on automorphic l-functionscass/beijing/pdf/ngo-sakellaridis-hue.pdf · 2014. 5. 14. · on...

On automorphicL-functions

Ngô Bảo Châu

Classical zetafunctions andL-functions

Adèles and Tate’sthesis

AutomorphicL-functions

Vinberg’s theory offlat monoids

On automorphic L-functions

Ngô Bảo Châu

University of Chicago &Vietnam Institute for Advanced Study

Ngô Bảo Châu

The Riemann zeta function

I The seriesζ(s) =

∑n≥1

n−s

converges absolutely for 1 and uniformly on 1 + � for every � > 0.

I The function s 7→ ζ(s) is holomorphic on the half-plane 1.

I Over that domain, it has a development as Eulerianproduct

ζ(s) =∏p

(1− p−s)−1.

Ngô Bảo Châu

I The seriesζ(s) =

∑n≥1

n−s

ζ(s) =∏p

(1− p−s)−1.

Ngô Bảo Châu

I The seriesζ(s) =

∑n≥1

n−s

ζ(s) =∏p

(1− p−s)−1.

Ngô Bảo Châu

Meromorphic continuation and functionalequation

I ζ can be continued as a meromorphic function over Cwith only two simple poles at s = 0 and s = 1.

I It satisfies a functional equation ζ∗(1− s) = ζ∗(s)where ζ∗(s) = π−s/2Γ(s/2)ζ(s).

I Zeros and special values of the Riemann zeta functionhave deep arithmetic significance. It is conjectured byRiemann that except the "trivial" zeros at evennegative integers, all others zeros lie on the "critical"line =(s) = 1/2.

I The non-vanishing of ζ on the line =(s) = 1 is a crucialingredient in the proof of prime number theorem.

Ngô Bảo Châu

Dirichlet’s L-function

I For any Dirichlet character χ : (Z/mZ)× → C×, we canform the series

L(s, χ) =∑

(n,m)=1

χ(n)n−s .

I This series converges absolutely on the domain 1. For χ 6= 1, L(s, χ) can be continued asholomorphic function of s ∈ C satisfying a functionalequation relating its values at s and 1− s.

I The non vanishing L(1, χ) 6= 0 is a critical ingredient inthe proof of infiniteness of prime numbers in anarithmetic progression.

Ngô Bảo Châu

L(s, χ) =∑

(n,m)=1

χ(n)n−s .

Ngô Bảo Châu

L(s, χ) =∑

(n,m)=1

χ(n)n−s .

Ngô Bảo Châu

Completions of Q

I The ring of integers Z and the field of rational numbersQ can be endowed with the p-adic topology in whichtwo numbers are closed if their difference is divisible bya high power of p.

I The ring of p-adic integers Zp is the completion of Zwith respect to the p-adic topology. Idem, Qp is thecompletion of Q.

I Natural completions of Q include the field of realnumbers R as well as the fields of p-adic numbers Qpfor different primes p. It is convenient to look at R asthe completion of Q with respect to the infinite primep =∞.

Ngô Bảo Châu

Completions of Q

Ngô Bảo Châu

Completions of Q

Ngô Bảo Châu

Adèles

I An adèle is a sequence (xp) with xp ∈ Qp if p is a finiteprime and x∞ ∈ R if p =∞ such that xp ∈ Zp foralmost all finite primes p.

I The ring A of all adèles is a locally compact topologicalgroup with respect to the addition.

I The product over finite primes Ẑ =∏

p Zp is a compactsubgroup.

I Q is a discrete cocompact subgroup of A.I Q\A is the pro-universal covering of the circle R/Z.I More precisely, Q\A/Ẑ can be identified with R/Z, and

for every integer m, Q\A/mẐ can be identified with thecovering R/mZ of R/Z.

Ngô Bảo Châu

Adèles

Ngô Bảo Châu

Adèles

Ngô Bảo Châu

Adèles

I Q is a discrete cocompact subgroup of A.

I Q\A is the pro-universal covering of the circle R/Z.I More precisely, Q\A/Ẑ can be identified with R/Z, and

Ngô Bảo Châu

Adèles

I Q is a discrete cocompact subgroup of A.I Q\A is the pro-universal covering of the circle R/Z.

I More precisely, Q\A/Ẑ can be identified with R/Z, andfor every integer m, Q\A/mẐ can be identified with thecovering R/mZ of R/Z.

Ngô Bảo Châu

Adèles

Ngô Bảo Châu

Idèles

I An idèle is a sequence (xp) with xp ∈ Q×p if p is a finiteprime and x∞ ∈ R× if p =∞ such that xp ∈ Z×p foralmost all finite primes p i.e. both xp and x

−1p are in Zp

I The group A× of all idèles is a locally compacttopological group.

I The product over finite primes Z× =∏

p Z×p is acompact subgroup.

I There is a decomposition A× = Q× × R×+ × Ẑ, eachidele can be written uniquely as x = αtu withα ∈ Q×, t ∈ R×+, u ∈ Ẑ×.

I Every quasi-character ω : A× → C×, trivial on Q×, canbe written as ω(αtu) = tsχ(u) where s ∈ C andχ : Z× → C× is a i.e. factoring through a Dirichletcharacter (Z/mZ)× → C×.

I The set of quasi-character can be endowed withcomplex analytic structure.

Ngô Bảo Châu

Idèles

−1p are in Zp

Ngô Bảo Châu

Idèles

−1p are in Zp

Ngô Bảo Châu

Idèles

−1p are in Zp

Ngô Bảo Châu

Idèles

−1p are in Zp

Ngô Bảo Châu

Idèles

−1p are in Zp

Ngô Bảo Châu

The Mellin transform

I If φ ∈ C∞(R×+) has rapid decay at 0 and ∞, its Mellintransform

φ̃(s) =

∫R×+

φ(t)tsd×t

defines a holomorphic function of variable s.

I If φ has rapid decay only at ∞, the above integralconverges only for 0. Nevertheless, the functions 7→ φ̃(s) can be continued as meromorphic function ofs ∈ C.

I If φ = e−t then φ̃(s) is the Gamma function

Γ(s) =

∫R×+

e−ttsd×t

that has simple poles at negative integers.

Ngô Bảo Châu

φ̃(s) =

∫R×+

φ(t)tsd×t

Γ(s) =

∫R×+

e−ttsd×t

Ngô Bảo Châu

φ̃(s) =

∫R×+

φ(t)tsd×t

Γ(s) =

∫R×+

e−ttsd×t

Ngô Bảo Châu

p-adic Mellin transform

I We normalize the Haar measure d×x on Q×p so that Z×phas volume one.

I For every locally constant with compact supportφ ∈ C∞c (Q×p ), the integral

φ̃(ω) =

∫Q×p

φ(x)ω(x)d×x

for every quasi-character ω : Q×p → C×.I The function ω 7→ φ̃(ω) is analytic.I For instant, if φ = 1pZ×p , then for unramified

quasi-character ω : Q×p → C× (i.e. trivial on Z×p ),φ̃(ω) = ω(p).

Ngô Bảo Châu

φ̃(ω) =

∫Q×p

φ(x)ω(x)d×x

for every quasi-character ω : Q×p → C×.

I The function ω 7→ φ̃(ω) is analytic.I For instant, if φ = 1pZ×p , then for unramified

Ngô Bảo Châu

φ̃(ω) =

∫Q×p

φ(x)ω(x)d×x

for every quasi-character ω : Q×p → C×.I The function ω 7→ φ̃(ω) is analytic.

I For instant, if φ = 1pZ×p , then for unramified

Ngô Bảo Châu

φ̃(ω) =

∫Q×p

φ(x)ω(x)d×x

for every quasi-character ω : Q×p → C×.I The function ω 7→ φ̃(ω) is analytic.I For instant, if φ = 1pZ×p , then for unramified

Ngô Bảo Châu

I Let us allow the test function φ to have compactsupport in Qp instead of Q×p . Then the integral∫Q×p φ(x)ω(x)d

×x may not converge.

I If ω is unitary, the integral

φ̃(s, ω) =

∫Q×p

φ(x)ω(x)|x |sd×x

converge for 0. Moreover it can bemeromorphically continued to all ω.

I For instant, if φ = 1Zp , and for unramified character ω,we have

φ̃(s, ω) =∞∑i=0

ω(p)ip−is = (1− ω(p)p−s)−1

Ngô Bảo Châu

φ̃(s, ω) =

∫Q×p

φ(x)ω(x)|x |sd×x

φ̃(s, ω) =∞∑i=0

ω(p)ip−is = (1− ω(p)p−s)−1

Ngô Bảo Châu

φ̃(s, ω) =

∫Q×p

φ(x)ω(x)|x |sd×x

φ̃(s, ω) =∞∑i=0

ω(p)ip−is = (1− ω(p)p−s)−1

Ngô Bảo Châu

Adelic Mellin transform

I We consider functions on A of the form φ = ⊗pφp withφ∞ ∈ S(R) and φp ∈ C∞c (Qp), φp = 1Zp for almost allp.

I For every character ω : A× → C×, the integral

φ̃(s, ω) =

∫A×

φ(x)ω(x)|x |sd×x

converges for 1.I On this region, the above integral defines a holomorphic

function with development as Euler product

φ̃(s, ω) =∏p

φ̃p(s, ωp)

with φ̃p(s, ωp) = (1− ωp(p)p−s)−1 for almost all p.

Ngô Bảo Châu

φ̃(s, ω) =

∫A×

φ(x)ω(x)|x |sd×x

converges for 1.

I On this region, the above integral defines a holomorphicfunction with development as Euler product

φ̃(s, ω) =∏p

φ̃p(s, ωp)

Ngô Bảo Châu

φ̃(s, ω) =

∫A×

φ(x)ω(x)|x |sd×x

converges for 1.I On this region, the above integral defines a holomorphic

function with development as Euler product

φ̃(s, ω) =∏p

φ̃p(s, ωp)

Ngô Bảo Châu

Characters of the idèle class group

I If ω is a character of the idèle class groupω : Q×\A× → C×, the function φ̃(s, ω) can bemeromorphically continued to s ∈ C.

I It is closely related to the Dirichlet L-function and theRiemann zeta function.

I It satisfies the functional equation

φ̃(s, ω) = ˜̂φ(1− s, ω−1)

where φ̂ is the Fourier transform of φ.

I The above formula derives from the Poisson summationformula.

Ngô Bảo Châu

φ̃(s, ω) = ˜̂φ(1− s, ω−1)

Ngô Bảo Châu

φ̃(s, ω) = ˜̂φ(1− s, ω−1)

Ngô Bảo Châu

φ̃(s, ω) = ˜̂φ(1− s, ω−1)

Ngô Bảo Châu

Tate’s thesis: upshot

I Tate’s proof is not fundamentally different fromRiemann’s proof of the functional equation satisfied byζ and L-functions. Both relied ultimately on thePoisson summation formula.

I However, the adelic language allows us to express thevalue of L-functions as the trace of certain operator onone-dimensional representation of idèle class group.

I The operator 1Zp is the shadow of of the affine line A1acted on by the multiplicative group Gm.

I The functional equation relies on the additive structureof the affine line: the Fourier transform and the Poissonsummation formula.

Ngô Bảo Châu

Automorphic representations

I Automorphic representation is non-abeliangeneralization of characters of idèle class group.

I Let G be a reductive group defined over Q. Anautomorphic representation of G is an irreducibleunitary representation π of G (A) occurring inL2(G (Q)\G (A)).

I π decomposes as tensor product π = ⊗pπp for finiteand infinite primes p, πp is an irreducible unitaryrepresentation of G (Qp).

Ngô Bảo Châu

Unramified representation at an unramified prime

I A prime p is said to be unramified prime for G if G hasa reductive model over Zp.

I The space of fixed vectors of G (Zp) in πp is either zeroor one dimensional.

I A representation πp of G (Qp) is unramified if G (Zp)has a non-zero fixed vector.

Ngô Bảo Châu

Classification of unramified representation andthe Langlands dual group

I Unramified representations of G (Qp) has a niceclassification.

I Assume from now on that G is semi-simple and split. Itis determined by a by a combinatorial datum consistingof the group of characters, of cocharacters and the setof roots and coroots.

I By inverting the role of characters and cocharacters,roots and coroots, we define a semi-simple group Ĝover C.

I There is natural 1-1 correspondence between unramifiedrepresentations of G (Qp) and semi-simple conjugacyclasses of Ĝ (C)

πv ↔ α(πv ) ∈ Ĝ/ ∼

Ngô Bảo Châu

πv ↔ α(πv ) ∈ Ĝ/ ∼

Ngô Bảo Châu

πv ↔ α(πv ) ∈ Ĝ/ ∼

Ngô Bảo Châu

πv ↔ α(πv ) ∈ Ĝ/ ∼

Ngô Bảo Châu

Langlands’ automorphic L-function

I Let r be a finite dimensional representation of Ĝ .

I If πv is an unramified representation of G (Qp), wedefine the local L-factor as

Lv (s, π, r) = det(1− r(α(πv ))p−s)−1.

I Local factors at ramified places and infinite places canalso be properly defined.

I The automorphic L-function is defined as an infiniteproduct

L(s, π, r) =∏p

Lv (s, π, r)

which is absolutely convergent > 0.

Ngô Bảo Châu

Lv (s, π, r) = det(1− r(α(πv ))p−s)−1.

L(s, π, r) =∏p

Lv (s, π, r)

Ngô Bảo Châu

Lv (s, π, r) = det(1− r(α(πv ))p−s)−1.

L(s, π, r) =∏p

Lv (s, π, r)

Ngô Bảo Châu

Lv (s, π, r) = det(1− r(α(πv ))p−s)−1.

L(s, π, r) =∏p

Lv (s, π, r)

Ngô Bảo Châu

Langlands’ conjecture

I It is conjectured that L(s, π, r) has meromorphiccontinuation and functional equation.

I This statement is surprisingly deep. It contains a lot ofarithmetic information about the arithmetic ofautomorphic representation.

I It is known in some cases where r is closed to thestandard representations by various method.

I There have been many attempts to generalize themethod of Tate to general automorphic L-function.Tamagawa, Godement, Jacquet succeeded completely inthe case of principal L-function.

Ngô Bảo Châu

Tamagawa-Godement-Jacquet principalL-function

I G = GLn, Ĝ = GLn, r is the tautologicalrepresentation.

I Let M = gln the space of n × n matrices equipped withaction of G by left and right translation.

I The trace of 1M(Zp) on the unramified representationπv ⊗ | det |s is the principal local L-factor

Lv (s, π) = (1− α(πv )p−s)−1.

I The additive structure on M permits to define theFourier transform and obtain the Poisson summationformula. Using this Tamagawa, Godement, Jacquetwere able to meromorphically continue the principalL-function and prove its functional equation.

Ngô Bảo Châu

Lv (s, π) = (1− α(πv )p−s)−1.

Ngô Bảo Châu

Lv (s, π) = (1− α(πv )p−s)−1.

Ngô Bảo Châu

Lv (s, π) = (1− α(πv )p−s)−1.

Ngô Bảo Châu

Search for a general answer

I We will give a conjectural description of M for generalG and r .

I In general, M has no additive structure, and is evensingular.

I The generalization of 1Zp has to take into account toincorporate singularities of M.

Ngô Bảo Châu

Monoids for a semi-simple group

I Let us fix a semi-simple (simply connected) group G .

I We will consider all reductive groups G ′ whose derivedG given together with an affine open embeddingG ′ ↪→ M ′ such that the actions of G ′ by left and righttranslations can be extended to M ′.

I Since M ′ is affine, this is equivalent to saying that thegroup law og G ′ can be extended as a monoid law onM ′.

I Typical example: G = SLn, G ′ = GLn and M = gln.

Ngô Bảo Châu

Flat monoids

I Let A′ be the invariant quotient M ′//G × G . This is anaffine embedding of the torus G ′/G . Following Vinberg,we call it the abelianization of M ′.

I M ′ is said to be a flat monoid G if the map M ′ → A′ isflat with geometric reduced fibers.

I According to Vinberg, the category of flat monoids of Ghas a pleasant description.

Ngô Bảo Châu

Flat monoids

Ngô Bảo Châu

Flat monoids

Ngô Bảo Châu

Universal flat monoidI Let G+ = (G × T )/Z where T is a maximal torus of G

and Z is the center of G , acting diagonally on G × T .

I Let α1, . . . , αr denote the simple roots attached to thechoice of a Borel subgroup containing T . Letω1, . . . , ωr denote the fundamental weights.

I We have one-dimensional αi : T → Gm of the maximaltorus and the fundamental representationsρi : G → GL(Vi ) of highest weight ωi .

I There is only one reasonable way to combine the αi andρi into a faithful representation of G

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

I Let M+ be the closure of G+ in Ar × End(Vi ).I The invariant quotient of M+ by the double action of G

is A+ = Ar which is a an affine embedding of the torusG+/G = T ad .

Ngô Bảo Châu

and Z is the center of G , acting diagonally on G × T .I Let α1, . . . , αr denote the simple roots attached to the

choice of a Borel subgroup containing T . Letω1, . . . , ωr denote the fundamental weights.

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

Ngô Bảo Châu

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

Ngô Bảo Châu

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

Ngô Bảo Châu

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

I Let M+ be the closure of G+ in Ar × End(Vi ).

I The invariant quotient of M+ by the double action of Gis A+ = Ar which is a an affine embedding of the torusG+/G = T ad .

Ngô Bảo Châu

+

(α+, ρ+) : (T × G )/Z → Grm ×r∏

i=1

GL(Vi ).

Ngô Bảo Châu

Universal flat monoid

I The invariant quotient of M+ by the double action of Gis A+ = Ar which is a an affine embedding of the torusG+/G = Grm.

I M+ is the universal flat monoid in the sense that everyother flat monoid M ′ of G can be obtained by formingfibered product over the abelinization A′ → A+.

Ngô Bảo Châu

Universal flat monoid

I The invariant quotient of M+ by the double action of Gis A+ = Ar which is a an affine embedding of the torusG+/G = Grm.

I M+ is the universal flat monoid in the sense that everyother flat monoid M ′ of G can be obtained by formingfibered product over the abelinization A′ → A+.

Ngô Bảo Châu

1-dimensional centered flat monoid

I We will restrict ourselves to the flat monoids M ′ whoseabelianization is A with action of G ′/G = Gm.

I By universal property, they are given by homomorphismλ : Gm → T ad that can be extended as a map A→ A+.

I These λ are exactly the highest weight in finitedimensional representation of Ĝ sc .

I Let the corresponding monoid Mλ = M+ ×A+ A shouldplay the role of the affine line in Tate’s thesis.

I The group Gλ has a determinant map det : Gλ → Gmwhose kernel is G .

Ngô Bảo Châu

The formal arc spaceI Let us consider the arc space of Mλ. This is an infinite

dimensional scheme LMλ such that

LMλ(Fp) = Mλ(Fp[[t]]).

I According to Grinberg-Kazhdan and Drinfeld,singularities of LMλ can be approximated by finitedimensional singularities i.e. for every point x ∈ LMλ,there is a scheme Y of finite type, y ∈ Y such thatthere is an isomorphism of formal schemes

(LMλ)x = Yy × D∞

where D is the formal disc.I It makes sense to talk about the intersection complex of

the formal arc space LMλ. The trace of Frobenius onstalks of LMλ defines a function

mλ : Mλ(Fp[[t]])→ Q` ∼ C.

Ngô Bảo Châu

LMλ(Fp) = Mλ(Fp[[t]]).I According to Grinberg-Kazhdan and Drinfeld,

singularities of LMλ can be approximated by finitedimensional singularities i.e. for every point x ∈ LMλ,there is a scheme Y of finite type, y ∈ Y such thatthere is an isomorphism of formal schemes

where D is the formal disc.

I It makes sense to talk about the intersection complex ofthe formal arc space LMλ. The trace of Frobenius onstalks of LMλ defines a function

mλ : Mλ(Fp[[t]])→ Q` ∼ C.

Ngô Bảo Châu

LMλ(Fp) = Mλ(Fp[[t]]).I According to Grinberg-Kazhdan and Drinfeld,

singularities of LMλ can be approximated by finitedimensional singularities i.e. for every point x ∈ LMλ,there is a scheme Y of finite type, y ∈ Y such thatthere is an isomorphism of formal schemes

where D is the formal disc.I It makes sense to talk about the intersection complex of

the formal arc space LMλ. The trace of Frobenius onstalks of LMλ defines a function

mλ : Mλ(Fp[[t]])→ Q` ∼ C.

Ngô Bảo Châu

Conjecture on local test function

I Following ideas of Sakellaridis, it is tempting toconjecture that for every unramified representation πpof Gλ(Fp((t))), the trace of mλ on πv ⊗ | det |s is thelocal L-factor Lp(s, πp, rλ).

I The function mλ can be transferred to Gλ(Qp) and the

similar spectral statement should be true for Gλ(Qp).

Ngô Bảo Châu

Conjecture on local test function

I Following ideas of Sakellaridis, it is tempting toconjecture that for every unramified representation πpof Gλ(Fp((t))), the trace of mλ on πv ⊗ | det |s is thelocal L-factor Lp(s, πp, rλ).

I The function mλ can be transferred to Gλ(Qp) and the

similar spectral statement should be true for Gλ(Qp).

Ngô Bảo Châu

Evidences

I It works in the case of Tate, and Godement, Jacquet.

I There is a closely related, but different geometricinterpretation of the local test function as sheaf on theformal arc space of Mλ that can actually be proved.

Ngô Bảo Châu

Evidences

I It works in the case of Tate, and Godement, Jacquet.

I There is a closely related, but different geometricinterpretation of the local test function as sheaf on theformal arc space of Mλ that can actually be proved.

Ngô Bảo Châu

Problems in increasing order of difficulty

I What is the local construction on the archimedeanplace?

I Is there an integral transform on Mλ(Qp) that plays thesame role as the Fourier transform on the line?

I Is there a Poisson summation formula for Mλ(Q) inMλ(A)?

Ngô Bảo Châu

Classical zeta functions and L-functionsAdèles and Tate's thesisAutomorphic L-functionsVinberg's theory of flat monoids

on automorphic l-functionscass/beijing/pdf/ngo-sakellaridis-hue.pdf · 2014. 5. 14. · on...

Documents