Introduction Motivation Results References

The Langlands Program: Beyond Endoscopy

Oscar E. Gonz´alez1, [email protected] Kevin Kwan2, [email protected]

1Department of Mathematics, University of Puerto Rico, R´ıoPiedras.

2Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.

Young Mathematicians Conference The Ohio State University August 21, 2016 Introduction Motivation Results References

What is the Langlands program?

Conjectures relating many areas of math. Proposed by Robert P. Langlands about 50 years ago. 1 a b
For any z in H and any matrix c d in SL2(Z), f satisfies the  az+b  k equation f cz+d = (cz + d) f (z). 2 f is a holomorphic (complex analytic) function on H. 3 f satisfies certain growth conditions at the cusps.

Automorphic forms generalize modular forms.

Introduction Motivation Results References

Automorphic Forms

A modular form of weight k is a complex-valued function f on the upper half-plane H = {z ∈ C, Im(z) > 0} (z = a + bi with b > 0), satisfying the following three conditions: Automorphic forms generalize modular forms.

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Automorphic Forms

A modular form of weight k is a complex-valued function f on the upper half-plane H = {z ∈ C, Im(z) > 0} (z = a + bi with b > 0), satisfying the following three conditions:

1 a b
For any z in H and any matrix c d in SL2(Z), f satisfies the  az+b  k equation f cz+d = (cz + d) f (z). 2 f is a holomorphic (complex analytic) function on H. 3 f satisfies certain growth conditions at the cusps. Introduction Motivation Results References

Automorphic Forms

A modular form of weight k is a complex-valued function f on the upper half-plane H = {z ∈ C, Im(z) > 0} (z = a + bi with b > 0), satisfying the following three conditions:

1 a b
For any z in H and any matrix c d in SL2(Z), f satisfies the  az+b  k equation f cz+d = (cz + d) f (z). 2 f is a holomorphic (complex analytic) function on H. 3 f satisfies certain growth conditions at the cusps.

Automorphic forms generalize modular forms. Introduction Motivation Results References

Langlands’ Functoriality Conjecture

Conjecture 1.1 Given two reductive groups H, G with a homomorphism LH →L G, there exists a related transfer of automorphic forms on H to automorphic forms on G Π(H) → Π(G). Introduction Motivation Results References

What are L-functions?

Diverse families of functions attached to arithmetic objects, for example: Riemann zeta function ∞ X 1 ζ(s) = for 1, ns n=1 useful in studying distribution of primes. Dirichlet L-function ∞ X χ(n) L(s, χ) = for 1, ns n=1 where χ is a Dirichlet’s character, i.e. homomorphism × 1 χ :(Z/NZ) → S . Useful in studying distribution of primes in arithmetic progression. X 1 ζ (s) = , E N (a)s a E/Q for 1, where the sum is over all non-zero integral ideals of E.

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What are L-functions?

Dedekind zeta function: associated to a number field E, i.e., finite degree field extension of Q. Introduction Motivation Results References

What are L-functions?

Dedekind zeta function: associated to a number field E, i.e., finite degree field extension of Q. X 1 ζ (s) = , E N (a)s a E/Q for 1, where the sum is over all non-zero integral ideals of E. Definition 2.1 Let E be a number field. Let ρ : Gal(E/Q) → GL(d, C) be a finite dimensional Galois representation. The Artin L-function associated to ρ is given in terms of the Euler product:

Y −s −1 L(s, ρ) = det(I − ρ(Frp)p ) , p

where Frp is the Frobenius element in Gal(E/Q).

Denote by σE the Artin representation obtained from the factorization ζE (s) = ζQ(s)L(s, σE ), where ζQ is the Riemann zeta function.

Introduction Motivation Results References

Artin L-function: associated to a Galois representation. Introduction Motivation Results References

Artin L-function: associated to a Galois representation.

Definition 2.1 Let E be a number field. Let ρ : Gal(E/Q) → GL(d, C) be a finite dimensional Galois representation. The Artin L-function associated to ρ is given in terms of the Euler product:

Y −s −1 L(s, ρ) = det(I − ρ(Frp)p ) , p

where Frp is the Frobenius element in Gal(E/Q).

Denote by σE the Artin representation obtained from the factorization ζE (s) = ζQ(s)L(s, σE ), where ζQ is the Riemann zeta function. Euler product for 1: Y ζ(s) = (1 − p−s )−1, p Y L(s, χ) = (1 − χ(p)p−s )−1, p Y 1 ζ (s) = . E 1 − N (p)s p E/Q

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Essential Properties of L-functions

Series expansion for 1. Introduction Motivation Results References

Essential Properties of L-functions

Series expansion for 1. Euler product for 1: Y ζ(s) = (1 − p−s )−1, p Y L(s, χ) = (1 − χ(p)p−s )−1, p Y 1 ζ (s) = . E 1 − N (p)s p E/Q Introduction Motivation Results References

Essential Properties of L-functions

Can be completed by ‘gamma factors’. The completed L-functions are meromorphic on C with at most poles at s = 0 and s = 1 and they satisfy functional equations. Completion: Λ(s) = γ(s)ζ(s). Functional equation: Λ(s) = Λ(1 − s). Λ(s) is analytic on C except with a simple pole at s = 1. Also, Ress=1 ζ(s) = 1. Riemann Hypothesis: all non-trivial zeros lie on the critical 1 line Re(s) = 2 . Such a result would provide results on the distribution of prime numbers.

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Functional Equations of ζ(s)

Gamma factor: γ(s) = π−s/2Γ(s/2). Functional equation: Λ(s) = Λ(1 − s). Λ(s) is analytic on C except with a simple pole at s = 1. Also, Ress=1 ζ(s) = 1. Riemann Hypothesis: all non-trivial zeros lie on the critical 1 line Re(s) = 2 . Such a result would provide results on the distribution of prime numbers.

Introduction Motivation Results References

Functional Equations of ζ(s)

Gamma factor: γ(s) = π−s/2Γ(s/2). Completion: Λ(s) = γ(s)ζ(s). Λ(s) is analytic on C except with a simple pole at s = 1. Also, Ress=1 ζ(s) = 1. Riemann Hypothesis: all non-trivial zeros lie on the critical 1 line Re(s) = 2 . Such a result would provide results on the distribution of prime numbers.

Introduction Motivation Results References

Functional Equations of ζ(s)

Gamma factor: γ(s) = π−s/2Γ(s/2). Completion: Λ(s) = γ(s)ζ(s). Functional equation: Λ(s) = Λ(1 − s). Introduction Motivation Results References

Functional Equations of ζ(s)

Gamma factor: γ(s) = π−s/2Γ(s/2). Completion: Λ(s) = γ(s)ζ(s). Functional equation: Λ(s) = Λ(1 − s). Λ(s) is analytic on C except with a simple pole at s = 1. Also, Ress=1 ζ(s) = 1. Riemann Hypothesis: all non-trivial zeros lie on the critical 1 line Re(s) = 2 . Such a result would provide results on the distribution of prime numbers. Completion Λ(s) = qs/2γ(s)L(s, χ), where q is the modulus of χ. τ(χ) √ Functional equation Λ(s, χ) = q Λ(1 − s, χ¯).

L(s, χ) is entire if χ 6= χ0. If χ = χ0, then L(s, χ) is analytic on C except having a simple pole at s = 1.

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Functional Equation of L(s, χ)

For a primitive character χ (i.e., not induced by any character of smaller modulus), we have the following. Gamma factor: γ(s) = π−s/2Γ((s + δ)/2), where δ = 0 if χ(−1) = 1 and δ = 1 if χ(−1) = −1. τ(χ) √ Functional equation Λ(s, χ) = q Λ(1 − s, χ¯).

L(s, χ) is entire if χ 6= χ0. If χ = χ0, then L(s, χ) is analytic on C except having a simple pole at s = 1.

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Functional Equation of L(s, χ)

For a primitive character χ (i.e., not induced by any character of smaller modulus), we have the following. Gamma factor: γ(s) = π−s/2Γ((s + δ)/2), where δ = 0 if χ(−1) = 1 and δ = 1 if χ(−1) = −1. Completion Λ(s) = qs/2γ(s)L(s, χ), where q is the modulus of χ. L(s, χ) is entire if χ 6= χ0. If χ = χ0, then L(s, χ) is analytic on C except having a simple pole at s = 1.

Introduction Motivation Results References

Functional Equation of L(s, χ)

For a primitive character χ (i.e., not induced by any character of smaller modulus), we have the following. Gamma factor: γ(s) = π−s/2Γ((s + δ)/2), where δ = 0 if χ(−1) = 1 and δ = 1 if χ(−1) = −1. Completion Λ(s) = qs/2γ(s)L(s, χ), where q is the modulus of χ. τ(χ) √ Functional equation Λ(s, χ) = q Λ(1 − s, χ¯). Introduction Motivation Results References

Functional Equation of L(s, χ)

For a primitive character χ (i.e., not induced by any character of smaller modulus), we have the following. Gamma factor: γ(s) = π−s/2Γ((s + δ)/2), where δ = 0 if χ(−1) = 1 and δ = 1 if χ(−1) = −1. Completion Λ(s) = qs/2γ(s)L(s, χ), where q is the modulus of χ. τ(χ) √ Functional equation Λ(s, χ) = q Λ(1 − s, χ¯).

L(s, χ) is entire if χ 6= χ0. If χ = χ0, then L(s, χ) is analytic on C except having a simple pole at s = 1. s/2 Completion: ΛE (s) = |DE | γ(s)ζE (s), where DE is the discriminant of the number field E.

Functional equation ΛE (s) = ΛE (1 − s). ζ(E) is analytic on C except with a simple pole at s = 1.

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Functional equation of ζE (s) (Hecke)

r1 2r2 Gamma factor: γ(s) = ΓR(s) ΓC(s) , where −s/2 −s ΓR(s) = π Γ(s/2), ΓC(s) = 2(2π) Γ(s), r1 and 2r2 are the number of real and complex places of the number field E. Functional equation ΛE (s) = ΛE (1 − s). ζ(E) is analytic on C except with a simple pole at s = 1.

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Functional equation of ζE (s) (Hecke)

r1 2r2 Gamma factor: γ(s) = ΓR(s) ΓC(s) , where −s/2 −s ΓR(s) = π Γ(s/2), ΓC(s) = 2(2π) Γ(s), r1 and 2r2 are the number of real and complex places of the number field E. s/2 Completion: ΛE (s) = |DE | γ(s)ζE (s), where DE is the discriminant of the number field E. ζ(E) is analytic on C except with a simple pole at s = 1.

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Functional equation of ζE (s) (Hecke)

r1 2r2 Gamma factor: γ(s) = ΓR(s) ΓC(s) , where −s/2 −s ΓR(s) = π Γ(s/2), ΓC(s) = 2(2π) Γ(s), r1 and 2r2 are the number of real and complex places of the number field E. s/2 Completion: ΛE (s) = |DE | γ(s)ζE (s), where DE is the discriminant of the number field E.

Functional equation ΛE (s) = ΛE (1 − s). Introduction Motivation Results References

Functional equation of ζE (s) (Hecke)

r1 2r2 Gamma factor: γ(s) = ΓR(s) ΓC(s) , where −s/2 −s ΓR(s) = π Γ(s/2), ΓC(s) = 2(2π) Γ(s), r1 and 2r2 are the number of real and complex places of the number field E. s/2 Completion: ΛE (s) = |DE | γ(s)ζE (s), where DE is the discriminant of the number field E.

Functional equation ΛE (s) = ΛE (1 − s). ζ(E) is analytic on C except with a simple pole at s = 1. r1 is the number of real places of E,

2r2 is the number of complex places of E (r1 + 2r2 = n),

hE is the class number of E,

RE is the regulator of E,

wE is the number of roots of unity in E Theorem 2.2 (Class Number Formula) r r 2 1 (2π) 2 hE RE Ress=1 ζE (s) = p . wE |DE |

Introduction Motivation Results References

Ress=1 ζE (s) is very interesting and it connects many fundamental quantities of algebraic number theory. Introduction Motivation Results References

Ress=1 ζE (s) is very interesting and it connects many fundamental quantities of algebraic number theory.

r1 is the number of real places of E,

2r2 is the number of complex places of E (r1 + 2r2 = n),

hE is the class number of E,

RE is the regulator of E,

wE is the number of roots of unity in E Theorem 2.2 (Class Number Formula) r r 2 1 (2π) 2 hE RE Ress=1 ζE (s) = p . wE |DE | Introduction Motivation Results References

Arthur-Selberg Trace Formula

Definition 2.3 (Right Regular Representation of G) The right regular representation of group G on L2(Γ \ G), denoted by R, is defined by R(y)φ(x) = φ(xy), where x, y ∈ G, φ ∈ L2(Γ \ G).

Definition 2.4 (Right Regular Operator)

Let f ∈ Cc (G). We define the right regular operator R(f ) on L2(Γ \ G) as follows: for φ ∈ L2(Γ \ G), we have Z Z R(f )φ(x) = f (x)R(y)φ(x)dy = f (y)φ(xy)dy. G G The ring of adeles is the restricted direct product of all Qp, where p ≤ ∞. We denote the ring of adeles by A. In this way A is locally compact and we can assign a Haar measure on G(A), where G = GL(n).

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Let Qp denote the p-adic field (p being a prime), Q∞ := R. Together they give all of the completions of Q. In this way A is locally compact and we can assign a Haar measure on G(A), where G = GL(n).

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Let Qp denote the p-adic field (p being a prime), Q∞ := R. Together they give all of the completions of Q. The ring of adeles is the restricted direct product of all Qp, where p ≤ ∞. We denote the ring of adeles by A. Introduction Motivation Results References

Let Qp denote the p-adic field (p being a prime), Q∞ := R. Together they give all of the completions of Q. The ring of adeles is the restricted direct product of all Qp, where p ≤ ∞. We denote the ring of adeles by A. In this way A is locally compact and we can assign a Haar measure on G(A), where G = GL(n). Denote by Gγ the centralizer of γ in G.

Theorem 2.5 (Elliptic Part of Arthur-Selberg Trace Tormula)

X Z tr R(f ) = meas(γ) f (g −1γg)dg + ··· γ ell Gγ (A)\G(A) Z X Y −1 = meas(γ) fv (g γg)dgv . + ··· γ ell v Gγ (Qv )\G(Qv )

Introduction Motivation Results References

Arthur-Selberg Trace Formula

γ ∈ G(Q) is said to be elliptic if its characteristic polynomial is irreducible over Q. Introduction Motivation Results References

Arthur-Selberg Trace Formula

γ ∈ G(Q) is said to be elliptic if its characteristic polynomial is irreducible over Q. Denote by Gγ the centralizer of γ in G.

Theorem 2.5 (Elliptic Part of Arthur-Selberg Trace Tormula)

X Z tr R(f ) = meas(γ) f (g −1γg)dg + ··· γ ell Gγ (A)\G(A) Z X Y −1 = meas(γ) fv (g γg)dgv . + ··· γ ell v Gγ (Qv )\G(Qv ) Let G, G 0 be reductive groups. If an automorphic form π on G is a functorial transfer from a smaller G 0, then one expects the L-function L(s, π, r) to have a pole at s = 1 for some representation r of LG. In particular, the order of pole of L(s, π, r) at s = 1, which we denote by mr (π), should be nonzero if and only if π is a transfer.

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Idea of Beyond Endoscopy and L-functions

Given an automorphic form π on G and a representation r of LG, one can define an automorphic L-function

∞ X aπ,r (n) L(s, π, r) = . ns n=1 In particular, the order of pole of L(s, π, r) at s = 1, which we denote by mr (π), should be nonzero if and only if π is a transfer.

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Idea of Beyond Endoscopy and L-functions

Given an automorphic form π on G and a representation r of LG, one can define an automorphic L-function

∞ X aπ,r (n) L(s, π, r) = . ns n=1

Let G, G 0 be reductive groups. If an automorphic form π on G is a functorial transfer from a smaller G 0, then one expects the L-function L(s, π, r) to have a pole at s = 1 for some representation r of LG. Introduction Motivation Results References

Idea of Beyond Endoscopy and L-functions

Given an automorphic form π on G and a representation r of LG, one can define an automorphic L-function

∞ X aπ,r (n) L(s, π, r) = . ns n=1

Let G, G 0 be reductive groups. If an automorphic form π on G is a functorial transfer from a smaller G 0, then one expects the L-function L(s, π, r) to have a pole at s = 1 for some representation r of LG. In particular, the order of pole of L(s, π, r) at s = 1, which we denote by mr (π), should be nonzero if and only if π is a transfer. Since in general L(s, π, r) is not a priori defined at s = 1, we account for the weight factor by taking the residue at s = 1 of the logarithmic derivative of L(s, π, r).

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Idea of Beyond Endoscopy and L-functions

Langlands’ idea is to weight the spectral terms in the stable trace formula by mr (π), resulting in a trace formula whose spectral side detects only π for which the mr (π) is nonzero. Introduction Motivation Results References

Idea of Beyond Endoscopy and L-functions

Langlands’ idea is to weight the spectral terms in the stable trace formula by mr (π), resulting in a trace formula whose spectral side detects only π for which the mr (π) is nonzero. Since in general L(s, π, r) is not a priori defined at s = 1, we account for the weight factor by taking the residue at s = 1 of the logarithmic derivative of L(s, π, r). This isolates the contribution from trivial representation, which should give us the major contribution. Give other versions of trace formula to be used in detecting the functorial transfer.

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Outline of Altu˘g’swork

Apply Poisson summation to the elliptic part of the trace formula. Give other versions of trace formula to be used in detecting the functorial transfer.

Introduction Motivation Results References

Outline of Altu˘g’swork

Apply Poisson summation to the elliptic part of the trace formula. This isolates the contribution from trivial representation, which should give us the major contribution. Introduction Motivation Results References

Outline of Altu˘g’swork

Apply Poisson summation to the elliptic part of the trace formula. This isolates the contribution from trivial representation, which should give us the major contribution. Give other versions of trace formula to be used in detecting the functorial transfer. Obstacle: We cannot directly apply Poisson summation formula! There are singularities from the real orbital integrals.

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Our goal: Carry out the analysis of Altu˘gto GL(n). There are singularities from the real orbital integrals.

Introduction Motivation Results References

Our goal: Carry out the analysis of Altu˘gto GL(n). Obstacle: We cannot directly apply Poisson summation formula! Introduction Motivation Results References

Our goal: Carry out the analysis of Altu˘gto GL(n). Obstacle: We cannot directly apply Poisson summation formula! There are singularities from the real orbital integrals. Introduction Motivation Results References

Our Work

By using the Class Number Formula, we have

meas(γ) = meas(Z+Gγ(Q)\Gγ(A)) × 1 = meas(E \IE ) 2r1 (2π)r2 h R = E E wE p = |DE | L(1, σE ). Introduction Motivation Results References

Using the CNF, we can write the elliptic part of the trace formula as follows:

X X 1 1/2 Y L(1, σE )|Dγ| Orb(f∞; γ) Orb(fq; γ). |sγ| ±pk tr(γ),...,tr(γn−1) q Introduction Motivation Results References

Approximate functional equation of Artin L-function

Assuming Artin’s conjecture, we can write the approximate functional equation of L(1, σE ). Introduction Motivation Results References

Z  −u X λρ(n) 1 n L(s, ρ) = √ G(u) ns 2πi X q n (3) + −  d +2dr2  d +2dr2 π−d(s+u)n/2 Γ s+u Γ s+u+1 2 2 du 1 2 −s + − + (ρ)q d +2dr2 d +2dr2 −d(s)n/2  s   s+1  u π Γ 2 Γ 2 + −  d +2dr2  d +2dr2 π−d(1−s)n/2 Γ 1−s Γ 2−s ¯ Z  −u 2 2 X λρ(n) 1 nX + − √  d +2dr2  d +2dr2 n1−s 2πi q −d(s)n/2 s s+1 n (3) π Γ 2 Γ 2 d++2dr d−+2dr −d(1−s+u)n/2  1−s+u  2  2−s+u  2 π Γ 2 Γ 2 du G(u) + − . d +2dr2 d +2dr2 −d(1−s)n/2  1−s   2−s  u π Γ 2 Γ 2 Introduction Motivation Results References

Our work

Recall our form of the elliptic part of the trace formula

X X 1 1/2 Y L(1, σE )|Dγ| Orb(f∞; γ) Orb(fq; γ). |sγ| ±pk tr(γ),...,tr(γn−1) q

By applying the AFE and using some results of Shelstad on real orbital integrals [Shelstad 1979], we get Theorem 1 Assume Artin’s conjecture. Then

1/2 L(1, σE )|Dγ| Orb(f∞; γ)

is smooth. This result is unconditional for n = 2, 3. Introduction Motivation Results References

What we’d like to do

Work with a general reductive group instead of GL(n). Study the product of p-adic orbital integrals. Apply Poisson summation to the resulting smooth expression. Introduction Motivation Results References

Thanks

The Ohio State University Rodica Costin, Craig Jackson, Bart Snapp Williams College NSF grants DMS1347804, DMS1561945 and DMS1265673 The Chinese University of Hong Kong Steven J. Miller and Tian An Wong Roger Van Peski Introduction Motivation Results References

References

S. A. Altu˘g, Beyond Endoscopy via the Trace Formula - I: Poisson Summation and Contributions of Special Representations. Composito Mathematica 151 (2015), 1791-1820. O. E. Gonz´alez,C. H. Kwan, S. J. Miller, R. Van Peski and T. A. Wong, On smoothing singularities of elliptic orbital integrals on GL(n) and Beyond Endoscopy. Available at http://web.williams.edu/Mathematics/sjmiller/ public_html/math/papers/BeyondEndoSMALL21.pdf. H. Iwaniec and E. Kowalski, Analytic number theory, AMS Colloquium Publications, vol. 53, 2004. Introduction Motivation Results References

References

R. P. Langlands, Beyond Endoscopy, in Contributions to automorphic forms, geometry and number theory, pp. 611-697, Johns Hopkins University Press, Baltimore, MD, 2004. B. C. Ngˆo,Le lemme fondamental pour les alg`ebres de Lie. Publ. Math. Inst. Hautes Etudes´ Sci. 111 (2010), 1-169.

D. Shelstad, Orbital integrals for GL2(R), in Automorphic forms, representations and L-functions, pp. 107-110, Proceedings of Symposia in Pure Mathematics, vol. 33 part I, American Mathematical Society, Corvallis, OR, 1979.