On the Langlands Program

Transcription

1 On the Langlands Program John Rognes Colloquium talk, May 4th 2018

2 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for Advanced Study, Princeton, USA for his visionary program relating representation theory to number theory.

3 Robert P. Langlands (1967)

4 The Langlands Conjectures (ca. 1967) Conjecture (Reciprocity) To each Galois representation there corresponds an automorphic representation with the same L-function. Conjecture (Functoriality) To each homomorphism L G(C) L G (C) and each automorphic G-representation there corresponds an automorphic G -representation with the same L-function.

5 Quadratic Reciprocity (1801) p, l odd primes l = ( 1) l 1 2 l Theorem (Gauss) l is a quadratic residue modulo p if and only if p is a quadratic residue modulo l. For fixed l and varying p, the solvability of l x 2 mod p only depends on the residue class of p mod l.

6 Frobenius Automorphism ζ l = e 2πi/l root of unity Number fields Q Q( l ) Q(ζ l ) Frobenius automorphism: Frob p Gal(Q(ζ l )/Q) Frob p (ζ l ) = ζ p l x 2 p mod l solvable Frob p fixes l.

7 Abelian Number Fields Q Q algebraic closure Gal Q = Gal( Q/Q) absolute Galois group Theorem (Kronecker, Weber, Hilbert) For each homomorphism ρ: Gal Q C there exists a Dirichlet character χ: (Z/m) C such that for all p m. ρ(frob p ) = χ(p) Calculates Gal Q modulo commutators.

8 The Nonabelian Case A homomorphism ρ: Gal Q GL n (C) is a rank n complex representation. Nonabelian target for n 2. ρ(frob p ) GL n (C) is defined up to conjugacy. Well-defined characteristic polynomial Modified form P(t) = det(ti ρ(frob p )) Q(t) = det(i ρ(frob p )t) 1 Question What replaces Dirichlet characters for n 2?

9 Robert P. Langlands (1971)

10 There are at least three different problems with which one is confronted in the study of L-functions: the analytic continuation and functional equation; the location of the zeroes; and in some cases, the determination of the values at special points. The first may be the easiest. It is certainly the only one with which I have been closely involved. Langlands (Helsinki ICM 1978)

11 The Riemann Zeta Function ζ(s) = for Re(s) > 1. n=1 1 n s = s s s s +... Euler product ζ(s) = p 1 1 p s Analytic continuation, simple pole at s = 1. Real factor ξ(s) = π s/2 Γ(s/2) ζ(s) Functional equation ξ(1 s) = ξ(s)

12 Two views of the zeta function (Derbyshire)

13 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 dt log t ( x log x ) Proof (sketch). ζ(s) 0 for Re(s) = 1.

14 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(x 1/2+ɛ ) for each ɛ > 0. Question What is the natural context for the zeta function?

15 Robert P. Langlands (2016)

16 There are two kinds of L-functions, and they will be described below: motivic L-functions which generalize the Artin L-functions and are defined purely arithmetically, and automorphic L-functions, defined by data which are largely transcendental. Within the automorphic L-functions a special class can be singled out, the class of standard L-functions, which generalize the Hecke L-functions and for which the analytic continuation and functional equation can be proved directly. Langlands (Helsinki ICM 1978)

17 Artin L-functions (1923) Gal Q = Gal( Q/Q) Galois representation ρ: Gal Q GL n (C). Modified characteristic polynomial L p (ρ, s) = det(i ρ(frob p )p s ) 1 Euler product L(ρ, s) = p L p (ρ, s) converges for Re(s) sufficiently large.

18 Artin Conjecture If ρ is the trivial rank 1 representation, then L(ρ, s) = ζ(s) Brauer: The Artin L-function L(ρ, s) admits a meromorphic continuation. With real and ramified factors it satisfies a functional equation. Conjecture (Artin) If ρ is nontrivial, then L(ρ, s) is entire (has no poles).

19 Modular Forms A modular form of weight k is a holomorphic function f : H = {z C : Im(z) > 0} C such that f ( az + b cz + d ) = (cz + d)k f (z) [ ] a b for all γ = SL c d 2 (Z). Here H = SL 2 (R)/SO 2 is a symmetric space.

20 Fundamental regions for SL 2 (Z) acting on H (Womack)

21 Hecke Theory, I The following are equivalent: (a) The holomorphic function f (z) = n a n e 2πinz is a modular form. (b) The Dirichlet series satisfies a functional equation. n a n n s

22 Modular Representations Gelfand-Fomin (1952): A modular form f of weight k defines a smooth function by φ f : SL 2 (Z)\SL 2 (R) C φ f (g) = (ci + d) k f ( ai + b ci + d ) [ ] a b for g = SL c d 2 (R). Let π f = φ f L 2 (SL 2 (Z)\SL 2 (R)) be the SL 2 (R)-representation generated by φ f.

23 Hecke Theory, II The following are equivalent: (a) The SL 2 (R)-representation π f = φ f is irreducible. (b) The modular form f (z) = n a n e 2πinz is an eigenfunction for the Hecke algebra. (c) The Dirichlet series has an Euler product expansion. n a n n s

24 Completions of Q Let v be a norm on Q, so that x y v defines a metric. Let Q Q v be the associated completion. v =, x = x. Field of real numbers: Q = R = 1 in R. v = p any prime, x p = 1/p n for x = ap n /b with p ab. Field of p-adic numbers: Q p. Ring of p-adic integers: Z p = {x Q p : x p 1}. 1 + p + p 2 + p 3 + = 1/(1 p) in Z p Q p.

25 The Adèle Ring Diagonal embedding Q v Q v = R p Q p The adèle ring A v Q v is locally compact. A contains R and Q p as subspaces. A contains Q as a discrete subring.

26 Automorphic Representations GL n (A) contains GL n (Q) as a discrete subgroup. GL n (A) acts on L 2 (GL n (Q)\GL n (A)) by right translation: (h f )(g) = f (gh) for f : GL n (Q)\GL n (A) C and g, h GL n (A). An automorphic representation π is an irreducible GL n (A)-representation π L 2 (GL n (Q)\GL n (A)) contained in the regular representation.

27 Größencharakteren (n = 1) GL 1 (Q)\GL 1 (A) = R >0 p Z p. Each automorphic GL 1 -representation π is a character GL 1 (Q)\GL 1 (A) C of finite order. It factors uniquely through a Dirichlet character χ: (Z/m) C and vice versa.

28 Parabolic Induction, I Parabolic subgroup g 1... { 0 g 2... } P = p = : g i GL ni GL n g r Given automorphic GL ni -representations π i, get P-representation p π 1 (g 1 ) π r (g r ) by restriction along P GL n1 GL nr. Extend this P-representation to a GL n -representation π by induction along P GL n.

29 Parabolic Induction, II Theorem (Langlands) The automorphic representations of GL n (A) are precisely the irreducible constituents π of the representations where π 1,..., π r are cuspidal. π = Ind GLn Res P (π 1 π r ), Proof depends on Langlands theory (1965) of Eisenstein series for GL n, started by Selberg (1962) for SL 2. Gelfand (1962) clarified role of cusp forms for rank r 2.

30 Local Components Any automorphic GL n -representation π factors as π = v π v where each π v is an irreducible GL n (Q v )-representation. Almost all π p are constituents of π p = Ind GLn Res P (χ z ) for some z = (z 1,..., z n ) C n. Here r = n, each n i = 1, and χ z (g 1,..., g n ). = g 1 z 1 p g n zn p.

31 Satake Isomorphism Almost all π p are unramified, so that Hecke algebra dim π GLn(Zp) p = 1 H p = C c (GL n (Z p )\GL n (Q p )/GL n (Z p )) acts naturally on π GLn(Zp) p, multiplying by χ p : H p C. Satake/Langlands (1970): H p is the representation ring of the complex Lie group called the L-group of GL n. L GL n (C),

32 Langlands Parameter Dually, χ p is evaluation at a semisimple conjugacy class σ(π p ) L GL n (C)/ For π p in π p = Ind GLn Res P (χ z ), with z = (z 1,..., z n ) p z σ(π p ) p zn

33 Automorphic L-Functions Local L-function L p (s, π) = det(i σ(π p )p s ) 1 Also real and ramified cases. (Standard) automorphic L-function L(s, π) = v L v (s, π) Theorem (Godement Jacquet (1972)) Let π be an automorphic GL n -representation. Then L(s, π) has analytic continuation to a meromorphic function of s C, which satisfies a functional equation.

34 Reciprocity Conjecture Conjecture (Langlands) For each rank n Galois representation ρ: Gal Q L GL n (C) there exists an automorphic GL n -representation π such that ρ(frob p ) = σ(π p ) for almost all p. This will determine π uniquely, and L(ρ, s) = L(s, π).

35 Local Langlands So far we only discussed unramified π v for v = p. To parametrize other irreducible GL n (Q v )-representations, more information is needed. The Weil Deligne group L Qv is a variant of the absolute Galois group Gal Qv. Conjecture (Langlands) Irreducible GL n (Q v )-representations correspond to conjugacy classes of homomorphisms called Langlands parameters. φ v : L Qv L GL n (C),

36 Robert P. Langlands (2013)

37 For the other L-functions the analytic continuation is not so easily effected. However all evidence indicates that there are fewer L-functions than the definitions suggest, and that every L-function, motivic or automorphic, is equal to a standard L-function. Such equalities are often deep, and are called reciprocity laws, for historical reasons. Once a reciprocity law can be proved for an L-function, analytic continuation follows, and so, for those who believe in the validity of the reciprocity laws, they and not analytic continuation are the focus of attention, but very few such laws have been established. Langlands (Helsinki ICM 1978)

38 Reductive Groups Harish-Chandra: What can be done for GL n should be done for each reductive group G. Each algebraic representation of a reductive group is a direct sum of irreducible representations. An automorphic representation of G is an irreducible G(A)-representation π L 2 (G(Q)\G(A)) contained in the regular representation.

39 The L-Group The Hecke algebra for (G, K ) is the representation ring of a complex Lie group L G(C) called the L-group, or Langlands dual, of G. The maximal torus of L G is dual to that of G. Automorphic representations of G have Langlands parameters σ(π p ) and φ v in L G(C).

40 Functoriality Conjecture Conjecture (Langlands) For each homomorphism h : L G L G and automorphic representation π of G there exists an automorphic representation π of G such that for almost all p. h(σ(π p )) = σ(π p) This will determine π uniquely, and L(s, π, r) = L(s, π, rh) for each finite-dimensional representation r of L G.

41 The Rosetta Stone (196 BC)

42 Global Fields The three columns of the Rosetta stone: Number field: finite extension F Q. Function field: finite extension E F p (t). Riemann surface: finite cover X CP 1. Weil: What can be done for number fields should also be done for function fields and Riemann surfaces.

43 Local Fields The local Langlands conjecture for a reductive group G over a local field F v has been proved: For GL 1 by local class field theory. Over F v = R or C by Langlands (1973). For GL 2 by Jacquet Langlands (1970) and Kutzko (1980). For GL n with char(f v ) = p by Laumon Rapoport Stuhler (1993). For GL n with char(f v ) = 0 by Harris Taylor (2001), Henniart (2000) and Scholze (2013). For general G, ongoing work by Fargues, Scholze.

44 Number Fields The Langlands reciprocity conjecture for a reductive group G over a number field F has been proved: For GL 1 by global class field theory. For tori by Langlands (1968). Partial results for GL 2 by Langlands (1980), Tunnel (1981), Wiles (1995) and Breuil Conrad Diamond Taylor (2001). Artin conjecture and Riemann hypothesis open (!)

45 Function Fields over Curves The Langlands reciprocity conjecture for a reductive group G over a function field E has been proved: For GL 1 by global class field theory. For GL 2 by Drinfeld (1974), introducing shtukas. For GL n by Laurent Lafforgue (1998). For general G, automorphic to Galois direction, by Vincent Lafforgue (2014). Artin conjecture and Riemann hypothesis proved by Weil.

46 Geometric Langlands Translation from curves over finite fields to curves over C, using ideas of Deligne, Drinfeld, Laumon and Beilinson. Seek equivalence between a category of D-modules on BunG (X); a category of quasi-coherent sheaves on LocSysL G(X). More precisely, these should be -categories. Witten: Langlands duality G L G is parallel to S-duality in supersymmetric gauge theories.

47 Robert P. Langlands (2015)