository paper is an instructive reference that is particularly recommended. Yihang Zhu’s notes are based on a topics course on Shimura varieties at the University of Maryland in Spring of 2022.

• Olivier Taïbi (ENS Lyon) – The Local Langlands Conjecture (3/3). (video, live-TeXed notes, notes)
  This is a follow-up to the yesterday session. The most recent version of notes of all 3 lectures by Oliver Taïbi is now available.

July 13

• Pierre-Henri Chaudouard (IMJ-PRG) – Introduction to the (Relative) Trace Formula (2/2). (video, notes)
  This is a follow-up to the previous session.

• Lucas Mason-Brown (Oxford) – Arthur’s Conjectures and the Orbit Method for Real Reductive Groups. (video, notes)
  The most fundamental unsolved problem in the representation theory of Lie groups is the Problem of the Unitary Dual: given a reductive Lie group G, this problem asks for a parameterization of the set of irreducible unitary G-representations. There are two big “philosophies” for approaching this problem. The Orbit Method of Kostant and Kirillov seeks to parameterize irreducible unitary representations in terms of finite covers of co-adjoint G-orbits. Arthur’s conjectures suggest a parameterization in terms of certain combinatorial gadgets (i.e. Arthur parameters) related to the Langlands dual group G^{\vee} of G. In this talk, We will define these correspondences precisely in the case of complex groups. We will also define a natural duality map from Arthur parameters (for G^{\vee}) to co-adjoint covers (for G) which, in a certain precise sense, intertwines these correspondences. This talk is partially based on joint work with Ivan Losev and Dmitryo Matvieievskyi.

• Zhiwei Yun (MIT) – Introduction to Shtukas and their Moduli (1/3). (video, notes)
  We will start with basic definitions of Drinfeld Shtukas and their moduli stacks. Then we will talk about its geometric and cohomological properties, and important constructions such as Hecke correspondences and partial Frobenius. We will also mention its relation with Drinfeld modules and analogy with motives.

• Sophie Morel (ENS Lyon) – Shimura Varieties (3/3). (video, live-TeXed notes, notes)
  To continue the previous lecture. See the most recent version of notes for all 3 lectures by Sophie Morel on Shimura varieties.

July 14

• Raphaël Beuzart-Plessis (Univ. Aix-Marseille) – The Relative Langlands Program (1/3). (video, notes)
  Broadly interpreted, the Relative Langlands Program (RLP) is an attempt to unify those somehow disparate phenomena by systematically relating them to a dual Galois picture similar to that of the mainstream Langlands program. As we will see, it turns out that the RLP can actually be regarded as a direct enhancement of the usual LP and not merely as a separately connected subject.

• Xinwen Zhu (Stanford) – Coherent Sheaves on the Stack of Langlands Parameters (1/3). (video, notes)
  The speaker gives an impression of some recent new ideas appearing in the arithmetic Langlands program, with an emphasis on coherent sheaves on moduli spaces of Langlands parameters.

• Zhiwei Yun (MIT) – Introduction to Shtukas and their Moduli (2/3). (video, notes)
  This is a follow-up to the previous session.

• Zhiwei Yun (MIT) – Introduction to Shtukas and their Moduli (3/3). (video, notes)
  This is a follow-up to the previous session.

July 15

• Raphaël Beuzart-Plessis (Univ. Aix-Marseille) – The Relative Langlands Program (2/3). (video, notes)
  This is a follow-up to the previous session.

• Xinwen Zhu (Stanford) – Coherent Sheaves on the Stack of Langlands Parameters (2/3). (video, notes)
  This is a follow-up to the previous session.

• Xinwen Zhu (Stanford) – Coherent Sheaves on the Stack of Langlands Parameters (3/3). (video, notes)
  This is a follow-up to the previous session.

• Raphaël Beuzart-Plessis (Univ. Aix-Marseille) – The Relative Langlands Program (3/3). (video, notes)
  This is a follow-up to the previous session.

July 18

• Tasho Kaletha (Univ. Michigan) – A Brief Introduction to the Trace Formula and its Stabilization (1/2). (video, notes)
  We will discuss the derivation of the stable Arthur-Selberg trace formula. In the first lecture we will focus on anisotropic reductive groups, for which the trace formula can be derived easily. We will then discuss the stabilization of this trace formula, which is unconditional on the geometric side, and relies on the Arthur conjectures on the spectral side. In the second lecture we will sketch the case of an arbitrary reductive group, which causes many analytic difficulties. We will briefly describe the various stops on the road to the stable trace formula, including the coarse and fine expansions of the non-invariant trace formula, as well the invariant trace formula. Examples will be given for the group SL(2). Towards the end, we will discuss the application of the stable trace formula to the classification of representations of classical groups.

• Jared Weinstein (Boston Univ.) – Local Shtukas and the Langlands Program (1/2). (video, notes)
  In the Langlands program over number fields, automorphic representations and Galois representations are placed into correspondence, using the cohomology of Shimura varieties as an intermediary. Over a function field, the appropriate intermediary is a moduli space of shtukas. We introduce the shtukas and their local analogs, which play a similar role in the local Langlands program. Along the way, we construct the Fargues-Fontaine curve and discuss perfectoid spaces and diamonds. This survey may be seen as preparatory for the lectures of Fargues-Scholze.

• Bao Châu Ngô (Chicago Univ.) – Orbital Integrals, Moduli Spaces and Invariant Theory (1/3). (video, notes)
  The goal of these lectures is to sketch a general framework to study orbital integrals over equal characteristic local fields by means of moduli spaces of Hitchin type following the main lines of the proof of the fundamental lemma for Lie algebras. After recalling basic elements of the proof of the fundamental lemma for Lie algebras as well as recent related developments, we will explain an invariant-theoretic construction which should be a basic tool to understand general orbital integrals.

• Bao Châu Ngô (Chicago Univ.) – Orbital Integrals, Moduli Spaces and Invariant Theory (2/3). (video, notes)
  This is a follow-up to the previous session.

July 19

• Jared Weinstein (Boston Univ.) – Local Shtukas and the Langlands Program (2/2). (video, notes)
  This is a follow-up to the previous session.

• Tasho Kaletha (Univ. Michigan) – A Brief Introduction to the Trace Formula and its Stabilization (2/2). (video, notes)
  This is a follow-up to the previous session.

• Cong Xue (IMJ-PRG) – Cohomology Sheaves of Stacks of Shtukas (1/2). (video, notes)
  Cohomology sheaves and cohomology groups of stacks of shtukas are used in the Langlands program for function fields. We will explain (1) how the Eichler-Shimura relations imply the finiteness property of the cohomology groups, (2) how the finiteness and Drinfeld’s lemma imply the action of the Weil group of the function field on the cohomology groups, and (3) how this action and the “Zorro lemma” imply the smoothness of the cohomology sheaves. The smoothness will be used in Sam Raskin’s lecture.

• Cong Xue (IMJ-PRG) – Cohomology Sheaves of Stacks of Shtukas (2/2). (video, notes)
  This is a follow-up to the previous session.

July 20

• Eugen Hellman (Univ, Münster) – An Introduction to the Categorical p-adic Langlands Program (1/4). (video, notes)
  An introduction to the “categorical” approach to the p-adic Langlands program, in both the “Banach” and “analytic” settings.

• Bao Châu Ngô (Chicago Univ.) – Orbital Integrals, Moduli Spaces and Invariant Theory (3/3). (video, notes)
  This is a follow-up to the previous session.

• Sam Raskin (Univ. Texas) – What does geometric Langlands mean to a number theorist? (1/2). (video, notes)
  We give a brief survey about the work series by Arinkin-Kazhdan-Gaitsgory-Raskin-Rozenblyum-Varshavsky.

• Sam Raskin (Univ. Texas) – What does geometric Langlands mean to a number theorist? (2/2). (video, notes)
  This is a follow-up to the previous session.

• Panel Discussion with Participants of the Corvallis Conference.
  J. Arthur (Univ. Toronto), B. Casselman (Univ. of British Columbia), B. Gross (Harvard Univ.), M. Harris (Columbia Univ.), G. Henniart (Univ. Paris-Saclay), H. Jacquet (Columbia Univ.), J.P. Labesse (Aix-Marseille Université), K. Ribet (Berkeley Univ.), C. Soulé (CNRS-IHES), M.F. Vignéras (IMJ-PRG).

July 21

• Peter Scholze (Univ. Bonn) – The Langlands Program and the Moduli of Bundles on the Curve (1/3). (video, notes)
  The speaker will talk about his joint work on the geometrization of the local Langlands correspondence.

• Wee Teck Gan (Nat. Univ. Singapour) – Explicit Constructions of Automorphic Forms (1/2). (video, notes)
  We will discuss the theory of theta correspondence, highlighting basic principles and recent results, before explaining how theta correspondence can now be viewed as part of the relative Langlands program. We will then discuss other methods of construction of automorphic forms, such as automorphic descent and its variants and the generalized doubling method.

• Toby Gee (Imperial College) – An Introduction to the Categorical p-adic Langlands Program (2/4). (video, notes)
  This is a follow-up to the previous session.

• Matthew Emerton (Chicago Univ.) – An Introduction to the Categorical p-adic Langlands Program (3/4). (video, notes)
  This is a follow-up to the previous session.

July 22

• Jean-François Dat (IMJ-PRG) – On Moduli Spaces of Local Langlands Parameters (1/2). (video, notes)
  The moduli space of local Langlands parameters plays a key role in the formulation of some recent enhancements of the original local Langlands correspondence, such as the “local Langlands correspondence in families” and various “categorifications/geometrizations of LLC”. We will explain their construction and basic properties, with special emphasis on the coarse moduli spaces.

• Wee Teck Gan (Nat. Univ. Singapour) – Explicit Constructions of Automorphic Forms (2/2). (video, notes)
  This is a follow-up to the previous session.

• Jessica Fintzen (Duke Univ. & Cambridge Univ.) – Supercuspidal Representations: Construction, Classification, and Characters (1/2). (video, notes)
  We have seen in the first week of the summer school that the buildings blocks for irreducible representations of p-adic groups are the supercuspidal representations. In these talks we will explore explicit exhaustive constructions of these supercuspidal representations and their character formulas and observe a striking parallel between a large class of these representations in the p-adic world and discrete series representations of real algebraic Lie groups. A key ingredient for the construction of supercuspidal representations is the Bruhat-Tits theory and Moy-Prasad filtration, which we will introduce in the lecture series.

• Peter Scholze (Univ. Bonn) – The Langlands Program and the Moduli of Bundles on the Curve (2/3). (video, notes)
  This is a follow-up to the previous session.

July 25

• Matthew Emerton (Chicago Univ.) – An Introduction to the Categorical p-adic Langlands Program (4/4). (video, notes)
  This is a follow-up to the previous session.

• Jessica Fintzen (Duke Univ. & Cambridge Univ.) – Supercuspidal Representations: Construction, Classification, and Characters (2/2). (video, notes)
  This is a follow-up to the previous session.

• Dipendra Prasad (IIT Bombay) – Branching Laws: Homological Aspects. (video, notes)
  We will see the question about decomposing a representation of a group when restricted to a subgroup which is referred to as the branching law. In this lecture, we focus attention on homological aspects of the branching law. The lecture will survey this topic beginning from the beginning going up to several results which have recently been proved.

• Jean-François Dat (IMJ-PRG) – On Moduli Spaces of Local Langlands Parameters (2/2). (video, notes)
  This is a follow-up to the previous session.

July 26

• Laurent Fargues (IMJ-PRG) – The Langlands Program and the Moduli of Bundles on the Curve (3/3). (video, notes)
  This is a follow-up to the previous session.

• Sug Woo Shin (UC Berkeley) – Shimura Varieties and Modularity (1/3). (video, slides for the first part, notes for the second part)
  We describe the construction of Galois representations associated to regular algebraic cuspidal automorphic representations of GL(n) over a CM field, as well as those Galois representations associated to torsion classes that occur in the Betti cohomology of the corresponding locally symmetric spaces. The emphasis will be on Scholze’s proof, which applies to torsion classes and which uses perfectoid Shimura varieties and the Hodge-Tate period morphism.

• Ana Caraiani (Imperial College) – Shimura Varieties and Modularity (2/3). (video, notes)
  This is a follow-up to the previous session.

• Chao Li (Columbia Univ.) – Geometric and Arithmetic Theta Correspondences (1/2). (video, slides)
  Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. I will give an introduction focusing on the example of unitary groups and highlight recent advances in the arithmetic theory (also known as the Kudla program) and their applications.

July 27

• Ana Caraiani (Imperial College) – Shimura Varieties and Modularity (3/3). (video, notes)
  This is a follow-up to the previous session.

• Chao Li (Columbia Univ.) – Geometric and Arithmetic Theta Correspondences (2/2). (video, slides)
  This is a follow-up to the previous session.

• Akshay Venkatesh (IAS) – Hamiltonian Actions and Langlands Duality (1/2).
  We will give a gentle introduction to my joint work with Ben-Zvi and Sakellaridis, in which we seek to formulate various phenomena in the Langlands program in terms of Hamiltonian actions of reductive groups. In particular, this makes visible a duality underlying the relative Langlands program.

• Akshay Venkatesh (IAS) – Hamiltonian Actions and Langlands Duality (2/2).
  This is a follow-up to the previous session.

July 28

• Wei Zhang (MIT) – High-dimensional Gross–Zagier Formula (1/2). (video, notes)
  We discuss various generalizations of the Gross-Zagier formula to high-dimensional Shimura varieties, with an emphasis on the AGGP conjecture and the relative trace formula approach. Roughly the first lecture will be devoted to the global aspect and the second one to the local aspect.

• David Ben-Zvi (UT Austin) – Between Coherent and Constructible Local Langlands Correspondences. (video, notes)
  (Joint with Harrison Chen, David Helm, and David Nadler.) Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part. In this talk, we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the archimedean case, we explain a conjectural realization of coherent local Langlands as geometric Langlands on the twistor line, the real counterpart of the Fargues-Fontaine curve, and its relation to constructible local Langlands via circle actions. In the non-archimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of GL(n), and propose a mechanism to relate the two settings in general.

• Tony Feng (MIT) – Derived Aspects of the Langlands Program (1/3). (video, notes)
  We discuss ways in which derived structures have recently emerged in connection with the Langlands correspondence, with an emphasis on derived Galois deformation rings and derived Hecke algebras.

• Yiannis Sakellaridis (Johns Hopkins Univ.) – Local and Global Questions “Beyond Endoscopy” (1/2). (video, notes)
  The near completion of the program of endoscopy poses the question of what lies next. These talks will take a broad view of ideas beyond the program of endoscopy, highlighting the connections among them, and emphasizing the relationship between local and global aspects. Central among those ideas is the one proposed in a 2000 lecture of R. P. Langlands, aiming to extract from the stable trace formula of a group G the bulk of those automorphic representations in the image of the conjectural functorial lift corresponding to a morphism of L-groups. With the extension of the problem of functionality to the “relative” setting of spherical varieties and related spaces, some structure behind such comparisons has started to reveal itself. In a seemingly unrelated direction, a program initiated by Braverman-Kazhdan, also around 2000, to generalize the Godement-Jacquet proof of the functional equation to arbitrary L-functions, has received renewed attention in recent years. We survey ideas and developments in this direction, as well, and discuss the relationship between the two programs.

July 29

• Tony Feng (MIT) – Derived Aspects of the Langlands Program (2/3). (video, notes)
  This is a follow-up to the previous session.

• Wei Zhang (MIT) – High-dimensional Gross–Zagier Formula (2/2). (video, notes)
  This is a follow-up to the previous session.

• Yiannis Sakellaridis (Johns Hopkins Univ.) – Local and Global Questions “Beyond Endoscopy” (2/2). (video, notes)
  This is a follow-up to the previous session.

• Michael Harris (Columbia Univ.) – Derived Aspects of the Langlands Program (3/3). (video, notes)
  This is a follow-up to the previous session.