Functoriality and Beyond Endoscopy · 2018-11-17 · General case, by comparing stable trace...

$: Functoriality and Beyond Endoscopy · 2018-11-17 · General case, by comparing stable trace formula for Sp(2n) to twisted trace formula for GL(2n + 1) \The endoscopic classi cation$
Functoriality and Beyond Endoscopy

S. Ali Altug

Abel Conference

November 14, 2018

S. Ali Altug Beyond Endoscopy November 14, 2018 1 / 35

Outline

1 Functoriality and Beyond Endoscopy

2 Brief History and results

3 Further directions


Principle of Functoriality

Let G ,H be reductive groups over a global field (G quasi-split), and

LHψ−−−→ LG

be a (nice) homomorphism.

Functoriality Conjecture (Langlands, ’67)

There is a transfer of (packets of) automorphic representations of H tothose of G ,

πHΨ−−−→ πG ,

such that for every finite dimensional representation ρ : LG → GL(V ),

L(s, πH , ρ ◦ ψ) = L(s, πG , ρ)

(and many other compatibilities...).


Several implications

Artin conjecture (holomorphy of Artin L-functions)

Ramanujan and Selberg conjectures in their general form for GL(N)

Sato-Tate and its generalizations

Virtual Haken conjecture for arithmetic hyperbolic 3-manifolds


An exampleExample

H = Sp(2n). The problem is to show functorial transfer in the case ofidentity map. i.e.

LH = SO(2n + 1,C) ↪→ GL(2n + 1,C)

General case, by comparing stable trace formula for Sp(2n) to twistedtrace formula for GL(2n + 1)“The endoscopic classification of representations: orthogonal andsymplectic groups” (Arthur, 2013)

Globally generic representations, by the converse theorem.“Functoriality for the classical groups” (CKPSS, 2004)

General case by doubling method. “Doubling constructions: local andglobal theory, with an application to global functoriality fornon-generic cuspidal representations” (Cai, Friedberg, Kaplan + D.Ginzburg 2018)


General methods

Eisenstein series and the Langlands-Shahidi method

Converse theorems

Theta correspondence

Trace formulae


Beyond Endoscopy (BE)

“Beyond Endoscopy” refers to a proposal of Langlands to tackle thefunctoriality conjectures in full generality.

ENDOSCOPY AND BEYOND†

I begin with a description of some of the basic general problems in the theory of automorphic

forms. They are already familiar to the participants in this seminar.

Functoriality and Related Matters

The following statement is a rough formulation of the problem of functoriality.

(I) If H and G are two reductive groups over the global field F and the group G

is quasi-split then to each homomorphism

φ : LH −→ LG

there is associated a transfer of automorphic representations of H to automorphic

representations of G.

The word transfer anticipates the complete solution of the problem; it has not been generally accepted,

and the word lift is frequently used instead. Whether those who are fond of this terminology would

want to use it in all cases is not clear. The transfer associated to the determinant, which maps the

L-group ofGL(n) to theL-group ofGL(1), takes a representation π to its central character, an unlikely

lift. At all events, transfer would have the property that

L(s, π, r ◦ φ) = L(s,Π, r),

if Π is the transfer of π. The formulation (I) does not take difficulties associated with L-packets and

endoscopy into account.

A second problem that arose some time after functoriality is that of associating to an automorphic

representation π, now on the groupG, a subgroupHπ of LG that would only be defined up to conjugacy

but would have the following property.

†Notes for a lecture delivered by R. Langlands, IAS, March 30, 2000. I have left these notes as they

were when distributed at the lecture. They are not free of errors and false starts.

BEYOND ENDOSCOPYl

R. P. Langlands2

Ya tutarsa – Nasreddin Hoca

Introduction3

0.1 Functoriality and related matters. The notion of L-group and the principle of functoriality

appeared in [L] and were explained at more length in [Cor] and elsewhere. The principle of

functoriality, which is now widely believed but is very far from being established in general,

can be roughly stated as follows.

(I) If H and G are two reductive groups over the global field F and the group G is quasi-split

then to each homomorphism

φ : LH −→ LG

there is associated a transfer of automorphic representations of H to automorphic repre-

sentations of G.

A second problem that arose some time after functoriality is that of associating to an

automorphic representation π, now on the group G, an algebraic subgroup λHπ of LG that

l Dedicated to Joseph Shalika on the occasion of his sixtieth birthday.2 I would like to thank James Arthur, who once again guided me through the subtleties

of weighted orbital integrals, Erez Lapid and Peter Sarnak for useful conversations related to

the material of this paper and Werner Hoffmann for his comments on [H] and on Appendix C

and D.3 There is available at http://SunSITE.UBC.CA/DigitalMathArchive/Langlands the

text of a lecture Endoscopy and beyond that can also serve as an introduction to this paper. Ithas the advantage of being informal, but there are misprints and some suggestions towardsthe end are red herrings. The present paper may well turn out to have the same defects!

It was first formulated in a set of lectures titled “Endoscopy andBeyond” at the IAS (∼ 2000). It then evolved into “BeyondEndoscopy” (∼ 2004).


“Ya Tutarsa!”

Nasreddin Hodja (∼ 13. century A.D.): A Seljuq satirical, considereda populist phiolsopher and a wise man. (Wikipedia)

The story is about Nasreddin Hodjastirring yoghurt into a lake to turn the wholelake into yoghurt, and the townspeople mockinghis eccentric try. “Ya tutarsa!” (“What if itworks!”) is his response to the townsmen.


Overview of the strategyOne expects automorphic representations π of G to be classified by(nice) maps, i.e. Langlands parameters

L φπ−−−−→ LG

from (the automorphic Galois group, or the Langlands group) L toLG . (The existence of L is expected to follow from functoriality...)

The idea underlying BE is to group together the automorphicrepresentations π of G with respect to the Zariski closure of theimage of their Langlands parameters Im(φπ) ⊂ LG .Ignoring all thesubtleties, the idea is to give a “stratification” which looks like

⊕

(LH,σ)/∼ψ:LH→LG

⊕

πIm(φπ)∼ψ(LH)

m(π)πG

The aim of BE is to establish a candidate stratification without theparameters φπ, and to compare it for different groups.


Functorial transfers and poles of automorphicL-functions

To establish such a decomposition one needs a gadget that detectsfunctorial transfers.

Poles and functorial transfers

Let S be a finite set of places including all the archimedean places and letπ be a cuspidal automorphic representation of G unramified outside of S .

If there is an irreducible finite dimensional ρ : LG → GL(V ) such thatLS(s, π, ρ) has a pole at s = 1 then π is expected to be a functorialtransfer.The converse also is expected to hold. i.e. If π is a functorial transferthen one should be able to find such a ρ.


Poles of automorphic L-functionsRemark

The situation is analogous to that of Artin L-functions. Given,

LH

LGL

ψφπH

φπG

(with Im(φπH ) = LH) one expects that

mπ(ρ) = m(1, ρ|ψ(LH)),

where m(π, ρ) denotes the order of the pole of LS(s, π, ρ) at s = 1 andm(1, ρ|ψ(LH)) denotes the multiplicity of the trivial representation in the

restriction of ρ to ψ(λH).


A Pipe Dream

Figure: Endoscopy andBeyond pg. 20.

Fix G and ρ : LG → GL(V ).Langlands’ idea is to use the (stable)trace formula to obtain a trace-like formula thatisolates π for which LS(s, π, ρ) has a pole.

∑

π

tr(πS(fS))m(π, ρ).

m(π, ρ) is the order of the pole at s = 1,

π, in theory, is running over representationsthat are of Ramanujan type (more on thislater...),

fS ∈ C∞c (G (AS)) is a test function.


Poles and partial averages

To arrive at such a formula one can try analyzing the logarithmicderivative of the L-function:

m(π, ρ) = −Ress=1d

dslog(LS(s, π, ρ))

= limX→∞

1X

′∑

p<X

aπ(p, ρ) log(p),

where LS(s, π, ρ) =∑′

naπ(n,ρ)

ns .

NB. This should be taken as a guide rather than a formula, since we don’tknow the analytic continuation of a general automorphic L-function.Moreover, even if we were given the analytic continuation for the equalityof the limit definition to hold π needs to satisfy a suitable temperedness(Ramanujan) condition.


Enter trace formula

Langlands, then, suggests bring the (stable) trace formula to get anexplicit expression for the a(p, ρ). To summarize:

Want∑

π tr(πS(fS))m(π, ρ),

For each p, trace formula “gives an expression for”∑π tr(πS(fS))aπ(p, ρ).

Beyond Endoscopy

For fixed G and ρ : LG → GL(V ), calculate the limit

limX→∞

1X

∑

p<X

log(p)∑

π

tr(πS(fS))aπ(p, ρ). (BE)

using the trace formula.


Langlands’ initial analysis

In the paper “Beyond Endoscopy” Langlands takes

G = GL(2) and ρ = Symk , the symmetric k ’th power of the standardrepresentation of LG = GL(2,C), and uses the trace formula inJacquet-Langlands.

He, essentially, analyzes the contributions of all the terms but (ii), theelliptic term, and (vi), a discrete contribution from the continuousspectrum, as well as the contribution of the trivial representation.

Resolution for GL(2)


Interlude: Sarnak’s letter and the Kuznetsov formula

Following the IAS lectures Sarnak wrote a detailed letter to Langlands on(BE) for GL(2).

Comments onRobert Langland’s Lecture: “Endoscopy and Beyond”

by

Peter Sarnak⇤

Letter to Robert Langland’s - April, 2001

Dear Bob,

Below I have put into writing my thoughts in connection with your lecture “Endoscopy andBeyond”† and especially the Section called “a pipe dream”. Also I have tried to digest your morerecent letters and numerical experiment and I address at least some of the points that you raisein the last section. There are no doubt a number of inaccuracies below but I think most of themare good approximations to the correct statements.

First, a comment about numerical testing your key problems T1 and P2 (the latter in generalconcerns the computation of the Hasse-Weil zeta function for a variety V over a number field whilethe former is functoriality for higher symmetric powers). The test that Booker has been using‡ isas follows:

Let

L(s, ⇡) =1X

n=1

�⇡(n)n�s . (1)

We normalize so that �⇡(1) = 1 and |�⇡(n)| is on average of size 1. Now if L(s, ⇡) satisfiesthe expected analytic properties then one can show that if N⇡ is the usual conductor of ⇡ andC⇡ the analytic conductor (as defined in Iwaniec-Sarnak “Perspectives on the analytic theory ofL-functions” [I-S]) then the following smooth sums S(X, ⇡) have a certain behavior. For 2C1

0 (0,1) set

S(X,⇡) :=X

(n,N⇡)=1

�⇡(n) ⇣ n

X

⌘(2)

For X C⇡ these sums are of sizep

X but as X increases beyond C⇡ these sums dropabruptly to zero. The latter can be used as a robust test of the analytic continuation of L(s, ⇡) to

⇤Department of Mathematics, Princeton University and the Courant Institute of Mathematical Sciences.†The complete version of this paper will appear in the American Journal of Math.‡Added in 2004: These appear in his Princeton thesis 2003

Uses Kuznetsov formula insteadof the trace formula: Readily available for GL(2)and avoids certain initial complications inherentin the trace formula (more on this later...).

Uses the L-function instead of the logarithmicderivative: Has the practical advantage thatthe sums over primes are replaced by sums overintegers, however one now gets a sum weightedby the residues instead of the orders of the poles.

For higher symmetric powers (k ≥ 4) highlights an essential problem,wich seem to need innovations in analytic number theory.

Results for GL(2)


Interlude: Results using Kuznetsov formula

Sarnak, GL(2), k = 1 (2001),

Venkatesh, GL(2), k = 2 (2002),

Herman, Asai L-function (quadratic base change) for GL(2) (2010),Rankin-Selberg for GL(2) (2012),

Getz, an approach towards base change for GL(N) (2012),

Sakellaridis, (relative) comparison for general rank 1 spherical varietieswith ρ = id (2018).


Back to the trace formula

After Beyond Endoscopy, Langlands studied the problem further fromvarious perspectives:

“Un nouveau point de repere dans la theorie des forme automorphes”((BE) over function fields, 2007)

“Formule des traces et fonctorialite: Le debut d’un programme” (w/Frenkel and Ngo, 2010)(Stenberg-Hitchin base and isolation of the contribution of the trivialrepresentation)

“Singularites et transfert” (Study of the singularities of orbitalintegrals, Poisson summation on the S-H base, and stable transfer,2013)


Steinberg-Hitchin base

Stable conj. classes are parametrized by an almost linear space, BG .

Example

For G = GL(N), the S-H base is given by the coefficients of thecharacteristic polynomial of the conjugacy class, i.e. BG = AN−1 oGm.

Idea: Express the geometric side of the stable trace formula as a functionon BG and use harmonic analysis.

∑

γ∈G ](Q)

vol(γ)O(st)γ (f ) //

∑

b∈BG (Q)

θ(f )

For G = GL(N) and elliptic γ:

vol(γ) = vol(Z (A)Gγ(Q)\Gγ(A)),

Oγ(f ) =∫Gγ(A)\G(A) f (g−1γg)dg .


Immediate issues

There are various obstacles one faces even before trying to implement thisstrategy:

Orbital integrals have singularities.

Volumes of centralizers are subtle arithmetic invariants, and are veryhard to control as γ moves around conjugacy classes.

Example (G = GL(2))

Let γ have tr(γ) = t, det(γ) = d .

I vol(γ) ∼ reg(√t2 − 4d)h(

√t2 − 4d),

I The singularity of Oγ(f ) is the same as that of the function√|t2 − 4d |.

Another problem is that the trace formula does not immediately givea sum over cuspidal representations!


Interlude: Trace formula for GL(N)

For a test function f ∈ C∞c (GLN(A)) it gives a distributional identity

Jspec(f ) = Jgeom(f ).

Principal contribution to the spectral side is from the “discrete part”,Idisc(f ), and the geometric side is from the “regular elliptic part”,Iell ,reg (f ).

Roughly, for the purposes of (BE), can think of the trace formula asgiving an formula for Idisc(f ),

Idisc(f ) = Iell ,reg (f ) + error .


Trace formula for GL(N)Source of quite a bit of pain

Trace formula, naturally, gives an (geometric) expression for Idisc(f ),

Idisc(f ) =∑

π:cuspidal

tr(π(f )) +∑

M(G

Idisc(f ,M).

NB. Many of the contributions in∑

M(G Idisc(f ,M) are non-tempered(e.g. trivial one dimensional representation). The importance of this to(BE) is that for the limit

limX→∞

1X

∑

p<X

log(p)∑

π

tr(πS(fS))aπ(p, ρ)

to exist, the double sum should be O(X ). For non-tempered π, thecoefficients aπ(p, ρ) are large (e.g. π : trivial rep. of GL(2),

a(p,Symk) ∼ pk2 ). They need to be taken out before the trace formula

can be used.S. Ali Altug Beyond Endoscopy November 14, 2018 22 / 35

Trace formula for GL(N)

For (BE) (and many other analytic problems) one wants an explicitexpression for

∑π:cuspidal tr(π(f )).

Problem (Isolation of Icusp(f ))

Develop an explicit (geometric) expression for

∑

π:cuspidal

tr(π(f )) = Idisc(f )−∑

M(G

Idisc(f ,M)

= Iell ,reg (f )−∑

M(G

Idisc(f ,M) + error .


Back to GL(2)

For G = GL(2) we have two conjugacy classes of Levi subgroups M = Gand the diagonal torus A. Therefore, Idisc(f ) is

∑

π:cuspidal

tr(π(f )) = Idisc(f )− Idisc(f ,A).

For the test functions f chosen by Langlands, Idisc(f ,A) itself is a sum oftwo terms: the trivial representation and term (vi) of Jacquet-Langlands .


Explicit example for GL(2)

For a specific choice of test function f, Idisc(f ,A) and Iell ,reg (f ) look likethe following:

Example

Idisc(f ,A) ∼ 2(√

7 + 1√7

) ∫Ff (x)dx +

∫|x |>1

Ff (x)√x2−1

dx

Iell ,reg (f ) ∼′∑

m

Ff

(m

2√

7

)L(

1,(m2−28·

))

L(

1,(m2−28·

)), is more or less the Hurwitz class number of the

corresponding order.

Ff is of compact support (and vanishes at |x | = 1).

First term of Idisc(f ,A) is the contribution of the trivialrepresentation, and the second one is (part of) the term (vi).


Results with the trace formula

Frenkel, Langlands, Ngo (2010)

Isolation of the trivial representation in Iell ,reg for a general class ofgroups (semi simple, simply connected, G = Gder ).

I Stable trace formula, S-H base, Poisson summation.I Utilizing the resulting formula in (BE) is left open (resulting formulas

are quite complicated)

‘‘Beyond endoscopy via the trace formula I (’15), II (’17), III (’18)”(A.)

For G = GL(2),Isolation of Idisc(f ,A) in Iell ,reg (f ),Analysis of the difference Iell ,reg (f )− Idisc(f ,A),(BE) for the standard L-function (k = 1).

I Main inputs, S-H base, approximate functional equation, Poissonsummation (methods different than that of FLN).

I Executes the limit over integers, following Sarnak’s suggestion.


Further directions

Recall that for a given group G (BE) is about understanding thelimit

limX→∞

1X

∑

p<X

log(p)∑

π

tr(πS(fS))aπ(p, ρ).

Here ρ : LG → GL(V ).

Two moving parameters

Change G keeping ρ fixed, e.g. G = GL(N), ρ = Std. (horizontal)Change ρ keeping G fixed, e.g. G = GL(2), ρ = Symk (vertical)

Vertical direction is very exciting (would, for instance, implyRamanujan for GL(2)) and wide open. The problems Sarnakhighlighted persist (for k � 1). Need new ideas in analytic numbertheory.

The horizontal direction seems (somewhat) more approachable.


Horizontal direction

“On smoothing singularities of elliptic orbital integrals on GL(n) andBeyond Endoscopy” (Gonzales et.al., 2018)(Partial results on extending methods (approximate functionalequation) of (A.’15) to GL(N))

Arthur et. al., 2018 (in preparation)(Extending the methods and results of (A.’15) to GL(3))


Further directions: GL(N)

Extending (and improving) the GL(2) methods to GL(N) may be withinreach. It requires control over orbital integrals and volumes of tori.

“On some class number relations for algebraic tori” (Shyr, 1977)(Class number formula for general algebraic tori.)

“Orbital integrals and Dedekind zeta functions” (Yun, 2013)(Relates orbital integrals for GL(N) to Dedekind zeta functions ofcertain orders. These zeta functions behave well: they have analyticcontinuation and functional equation.)

“Eisenstein series and the Selberg trace formula I” (Zagier, 1981)“Eisenstein series and the Selberg trace formula II”(Jacquet-Z., 1987)(Selberg trace formula as the residue of a Rankin-Selberg integral.)


Further directions: Arthur’s stratification

To go further in the horizontal direction one, in particular, needs aformula: ∑

π:cuspidal

tr(π(f )) = Ireg ,ell(f )−∑

M(G

Idisc(f ,M)

Arthur recently came with a conjectural stratification of the S-H base toisolate the contributions in

∑M(G Idisc(f ,M) for G = GL(N).

“A stratification related to characteristic polynomials” (Arthur, 2017)

The construction is inductive, reflecting the structure of the termscontributing to Idisc as induced from the L2-spectra of the Levi subgroupsM ⊂ G with equal sized blocks (e.g.M = GL(1)× GL(1)× GL(1) ↪→ GL(3), orM = GL(2)× GL(2) ↪→ GL(4)).


Arthur’s stratification

Let G = GL(n + 1) and let Ξ(n) = An.

The construction starts with the trace formula and the S-H base,

∑

b∈B(Q)

θ(b)Poisson−−−−−−−−→

∑

ξ∈B(Q)

θ(ξ).

For each decomposition (n + 1) = (m + 1)(d + 1), one inductivelydefines open subsets Ξ0(m) of Ξ(m) (in the ξ-side) as follows:

I Ξ0(0) = ∅.I Ξ0(m, n) :=

{(ξm, 0, ξm, 0, · · · , ξm, 0, ξm) | ξm ∈ Ξ0(m)

},

where d coordinates are 0 and d + 1 coordinates are ξm.

I Ξ(n) =⊔

(m+1)|(n+1)

Ξ0(m, n) and Ξ0(n) = Ξ(n)\⊔

(m+1)|(n+1)m 6=n

Ξ0(m, n).


Arthur’s stratification

Example

GL(2), i.e. n = 1. Ξ(1) = Ξ0(1) t Ξ0(0, 1) = (A1\{0} t {0}.GL(3), i.e. n = 2. Ξ(2) = Ξ0(2)tΞ0(0, 2) = (A2\{(0, 0)})t{(0, 0)}.GL(4), i.e. n = 3. Two factorizations (n + 1) = 1 · 4 = 2 · 2.

I Ξ0(0, 3) = {(0, 0, 0)},I Ξ0(1, 1) =

{(ξ, 0, ξ) | ξ ∈ A1\{0}

}

I Ξ0(3) = A3\{

(ξ, 0, ξ) | ξ ∈ A1}

These open strata Ξ0(n) is to correspond to∑

π:cuspidal tr(π(f )), and the

Ξ0(m) for the divisors (m + 1) | (n + 1) are to correspond to contributionsfrom the other Levi subgroups.


Future directions: Trace formula ↔ Kuznetsov

∑π aπ(p, ρ)

KuznetsovArthur-Selberg

(S-H)

Rankin-Selberg+P.S.

Poisson

“Transfer operators and Hankel transforms between relative traceformulas, II: Rankin-Selberg theory (Sakellaridis, 2018)

“Poincare Series” (Rudnick, PhD Thesis, 1990)

A.’15 and A’17, and earlier calculations point in this direction. UsingZagier’s kernel, from Selberg to Peterson. (In progress)


Figure: Istanbul, September 2018


Thank You!


Functoriality and Beyond Endoscopy · 2018-11-17 · General case, by comparing stable trace...

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