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The Langlands Program Seminar

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The Langlands Program Seminar Tian An Wong July 29, 2016 Abstract These are notes from the ongoing Langlands Program Seminar organized since Spring 2014. What is known today as the Langlands Program, originating in a letter from Langlands to Weil in 1967, is now a vast area of research. While it asks questions of largely number theoretic interest, its methods have required input from harmonic analysis, representation theory, algebraic geometry, algebraic number theory, and more. Regarding emphasis, I have used as a guide the expository writing of Langlands, and as a result we have focused on functoriality via the trace formula, and reciprocity in terms of motives. These notes are produced by a naive graduate student and there are mistakes scattered throughout, so use them at your own risk! Please direct feedback and comments to twong at gradcenter.cuny.edu. Contents 1 Basic objects 4 1.1 Automorphic Representations . .4 1.2 The L-group . .5 1.3 The Satake Isomorphism . .6 1.4 The Weil-Deligne group . .7 1.5 Touchstone I: Local Langlands . .8 1.6 The automorphic L-function . 11 1.7 Reductive groups . 11 1.8 Touchstone II: Functoriality principle . 13 2 The Arthur-Selberg trace formula 14 2.1 A first look . 14 2.2 Coarse expansion . 15 2.3 Jacquet Langlands correspondence . 17 2.4 Refinement . 20 2.5 Invariance . 21 2.6 The unramified case . 22 3 Base change 23 3.1 GL(2) base change . 23 3.2 Aside I: Weil restriction of scalars . 26 3.3 Aside II: Chevalley and Satake Isomorphism. 26 3.4 Aside III: Conjugacy classes in reductive groups . 27 3.5 Noninvariant base change for GL(n) . 27 3.6 Aside IV: L-group exercise . 30 1 4 Endoscopy theory 31 4.1 Warmup on group cohomology . 31 4.2 Endoscopic character . 32 4.3 Rational representative . 32 4.4 Endoscopic group . 33 4.5 Stable distributions . 34 5 Fundamental lemma 35 5.1 Transfer to characteristic p .................................. 35 5.2 A first example: SO5 and Sp4 ................................. 36 5.3 Examples of GLn Hitchin fibres . 36 5.4 An extended example: GLn .................................. 36 5.5 Another example: SL2 ..................................... 40 5.6 From GLn to reductive groups . 41 5.7 Aside I: some geometric representation theory . 43 5.8 Cohomology of the Hitchin base . 44 5.9 Aside II: Cohomological definitions . 47 6 Beyond Endoscopy 51 6.1 Introduction (Altuˇg) . 51 6.2 Endoscopy and Beyond . 53 6.3 The test case: GL2 ....................................... 55 6.4 The elliptic terms . 56 6.5 Geometrization: Frenkel-Langlands-Ng^o . 61 6.6 Aside: Poisson summation . 65 6.7 The story so far (Altuˇg) . 66 6.8 The r-trace formula . 67 6.9 Singularities and transfer . 69 7 Galois Representations 72 7.1 What are they . 72 7.2 L-functions for Galois representations . 73 7.3 Automorphy . 75 7.4 An infinite fern . 77 7.5 The eigencurve . 79 7.6 p-adic Galois representations and p-adic differential equations (Kramer-Miller) . 80 8 Motives 84 8.1 Grothendieck motives . 84 8.2 The Standard Conjectures (Ray Hoobler) . 85 8.3 Aside I: Fundamental groups and Galois groups . 88 8.4 Aside II: Tannakian categories . 90 8.5 The motivic Galois group: examples . 95 8.6 The Taniyama group . 96 8.7 The automorphic Langlands group . 99 8.8 Motivic homotopy theory . 103 9 Reciprocity 105 9.1 A historical overview . 105 9.2 Explicit reciprocity laws and the BSD conjecture (Florez) . 107 9.3 Non-abelian reciprocity laws on Riemann surfaces (Horozov) . 108 9.4 An n-dimensional Langlands correspondence . 110 9.5 A survey of the relationship between algebraic K-theory and L-functions (Glasman) . 112 2 9.6 The Stark conjectures . 115 9.7 Semistable abelian surfaces of given conductor (Kramer) . 118 10 Stable homotopy theory 120 10.1 Manifolds and modular forms . 120 10.2 Stable homotopy groups of spheres . 122 10.3 The Adams conjecture . 125 10.4 Chromatic homotopy theory . 127 10.5 Topological modular forms . 129 11 Appendix 130 11.1 Structure theory of algebraic groups . 130 12 Some references 130 3 1 Basic objects 1.1 Automorphic Representations 1.1. We begin with the classical theory of theta series: 1 X 2 θ(z) = eπin z n=−∞ with functional equation θ(−1=z) = (−iz)1=2θ(z), proven by the Poisson summation. Riemann used this to prove the functional equation for Riemann's zeta function: 1 X 1 Y 1 ζ(s) = = : ns 1 − p−s n=1 p Riemann showed that the completed zeta function ξ(s) is a Mellin transform: Z 1 − s s θ(it) − 1 s−1 ξ(s) = π 2 Γ( )ζ(s) = t dt 2 0 2 and using the functional equation of θ(z), the functional equation for ζ(s) follows: ξ(s) = ξ(1 − s). 1.2. Next we define a modular form of weight k for G = SL(2; R) with discrete subgroup Γ = SL(2; Z) or some congruence subgroup to be a holomorphic f(z) on the upper-half plane H with transformation law az + d a b f(γ(z)) = f = (cz + d)kf(z); γ = 2 Γ: cz + d c d Thus we can think of modular forms as representations of SL(2; Z)nSL(2; R) by lifting f(z) to φf (g), requiring φf to be left-invariant under SL(2; Z), right-SO(2; R) finite, an eigenfunction of the Casimir operator, and satisfying a certain slow-growth condition. More generally, an automorphic form of weight k for a topological group G with discrete subgroup Γ to be a function f(γ(z)) = j(γ; z)kf(z) where j(γ; z) is called a factor of automorphy, satisfying a similar growth, eigenfunction and invariance conditions. Denote A the space of automorphic forms. 1.3. Hecke generalized Riemann's proof as follows: let fang be a sequence of complex numbers with c an = O(n ) for c > 0, and let h > 0; k > 0; λ > 0;C = ±1, and define 1 1 X X an λ f(z) = a e2πinz=h; φ(s) = ; f(z); and Φ(s) = ( )sΓ(s)φ(s) n ns 2π n=0 n=1 Theorem 1.1.1. The following are equivalent: (i) Φ + a0=s + C=(k − s) is entire, bounded on vertical strips, and satisfies Φ(k − s) = CΦ(s) (ii) f(−1=z) = C(z=i)kf(z), i.e., f(z) is an automorphic form of weight k. P −s Q The next question asks when a Dirichlet series ann has an Euler product p Lp(s). Hecke showed that for Hecke operator Tp acting on the space A, we have the following P 2πinz Theorem 1.1.2. Assume, for convenience, that f(z) = ane is a modular form for SL(2; Z) and a1 = 1. Then f is an eigenfunction of Tp for all p, i.e., k−1 X k Tpf(z) = p j(γ; z) f(γ(z)) = apf(z) ΓnMm if and only if the an are multiplicative, i.e., if amn = aman whenever m; n are relatively prime. 4 As a corollary of the multiplicativity of the an, φ(s) will have an Euler product as follows: Y Y −s k−1−2s −1 φ(s) = Lp(s) = (1 − app + p ) p p 1.4. Tate, in a different setting, proved the functional equation for a Hecke L-function using adelic Q0 methods. Recall the adele ring of a number field F to be AF = v Fv, where Fv is a local field completed at the prime v, and product is the restricted direct product, where almost all factors are Ov, Q0 the ring of integers of Fv. In particular, is F = Q we have AQ = R × Qp, where p are finite primes. This analysis follows in the spirit of Hasse's local-global principle. × Now fix a Hecke character χ : F ×nA ! C× (a one dimensional representation of GL(1; A)) such Q F that χ(x) = χv(xv) where χv is trivial on units (unramified) for almost all v. Also define a Schwartz- Q Bruhat function f(x) = fv(xv) where fv is Schwartz if v is an infinite prime, and locally constant compactly supported when v is finite. In summary, Tate defined a local zeta function Z s−1 ζ(fv; χv; s) = fv(x)χv(x)jxj dx × Fv and using Poisson summation and an appropriately defined Fourier transform for functions on number ^ −1 fields, proved a functional equation and analytic continuation ζ(fv; χv; s) = γ(χv; s)ζ(fv; χv ; 1 − s) where γ(χv; s) (local gamma factor) is a meromorphic function of s. Then considering the global zeta Q function ζ(f; χ, s) = ζ(fv; χv; s) Tate showed that Hecke's L-series Y Y −s −1 L(s; χ) = L(s; χv) = (1 − χv(~!)N(p) ) v v unram has an analytic continuation and functional equation L(χ, s) = "(s; χ)L(1 − s; χ−1) with "(s; χ) = Q "(s; χv) and "(s; χv) = 1 when χv is unramified. 1.5. Now we define an automorphic representation by making A into a Hecke algebra H module as follows. For each finite place v let Hv be the convolution algebra of complex valued locally constant functions compactly supported onG(Fv), with Haar measure normalized to give G(Ov) measure 1, so that the characteristic function of G(Ov) is an idempotent Iv in Hv. Now form the restricted tensor product Hf of Hv with respect to Iv for all finite places v, by taking product functions that are Iv at almost.
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