The Langlands Program: Beyond Endoscopy

Transcription

1 The Langlands Program: Beyond Endoscopy Oscar E. González 1, Kevin Kwan 2, 1 Department of Mathematics, University of Puerto Rico, Río Piedras. 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. Young Mathematicians Conference The Ohio State University August 21, 2016

2 What is the Langlands program? Conjectures relating many areas of math. Proposed by Robert P. Langlands about 50 years ago.

3 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:

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

5 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 For any z in H and any matrix ( ) a b in SL2 (Z), f satisfies the equation f ( az+b cz+d ) c d = (cz + d) k 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.

6 Langlands Functoriality Conjecture Conjecture 1.1 Given two reductive groups H, G with a homomorphism L H L G, there exists a related transfer of automorphic forms on H to automorphic forms on G Π(H) Π(G).

7 What are L-functions? Diverse families of functions attached to arithmetic objects, for example: Riemann zeta function ζ(s) = n=1 1 n s for Rs > 1, useful in studying distribution of primes. Dirichlet L-function L(s, χ) = n=1 χ(n) n s for Rs > 1, where χ is a Dirichlet s character, i.e. homomorphism χ : (Z/NZ) S 1. Useful in studying distribution of primes in arithmetic progression.

8 What are L-functions? Dedekind zeta function: associated to a number field E, i.e., finite degree field extension of Q.

9 What are L-functions? Dedekind zeta function: associated to a number field E, i.e., finite degree field extension of Q. ζ E (s) = a 1 N E/Q (a) s, for Rs > 1, where the sum is over all non-zero integral ideals of E.

10 Artin L-function: associated to a Galois representation.

11 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: L(s, ρ) = p det(i ρ(fr p )p s ) 1, where Fr p is the Frobenius element in Gal(E/Q). Denote by σ E the Artin representation obtained from the factorization ζ E (s) = ζ Q (s)l(s, σ E ), where ζ Q is the Riemann zeta function.

12 Essential Properties of L-functions Series expansion for Rs > 1.

13 Essential Properties of L-functions Series expansion for Rs > 1. Euler product for Rs > 1: ζ(s) = p L(s, χ) = p ζ E (s) = p (1 p s ) 1, (1 χ(p)p s ) 1, 1 1 N E/Q (p) s.

14 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.

15 Functional Equations of ζ(s) Gamma factor: γ(s) = π s/2 Γ(s/2).

16 Functional Equations of ζ(s) Gamma factor: γ(s) = π s/2 Γ(s/2). Completion: Λ(s) = γ(s)ζ(s).

17 Functional Equations of ζ(s) Gamma factor: γ(s) = π s/2 Γ(s/2). Completion: Λ(s) = γ(s)ζ(s). Functional equation: Λ(s) = Λ(1 s).

18 Functional Equations of ζ(s) Gamma factor: γ(s) = π s/2 Γ(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, Res s=1 ζ(s) = 1. Riemann Hypothesis: all non-trivial zeros lie on the critical line Re(s) = 1 2. Such a result would provide results on the distribution of prime numbers.

19 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/2 Γ((s + δ)/2), where δ = 0 if χ( 1) = 1 and δ = 1 if χ( 1) = 1.

20 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/2 Γ((s + δ)/2), where δ = 0 if χ( 1) = 1 and δ = 1 if χ( 1) = 1. Completion Λ(s) = q s/2 γ(s)l(s, χ), where q is the modulus of χ.

21 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/2 Γ((s + δ)/2), where δ = 0 if χ( 1) = 1 and δ = 1 if χ( 1) = 1. Completion Λ(s) = q s/2 γ(s)l(s, χ), where q is the modulus of χ. Functional equation Λ(s, χ) = τ(χ) q Λ(1 s, χ).

22 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/2 Γ((s + δ)/2), where δ = 0 if χ( 1) = 1 and δ = 1 if χ( 1) = 1. Completion Λ(s) = q s/2 γ(s)l(s, χ), where q is the modulus of χ. Functional equation Λ(s, χ) = τ(χ) q Λ(1 s, χ). L(s, χ) is entire if χ χ 0. If χ = χ 0, then L(s, χ) is analytic on C except having a simple pole at s = 1.

23 Functional equation of ζ E (s) (Hecke) Gamma factor: γ(s) = Γ R (s) r 1 Γ C (s) 2r 2, where Γ R (s) = π s/2 Γ(s/2), Γ C (s) = 2(2π) s Γ(s), r 1 and 2r 2 are the number of real and complex places of the number field E.

24 Functional equation of ζ E (s) (Hecke) Gamma factor: γ(s) = Γ R (s) r 1 Γ C (s) 2r 2, where Γ R (s) = π s/2 Γ(s/2), Γ C (s) = 2(2π) s Γ(s), r 1 and 2r 2 are the number of real and complex places of the number field E. Completion: Λ E (s) = D E s/2 γ(s)ζ E (s), where D E is the discriminant of the number field E.

25 Functional equation of ζ E (s) (Hecke) Gamma factor: γ(s) = Γ R (s) r 1 Γ C (s) 2r 2, where Γ R (s) = π s/2 Γ(s/2), Γ C (s) = 2(2π) s Γ(s), r 1 and 2r 2 are the number of real and complex places of the number field E. Completion: Λ E (s) = D E s/2 γ(s)ζ E (s), where D E is the discriminant of the number field E. Functional equation Λ E (s) = Λ E (1 s).

26 Functional equation of ζ E (s) (Hecke) Gamma factor: γ(s) = Γ R (s) r 1 Γ C (s) 2r 2, where Γ R (s) = π s/2 Γ(s/2), Γ C (s) = 2(2π) s Γ(s), r 1 and 2r 2 are the number of real and complex places of the number field E. Completion: Λ E (s) = D E s/2 γ(s)ζ E (s), where D E is the discriminant of the number field E. Functional equation Λ E (s) = Λ E (1 s). ζ(e) is analytic on C except with a simple pole at s = 1.

27 Res s=1 ζ E (s) is very interesting and it connects many fundamental quantities of algebraic number theory.

28 Res s=1 ζ E (s) is very interesting and it connects many fundamental quantities of algebraic number theory. r 1 is the number of real places of E, 2r 2 is the number of complex places of E (r 1 + 2r 2 = n), h E is the class number of E, R E is the regulator of E, w E is the number of roots of unity in E Theorem 2.2 (Class Number Formula) Res s=1 ζ E (s) = 2r 1 (2π) r 2 h E R E w E DE.

29 Arthur-Selberg Trace Formula Definition 2.3 (Right Regular Representation of G) The right regular representation of group G on L 2 (Γ \ G), denoted by R, is defined by R(y)φ(x) = φ(xy), where x, y G, φ L 2 (Γ \ G). Definition 2.4 (Right Regular Operator) Let f C c (G). We define the right regular operator R(f ) on L 2 (Γ \ G) as follows: for φ L 2 (Γ \ G), we have R(f )φ(x) = f (x)r(y)φ(x)dy = f (y)φ(xy)dy. G G

30 Let Q p denote the p-adic field (p being a prime), Q := R. Together they give all of the completions of Q.

31 Let Q p 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 Q p, where p. We denote the ring of adeles by A.

32 Let Q p 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 Q p, 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).

33 Arthur-Selberg Trace Formula γ G(Q) is said to be elliptic if its characteristic polynomial is irreducible over Q.

34 Arthur-Selberg Trace Formula γ G(Q) is said to be elliptic if its characteristic polynomial is irreducible over Q. Denote by G γ the centralizer of γ in G. Theorem 2.5 (Elliptic Part of Arthur-Selberg Trace Tormula) tr R(f ) = meas(γ) γ ell = γ ell meas(γ) v G γ(a)\g(a) G γ(q v )\G(Q v ) f (g 1 γg)dg + f v (g 1 γg)dg v. +

35 Idea of Beyond Endoscopy and L-functions Given an automorphic form π on G and a representation r of L G, one can define an automorphic L-function L(s, π, r) = n=1 a π,r (n) n s.

36 Idea of Beyond Endoscopy and L-functions Given an automorphic form π on G and a representation r of L G, one can define an automorphic L-function L(s, π, r) = n=1 a π,r (n) n s. Let G, G be reductive groups. If an automorphic form π on G is a functorial transfer from a smaller G, then one expects the L-function L(s, π, r) to have a pole at s = 1 for some representation r of L G.

37 Idea of Beyond Endoscopy and L-functions Given an automorphic form π on G and a representation r of L G, one can define an automorphic L-function L(s, π, r) = n=1 a π,r (n) n s. Let G, G be reductive groups. If an automorphic form π on G is a functorial transfer from a smaller G, then one expects the L-function L(s, π, r) to have a pole at s = 1 for some representation r of L G. In particular, the order of pole of L(s, π, r) at s = 1, which we denote by m r (π), should be nonzero if and only if π is a transfer.

38 Idea of Beyond Endoscopy and L-functions Langlands idea is to weight the spectral terms in the stable trace formula by m r (π), resulting in a trace formula whose spectral side detects only π for which the m r (π) is nonzero.

39 Idea of Beyond Endoscopy and L-functions Langlands idea is to weight the spectral terms in the stable trace formula by m r (π), resulting in a trace formula whose spectral side detects only π for which the m r (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).

40 Outline of Altuğ s work Apply Poisson summation to the elliptic part of the trace formula.

41 Outline of Altuğ s work 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.

42 Outline of Altuğ s work 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.

43 Our goal: Carry out the analysis of Altuğ to GL(n).

44 Our goal: Carry out the analysis of Altuğ to GL(n). Obstacle: We cannot directly apply Poisson summation formula!

45 Our goal: Carry out the analysis of Altuğ to GL(n). Obstacle: We cannot directly apply Poisson summation formula! There are singularities from the real orbital integrals.

46 Our Work By using the Class Number Formula, we have meas(γ) = meas(z + G γ (Q)\G γ (A)) = meas(e \I 1 E ) = 2r 1 (2π) r 2 h E R E w E = D E L(1, σ E ).

47 Using the CNF, we can write the elliptic part of the trace formula as follows: ±p k tr(γ),...,tr(γ n 1 ) 1 s γ L(1, σ E ) D γ 1/2 Orb(f ; γ) q Orb(f q ; γ).

48 Approximate functional equation of Artin L-function Assuming Artin s conjecture, we can write the approximate functional equation of L(1, σ E ).

49 L(s, ρ) = n π d(s+u)n/2 Γ( s+u 2 λ ρ (n) 1 n s 2πi (3) d π Γ( d(s)n/2 s 2) + +2dr 2Γ ( s+1 2 ) d π Γ( + d(1 s)n/2 +2dr 1 s 2Γ ( 2 s 2 2 ( ) n u X G(u) q ) d + +2dr 2Γ ( ) d +2dr s+u du ) d +2dr 2 u + ɛ(ρ)q 1 2 s ) d +2dr 2 ) d +2dr 2 d π Γ( d(s)n/2 s 2) + +2dr 2Γ ( s+1 n 2 ) d π Γ( + d(1 s+u)n/2 +2dr 1 s+u 2Γ ( 2 s+u G(u) 2 π d(1 s)n/2 Γ( 1 s 2 ) d + +2dr 2Γ ( 2 s 2 ( ) λ ρ (n) 1 nx u q n 1 s 2πi (3) ) d +2dr 2 2 du ) d +2dr 2 u.

50 Our work Recall our form of the elliptic part of the trace formula ±p k tr(γ),...,tr(γ n 1 ) 1 s γ L(1, σ E ) D γ 1/2 Orb(f ; γ) q Orb(f q ; γ). 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 L(1, σ E ) D γ 1/2 Orb(f ; γ) is smooth. This result is unconditional for n = 2, 3.

51 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.

52 Thanks The Ohio State University Rodica Costin, Craig Jackson, Bart Snapp Williams College NSF grants DMS , DMS and DMS The Chinese University of Hong Kong Steven J. Miller and Tian An Wong Roger Van Peski

53 References S. A. Altuğ, Beyond Endoscopy via the Trace Formula - I: Poisson Summation and Contributions of Special Representations. Composito Mathematica 151 (2015), O. E. González, 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 public_html/math/papers/beyondendosmall21.pdf. H. Iwaniec and E. Kowalski, Analytic number theory, AMS Colloquium Publications, vol. 53, 2004.

54 References R. P. Langlands, Beyond Endoscopy, in Contributions to automorphic forms, geometry and number theory, pp , Johns Hopkins University Press, Baltimore, MD, B. C. Ngô, Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111 (2010), D. Shelstad, Orbital integrals for GL 2 (R), in Automorphic forms, representations and L-functions, pp , Proceedings of Symposia in Pure Mathematics, vol. 33 part I, American Mathematical Society, Corvallis, OR, 1979.