SEMINAR ON THE LOCAL LANGLANDS CONJECTURES FOR GL(n) OVER
A p-ADIC FIELD
Times: Mondays 2:30pm-4:00pm and Fridays 3:00pm-4:30pm.
Place: MCS B21.
Introduction
Summary. The goal of this seminar is to discuss the local Langlands conjectures for GL(n) over
a p-adic field. Roughly speaking, local Langlands matches up the smooth representation theory of
a p-adic reductive group, like GLn (Qp ), with certain n-dimensional linear representations of Weil
groups associated to p-adic fields, like Qp .
There are now three proofs of the correspondence. Historically, the first proof was due to Harris
and Taylor [8]. As the reference suggests, it relies on studying the geometry and cohomology of
Shimura varieties, essentially realizing the correspondence within the cohomology of certain local
systems. After their proof was announced, Henniart found a simpler proof [12] which bypasses the
high octane use of Shimura varieties but does not give the important realization of local Langlands
geometrically. The goal of our seminar will be to discuss this proof. There is a third, more recent,
proof due to Scholze [14] that revisits Harris-Taylor but makes significant simplifications. All three
proofs require global methods and every major prerequisite for Henniart’s proof is also a prerequisite
for one of the other proofs.
There are essentially three stages of the seminar.
- The first five weeks (Jan 17–Feb 28) will be spent on the background needed to understand
the objects involved in the local Langlands correspondence and its characterization. This
includes the cast of characters on each side of the correspondence and Henniart’s proof that
there is a unique correspondence matching L and ε-factors of pairs.
- The proof of the local correspondence, regardless of who has given it, exists in general
only by global methods. The second portion (Mar 4–Apr 4) constitutes a shift towards a
global perspective. We will begin with a discussion of the global theory of automorphic
representations of linear groups, including L-functions on the automorphic side. We will
also include global instance of functoriality (Arthur-Clozel) and the construction of Galois
representations attached to automorphic representations (Clozel), in special cases.
- The goal of the final month (Apr 7–May 2) is to explain Henniart’s proof in as much details
as possible. There are two majors results we need to cover: L and ε-factors of pairs are
determined once they are determined almost everywhere and the construction of non-Galois
automorphic induction. Henniart’s proof will be sandwiched between them.
Logistics. The seminar will begin meeting on Friday, January 17th at the time and dates above.
All meetings will take place in MCS B21. There are some exceptional dates where we will not meet:
• Mondays: 1/20 (MLK Day), 2/17 (President’s day/perfectoid conference), 3/10 (BU Spring
break) and 4/21 (Marathon day)
• Fridays: 2/21 (Perfectoid conference), 3/14 (BU spring break) and 3/28 (Number theory
colloquium).
On Mondays there will be tea and cookies at 4:00pm immediately preceding the research seminar.
On Fridays where we have colloquium there will also be food and we will end 15 minutes early.
1
Talks. Almost all of the talks will be given by graduate students. Each talk is slated for one
and a half hours but speakers may plan to speak for about an hour, with extra time available for
questions and interruptions. There are at least three days at the end of the semester which we have
no assigned talks for, so needing an extra day will not be a problem.
Note-taking. Hudson has kindly volunteered to help with note-taking efforts. He will TEXthe
notes live on the spot and it will be up to the speakers to then edit the notes for clarity and
completeness. Notes will be posted on the seminar website.
Schedule
The talks have already been assigned, with as many references include below as possible. There
are three secondary references which everyone should find useful: Carayol’s Bourbaki expose [4],
Wedhorn’s more recent notes [17] and the book of Bushnell and Henniart [3] (especially for the first
month and a half). If one of those references is listed below, it can be assumed the digging into
primary sources will be helpful for the speaker.
Jan 17 – Overview and motivation (Jared). Give an overview of the statement of local
Langlands with global motivation. Discuss the structure of Galois groups of number fields and
define the local Weil groups. Give the statements of local and global class field theory to realized
(local) Langlands for GL(1). Possibly include `-adic realizations of algebraic Hecke characters.
Jan 20 – No talk. Martin Luther King Day.
Jan 24 – Number theoretic background (Hudson). Discuss Weil groups of number fields and
virtual representations [15, Proposition 2.3.1]. Move on to Weil-Deligne representations motivated
by `-adic considerations [15, §4]. Discuss Galois representations and the relation to the monodromy
theorem [15, Theorem 4.2.1]. Specifically point out [15, (4.1.4)–(4.1.5)].
Jan 27 – L-functions on Galois side (Ben). Discuss L-functions of global Weil group representations, including meromorphy and the functional equation [15, Theorem 3.5.3]. Include specializations to the case of Hecke characters wherever interesting. State but do not prove the existence
of local ε-factors [15, Theorem 3.4.1].
Jan 31 – Local ε-factors (Aditya). Follow [6, §4] or [16] to prove the existence of local ε-factors
skipped in the previous talk. Focus on the technique of twisting by highly ramified characters.
Feb 3 – Introduction to representation theory of p-adic groups (David Corwin). Follow
[17, §2.1] to give an overview of the theory of smooth admissible representations of p-adic reductive
groups. Emphasize the definitions: admissible, smooth, Hecke algebras, distribution characters.
Define the Steinberg representation (in an ad hoc manner).
Feb 7 – Induction and Bernstein-Zelivinsky (James). Follow [17, §2.2] Discuss unimodularity issues in lctd groups and define parabolic induction. Define supercuspidal representations,
possibly using a discussion of matrix coefficients. Give the classification of Bernstein-Zelivinsky [1].
Feb 10 – Local Whittaker models for generic representations (Ben). Define generic representations. Give [17, Theorem 2.4.4] on which parabolic inductions are generic. Define Whittaker
models in general and show that local Whittaker models for generic representations are unique.
Feb 14 – L, ε-factors of pairs (Hudson). Follow [17, §2.5] to define the L and ε factors of a
pair. Explain interaction of Bernstein-Zelevinksy and the L, ε-factors [11, Lemma 3.3]. Make sure
to include computations or examples, e.g. [5, Theorem 3.3] or [5, Theorem 3.4].
Feb 17 – No meeting. Perfectoid conference at MSRI.
2
Feb 21 – No meeting. Perfectoid conference at MSRI.
Feb 24 – Characterization of local Langlands (John). Recall the statement of the local
Langlands correspondence for GL(n) over a p-adic field and L, ε-factors for pairs on GLn × GLn−1 .
Follow [11] to prove that a correspondence, if it exists, must be unique.
Feb 28 – First instances of the correspondence (Aditya). Finish anything unfinished by the
previous talk and then discuss the “unramified correspondence” as a reduction to the existence of
a supercuspidal/irreducible correspondence [17, Theorem 4.2.2]. Possibly mention numerical local
Langlands.
Mar 4 – Global theory for GL(2)/Q (David Rohrlich). Give an overview of automorphic
representations for GL(2)/Q and relate them to the classical definition of modular forms [13].
Mar 7 – Global theory for GL(2)/Q II (Jared). TBD. Possible topics: explicit discussion of
L-functions, Galois representations, local-global compatibility, etc.
Mar 10 – No meeting. BU Spring Break.
Mar 14 – No meeting. BU Spring Break.
Mar 17 – Automorphic representations for GL(n) (James). Discuss cuspidal automorphic
representations for GL(n) over a general number field. The main results we will want to hear about
are all contained in [2, §3.3]. In particular, you should explain the statements of [2, Theorem 3.3.2],
[2, Theorem 3.3.3] and [2, Theorem 3.3.6].
Mar 21 – Automorphic representations for GL(n) II (Ben). The goal here is to fill in details
of the previous talk and prepare for the construction of L-functions of automorphic representations
for GL(n) by discussing global Whittaker models. Thus discuss what you want about the proofs of
the previous results and then move on to Eulerian integral representations to explain as much as
possible from [5, Theorem 2.1] and [5, Theorem 2.2]. It is most important that we understand the
entire/meromorphic properties of the integral representations.
Mar 24 – L-functions of automorphic representations for GL(n) (Hudson). Continue
the previous talk in order to define the L-functions for automorphic representations of GL(n) via
integral representations, following [5, §4]. You should recall the relevant facts needed from [5, §3],
which should have been discussed already in the talk on local L and ε-pairs.
Mar 28 – No meeting. Jordan Ellenberg speaking at BU colloquium.
Mar 31 – Functoriality (Aditya). The goal here is to explain and state the results of ArthurClozel on automorphic induction and base change for solvable extensions of number fields. References TBD.
Apr 4 – Galois representations (John). The goal is to review and state the earliest results,
going back to Clozel (1991), on the construction of Galois representations attached to automorphic
representations in special cases. References TBD.
Apr 7 – Henniart’s Proof I (TBD). Not set in stone yet. Recall the full statement of the
local Langlands conjecture. Then explain Henniart’s numerical correspondence [10] to reduce the
question to construction an injective map σ 7→ π(σ).
3
Apr 11 – Equality of L and ε-factors a.e implies everywhere (Hudson). Recall the needed
facts about L-functions of global Weil groups representations and what it means for Weil group
representations to be associated to automorphic representations. Prove [12, Corollary 2.4] in as
much detail as times allows. In particular, explain [9, Theorem 4.1] and [9, Proposition 4.5]. Admit
as much as needed to give [12, Theorem 2.4] and the proof of the corollary.
Apr 14 – Henniart’s proof II (David Corwin). Sketch the complete proof of Henniart [12],
assuming the construction of a non-Galois automorphic induction [12, §3] (to be covered in the
next talk). Thus explain the use of Brauer induction and then focus on constructing the injective
map [12, §4] from local Weil group representations to the automorphic side.
Apr 18 – Automorphic induction in the non-Galois setting (Aditya). Explain Harris’
technique to produce automorphic induction of characters to non-Galois extensions following [7].
You should see as well [4, §3.5] and [8, Chapter VII]
Apr 21 – No meeting. Boston Marathon day.
Apr 25 – TBD.
Apr 28 – TBD.
May 2 – TBD.
References
[1] I. N. Bernstein and A. V. Zelevinsky. Induced representations of reductive p-adic groups. I. Ann. Sci. École
Norm. Sup. (4), 10(4):441–472, 1977.
[2] D. Bump. Automorphic forms and representations, volume 55 of Cambridge Studies in Advanced Mathematics.
Cambridge University Press, Cambridge, 1997.
[3] C. J. Bushnell and G. Henniart. The local Langlands conjecture for GL(2), volume 335 of Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin,
2006.
[4] H. Carayol. Preuve de la conjecture de Langlands locale pour GLn : travaux de Harris-Taylor et Henniart.
Astérisque, (266):Exp. No. 857, 4, 191–243, 2000. Séminaire Bourbaki, Vol. 1998/99.
[5] J. W. Cogdell. Analytic theory of L-functions for GLn . In An introduction to the Langlands program (Jerusalem,
2001), pages 197–228. Birkhäuser Boston, Boston, MA, 2003.
[6] P. Deligne. Les constantes des équations fonctionnelles des fonctions L. In Modular functions of one variable, II
(Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 501–597. Lecture Notes in Math., Vol.
349. Springer, Berlin, 1973.
[7] M. Harris. The local Langlands conjecture for GL(n) over a p-adic field, n < p. Invent. Math., 134(1):177–210,
1998.
[8] M. Harris and R. Taylor. The geometry and cohomology of some simple Shimura varieties, volume 151 of Annals
of Mathematics Studies. Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G.
Berkovich.
[9] G. Henniart. On the local Langlands conjecture for GL(n): the cyclic case. Ann. of Math. (2), 123(1):145–203,
1986.
[10] G. Henniart. La conjecture de Langlands locale numérique pour GL(n). Ann. Sci. École Norm. Sup. (4),
21(4):497–544, 1988.
[11] G. Henniart. Caractérisation de la correspondance de Langlands locale par les facteurs de paires. Invent. Math.,
113(2):339–350, 1993.
[12] G. Henniart. Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math.,
139(2):439–455, 2000.
[13] S. S. Kudla. From modular forms to automorphic representations. In An introduction to the Langlands program
(Jerusalem, 2001), pages 133–151. Birkhäuser Boston, Boston, MA, 2003.
[14] P. Scholze. The local Langlands correspondence for GLn over p-adic fields. Invent. Math., 192(3):663–715, 2013.
[15] J. Tate. Number theoretic background. In Automorphic forms, representations and L-functions (Proc. Sympos.
Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3–26.
Amer. Math. Soc., Providence, R.I., 1979.
4
[16] J. T. Tate. Local constants. In Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ.
Durham, Durham, 1975), pages 89–131. Academic Press, London, 1977. Prepared in collaboration with C. J.
Bushnell and M. J. Taylor.
[17] T. Wedhorn. The local Langlands correspondence for GL(n) over p-adic fields. In School on Automorphic Forms
on GL(n), volume 21 of ICTP Lect. Notes, pages 237–320. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.
5