On the Langlands Program

John Rognes

Colloquium talk, May 4th 2018

The Norwegian Academy of Science and Letters has decidedto award the Abel Prize for 2018 to Robert P. Langlands of the

Institute for Advanced Study, Princeton, USA

for his visionary program relating representationtheory to number theory.

Robert P. Langlands (1967)

The Langlands Conjectures (ca. 1967)

Conjecture (Reciprocity)

To each Galois representation there corresponds anautomorphic representation with the same L-function.

Conjecture (Functoriality)

To each homomorphism LG(C)→ LG ′(C) and eachautomorphic G-representation there corresponds anautomorphic G ′-representation with the same L-function.

Quadratic Reciprocity (1801)

I p, ` odd primes

I `∗ = (−1)`−1

2 `

Theorem (Gauss)

`∗ is a quadratic residue modulo p if and only if p is a quadraticresidue modulo `.

I For fixed ` and varying p, the solvability of

`∗ ≡ x2 mod p

only depends on the residue class of p mod `.

Frobenius Automorphism

I ζ` = e2πi/` root of unityI Number fields

Q ⊂ Q(√`∗) ⊂ Q(ζ`)

I Frobenius automorphism: Frobp ∈ Gal(Q(ζ`)/Q)

Frobp(ζ`) = ζp`

I x2 ≡ p mod ` solvable ⇐⇒ Frobp fixes√`∗.

Abelian Number Fields

I Q ⊂ Q̄ algebraic closureI GalQ = Gal(Q̄/Q) absolute Galois group

Theorem (Kronecker, Weber, Hilbert)

For each homomorphism ρ : GalQ → C× there exists a Dirichletcharacter χ : (Z/m)× → C× such that

ρ(Frobp) = χ(p)

for all p - m.

I Calculates GalQ modulo commutators.

The Nonabelian Case

I A homomorphism ρ : GalQ → GLn(C) is a rank n complexrepresentation.

I Nonabelian target for n ≥ 2.I ρ(Frobp) ∈ GLn(C) is defined up to conjugacy.I Well-defined characteristic polynomial

P(t) = det(tI − ρ(Frobp))

I Modified form

Q(t) = det(I − ρ(Frobp)t)−1

Question

What replaces Dirichlet characters for n ≥ 2?

Robert P. Langlands (1971)

There are at least three different problems with whichone is confronted in the study of L-functions: theanalytic continuation and functional equation; thelocation of the zeroes; and in some cases, thedetermination of the values at special points.

The first may be the easiest. It is certainly the onlyone with which I have been closely involved.

Langlands (Helsinki ICM 1978)

The Riemann Zeta Function

ζ(s) =∞∑

n=1

1ns = 1 +

12s +

13s +

14s +

15s + . . .

for Re(s) > 1.

I Euler product

ζ(s) =∏

p

11− p−s

I Analytic continuation, simple pole at s = 1.I Real factor

ξ(s) = π−s/2Γ(s/2) · ζ(s)

I Functional equation

ξ(1− s) = ξ(s)

Two views of the zeta function (Derbyshire)

The Prime Number Theorem (1896)

Let π(x) be the number of primes p ≤ x .

Theorem (Hadamard, de la Vallée-Poussin)

π(x) ∼ Li(x) =

∫ x

2

dtlog t

(∼ xlog x

)

Proof (sketch).

ζ(s) 6= 0 for Re(s) = 1.

The Riemann Hypothesis (1859)

Conjecture (RH)

The nontrivial zeros of ζ(s) lie on the critical line Re(s) = 1/2.

Theorem

If RH is true, then

π(x) = Li(x) + O(x1/2+ε)

for each ε > 0.

Question

What is the natural context for the zeta function?

Robert P. Langlands (2016)

There are two kinds of L-functions, and they will bedescribed below: motivic L-functions which generalizethe Artin L-functions and are defined purelyarithmetically, and automorphic L-functions, defined bydata which are largely transcendental.

Within the automorphic L-functions a special class canbe singled out, the class of standard L-functions,which generalize the Hecke L-functions and for whichthe analytic continuation and functional equation canbe proved directly.

Langlands (Helsinki ICM 1978)

Artin L-functions (1923)

I GalQ = Gal(Q̄/Q)

I Galois representation ρ : GalQ → GLn(C).I Modified characteristic polynomial

Lp(ρ, s) = det(I − ρ(Frobp)p−s)−1

I Euler productL(ρ, s) =

∏p

Lp(ρ, s)

converges for Re(s) sufficiently large.

Artin Conjecture

I If ρ is the trivial rank 1 representation, then

L(ρ, s) = ζ(s)

I Brauer: The Artin L-function L(ρ, s) admits a meromorphiccontinuation.

I With real and ramified factors it satisfies a functionalequation.

Conjecture (Artin)

If ρ is nontrivial, then L(ρ, s) is entire (has no poles).

Modular Forms

A modular form of weight k is a holomorphic function

f : H = {z ∈ C : Im(z) > 0} −→ C

such thatf (

az + bcz + d

) = (cz + d)k f (z)

for all γ =

[a bc d

]∈ SL2(Z).

Here H ∼= SL2(R)/SO2 is a symmetric space.

Fundamental regions for SL2(Z) acting on H (Womack)

Hecke Theory, I

The following are equivalent:

(a) The holomorphic function

f (z) =∑

n

an e2πinz

is a modular form.

(b) The Dirichlet series ∑n

an

ns

satisfies a functional equation.

Modular Representations

Gelfand-Fomin (1952): A modular form f of weight k defines asmooth function

φf : SL2(Z)\SL2(R)→ C

by

φf (g) = (ci + d)−k f (ai + bci + d

)

for g =

[a bc d

]∈ SL2(R). Let

πf = 〈φf 〉 ⊂ L2(SL2(Z)\SL2(R))

be the SL2(R)-representation generated by φf .

Hecke Theory, II

The following are equivalent:

(a) The SL2(R)-representation πf = 〈φf 〉 is irreducible.

(b) The modular form

f (z) =∑

n

ane2πinz

is an eigenfunction for the Hecke algebra.

(c) The Dirichlet series ∑n

an

ns

has an Euler product expansion.

Completions of Q

I Let | |v be a norm on Q, so that |x − y |v defines a metric.Let Q ⊂ Qv be the associated completion.

I v =∞, |x |∞ = |x |.Field of real numbers: Q∞ = R.0.9 + 0.09 + 0.009 + · · · = 1 in R.

I v = p any prime,|x |p = 1/pn

for x = apn/b with p - ab.Field of p-adic numbers: Qp.Ring of p-adic integers: Zp = {x ∈ Qp : |x |p ≤ 1}.1 + p + p2 + p3 + · · · = 1/(1− p) in Zp ⊂ Qp.

The Adèle Ring

I Diagonal embedding

Q ⊂∏

v

Qv = R×∏

p

Qp

I The adèle ringA ⊂

∏v

Qv

is locally compact.I A contains R and Qp as subspaces.I A contains Q as a discrete subring.

Automorphic Representations

I GLn(A) contains GLn(Q) as a discrete subgroup.I GLn(A) acts on L2(GLn(Q)\GLn(A)) by right translation:

(h · f )(g) = f (gh)

for f : GLn(Q)\GLn(A)→ C and g,h ∈ GLn(A).I An automorphic representation π is an irreducible

GLn(A)-representation

π ⊂ L2(GLn(Q)\GLn(A))

contained in the regular representation.

Größencharakteren (n = 1)

I GL1(Q)\GL1(A) ∼= R>0 ×∏

p Z×p .

I Each automorphic GL1-representation π is a character

GL1(Q)\GL1(A) −→ C×

of finite order.I It factors uniquely through a Dirichlet character

χ : (Z/m)× → C×

and vice versa.

Parabolic Induction, I

I Parabolic subgroup

P ={

p =

g1 ∗ . . . ∗0 g2 . . . ∗...

.... . .

...0 0 . . . gr

: gi ∈ GLni

}⊂ GLn

I Given automorphic GLni -representations πi , getP-representation

p 7→ π1(g1)⊗ · · · ⊗ πr (gr )

by restriction along P → GLn1 × · · · ×GLnr .I Extend this P-representation to a GLn-representation π̃ by

induction along P ⊂ GLn.

Parabolic Induction, II

Theorem (Langlands)

The automorphic representations of GLn(A) are precisely theirreducible constituents π of the representations

π̃ = IndGLn ResP(π1 ⊗ · · · ⊗ πr ) ,

where π1, . . . , πr are cuspidal.

I Proof depends on Langlands’ theory (1965) of Eisensteinseries for GLn, started by Selberg (1962) for SL2.

I Gelfand (1962) clarified role of cusp forms for rank r ≥ 2.

Local Components

I Any automorphic GLn-representation π factors as

π =⊗

v

πv

where each πv is an irreducible GLn(Qv )-representation.I Almost all πp are constituents of

π̃p = IndGLn ResP(χz)

for some z = (z1, . . . , zn) ∈ Cn.I Here r = n, each ni = 1, and

χz(g1, . . . ,gn).

= |g1|z1p · · · · · |gn|zn

p .

Satake Isomorphism

I Almost all πp are unramified, so that

dimπGLn(Zp)p = 1

I Hecke algebra

Hp = Cc(GLn(Zp)\GLn(Qp)/GLn(Zp))

acts naturally on πGLn(Zp)p , multiplying by χp : Hp → C.

I Satake/Langlands (1970): Hp “is” the representation ring ofthe complex Lie group

LGLn(C) ,

called the L-group of GLn.

Langlands Parameter

I Dually, χp “is” evaluation at a semisimple conjugacy class

σ(πp) ∈ LGLn(C)/∼

I For πp in π̃p = IndGLn ResP(χz), with z = (z1, . . . , zn)

σ(πp) 3

p−z1 . . . 0...

. . ....

0 . . . p−zn

Automorphic L-Functions

I Local L-function

Lp(s, π) = det(I − σ(πp)p−s)−1

I Also real and ramified cases.I (Standard) automorphic L-function

L(s, π) =∏

v

Lv (s, π)

Theorem (Godement–Jacquet (1972))

Let π be an automorphic GLn-representation. Then L(s, π) hasanalytic continuation to a meromorphic function of s ∈ C, whichsatisfies a functional equation.

Reciprocity Conjecture

Conjecture (Langlands)

For each rank n Galois representation

ρ : GalQ −→ LGLn(C)

there exists an automorphic GLn-representation π such that

ρ(Frobp) = σ(πp)

for almost all p.

This will determine π uniquely, and

L(ρ, s) = L(s, π) .

Local Langlands

I So far we only discussed unramified πv for v = p.I To parametrize other irreducible GLn(Qv )-representations,

more information is needed.I The Weil–Deligne group LQv is a variant of the absolute

Galois group GalQv .

Conjecture (Langlands)

Irreducible GLn(Qv )-representations correspond to conjugacyclasses of homomorphisms

φv : LQv −→ LGLn(C) ,

called Langlands parameters.

Robert P. Langlands (2013)

For the other L-functions the analytic continuation isnot so easily effected. However all evidence indicatesthat there are fewer L-functions than the definitionssuggest, and that every L-function, motivic orautomorphic, is equal to a standard L-function.

Such equalities are often deep, and are calledreciprocity laws, for historical reasons. Once areciprocity law can be proved for an L-function,analytic continuation follows, and so, for those whobelieve in the validity of the reciprocity laws, they andnot analytic continuation are the focus of attention, butvery few such laws have been established.

Langlands (Helsinki ICM 1978)

Reductive Groups

I Harish-Chandra: What can be done for GLn should bedone for each reductive group G.

I Each algebraic representation of a reductive group is adirect sum of irreducible representations.

I An automorphic representation of G is an irreducibleG(A)-representation

π ⊂ L2(G(Q)\G(A))

contained in the regular representation.

The L-Group

I The Hecke algebra for (G,K ) “is” the representation ring ofa complex Lie group

LG(C)

called the L-group, or Langlands dual, of G.I The maximal torus of LG is dual to that of G.I Automorphic representations of G have Langlands

parameters σ(πp) and φv in LG(C).

Functoriality Conjecture

Conjecture (Langlands)

For each homomorphism

h : LG −→ LG ′

and automorphic representation π of G there exists anautomorphic representation π′ of G ′ such that

h(σ(πp)) = σ(π′p)

for almost all p.

This will determine π′ uniquely, and

L(s, π′, r) = L(s, π, rh)

for each finite-dimensional representation r of LG ′.

The Rosetta Stone (196 BC)

Global Fields

The three columns of the Rosetta stone:

I Number field: finite extension F ⊃ Q.I Function field: finite extension E ⊃ Fp(t).I Riemann surface: finite cover X → CP1.

Weil: What can be done for number fields should also be donefor function fields and Riemann surfaces.

Local Fields

The local Langlands conjecture for a reductive group G over alocal field Fv has been proved:

I For GL1 by local class field theory.I Over Fv = R or C by Langlands (1973).I For GL2 by Jacquet–Langlands (1970) and Kutzko (1980).I For GLn with char(Fv ) = p by Laumon–Rapoport–Stuhler

(1993).I For GLn with char(Fv ) = 0 by Harris–Taylor (2001),

Henniart (2000) and Scholze (2013).I For general G, ongoing work by Fargues, Scholze.

Number Fields

The Langlands reciprocity conjecture for a reductive group Gover a number field F has been proved:

I For GL1 by global class field theory.I For tori by Langlands (1968).I Partial results for GL2 by Langlands (1980), Tunnel (1981),

Wiles (1995) and Breuil–Conrad–Diamond–Taylor (2001).

Artin conjecture and Riemann hypothesis open (!)

Function Fields over Curves

The Langlands reciprocity conjecture for a reductive group Gover a function field E has been proved:

I For GL1 by global class field theory.I For GL2 by Drinfeld (1974), introducing shtukas.I For GLn by Laurent Lafforgue (1998).I For general G, automorphic to Galois direction, by Vincent

Lafforgue (2014).

Artin conjecture and Riemann hypothesis proved by Weil.

Geometric Langlands

I Translation from curves over finite fields to curves over C,using ideas of Deligne, Drinfeld, Laumon and Beilinson.

I Seek equivalence betweenI a category of D-modules on BunG(X );I a category of quasi-coherent sheaves on LocSysLG(X ).

More precisely, these should be∞-categories.I Witten: Langlands duality G↔ LG is parallel to S-duality in

supersymmetric gauge theories.

Robert P. Langlands (2015)