Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets pr
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1 Langlands 1 1. Langlands p GL n Langlands [HT] The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets proved. ([HT], p.1) hydra [KP] Langlands Langlands Langlands G G = GL n Langlands 1 tetsushi@math.kyoto-u.ac.jp 1
2 Langlands Langlands 2 Langlands 2.1 G GL n U 2, U 3 [ ], [ ] G Langlands 3 (inner form) Langlands-Vogan Kottwitz, 3.1 Langlands 4 GL n (F q ) Deligne-Lusztig 5 Rapoport-Zink 6 Rapoport-Zink Kottwitz Rapoport-Zink Lubin-Tate Drinfeld L [RZ], [Ra], [Far1], [SW] Langlands 2. p Langlands Langlands Galois Weil Weil-Deligne Galois Langlands p F q ((t)) p R C Langlands Vogan [Vog] p F p Q p F p p F Galois Γ F := Gal(F /F ) F Galois G F Langlands G(F ) G F F Galois Γ F F F q Frob q Γ Fq := Gal(F q /F q ) 2
3 Frobenius Frob q Frob q (x) = x 1/q (x F q ) Γ Fq Γ F Γ Fq Frob i q Γ Fq U i Γ F I F := U 0 F W F := i Z U i F Weil U i Γ F U i W F W F I F = U 0 W F W F Artin Art F : F = W ab F U 1 W F W F [W F, W F ] G F G F - G(F ) G(F ) G(F ) p l mod p mod l Langlands C (admissible) [ ] Art F 1 1 1:1 GL 1 (F ) π ϕ: W F C / = : ϕ ϕ WF ab C GL 1 (F ) = F Art F WF ab ϕ C π π := ϕ Art F GL 1 (F ) = F GL 1 (F ) 1 G G(F ) W F Langlands : G(F ) π W F (Galois ) ϕ Langlands L G Ĝ G C C- [ ] [Bor], [Cog] Ĝ W F G L L G := Ĝ W F L G 1 Ĝ L G W F 1 ϕ: W F SL 2 (C) L G 4 ϕ G L Langlands ϕ(sl 2 (C)) Ĝ. ϕ SL 2 (C) ϕ SL2 (C) : SL 2 (C) Ĝ C (id,1) ϕ W F W F SL 2 (C) L G pr 2 W F pr i i Frobenius σ W F (pr 1 ϕ)(σ) Ĝ 3
4 L ϕ, ϕ Ĝ- gϕg 1 = ϕ g Ĝ G = GL n Ĝ = GL n(c), L G = GL n (C) W F GL n L W F SL 2 (C) SL 2 (C) W F 1 1 G = GL 1 Ĝ = C ϕ SL2 (C) : SL 2 (C) C C GL 1 L ϕ: W F C 1 1 Langlands 2.1 ( Langlands ). LLC G : G(F ) π G L ϕ / = /Ĝ- G(F ) L Ĝ- LLC G (π) π L Langlands ϕ Π G ϕ := LLC 1 G (ϕ) L G LLC G G LLC G [Bor] LLC G LLC G LLC G 2.1 LLC G (1) GL 1 LLC GL1 Langlands (2) GL n Zelevinsky π L ε n Zelevinsky [ ] GL n Langlands [ ] LLC GLn [He1] [He3] n LLC GLn [HT] [He2] [LRS] GL n Langlands LLC GLn [Laf] Scholze LLC GLn GL n Langlands [Scho] [Scho] LLC GLn [HT], [He2] LLC GLn (3) G GL n G(F ) GL n (F ) LLC G G Sp 2n SO n U n LLC G [MT], [Mœ] [MT], [Mœ] LLC G 4
5 G twisted endoscopy LLC G U 3 [Ro] [ ] Sp 2n, SO n [A3] U n [Mo] SO 2n [A4] G(F ) π LLC G G = SL n SL n GL n GL n G LLC SLn [LL], [HS] G = Sp 4 GSp 4 G(F ) GL 2 (F ) GL 4 (F ) LLC Sp4 LLC GSp4 [GT1], [GT2] (4) LLC G [DR] 0 LLC G Deligne-Lusztig Bruhat-Tits GL 2 0 [ ] LLC G 3. Langlands-Vogan Langlands G G (inner form) Langlands Vogan Langlands-Vogan [Vog] 2.1 Langlands G p F G F (quasi-split) 2.1 LLC G : G(F ) L Π G = ϕ G L ϕ /Ĝ- J G (inner form) G = GL n J = GL m (D) m n D F dim F D = n 2 /m 2 Ĝ = Ĵ, L G = L J G L J L J 2.1 LLC J : J(F ) L Π J ϕ G L ϕ /Ĝ- J J G = J LLC J 2 LLC 1 G LLC J : J(F ) L Π J ϕ G(F ) L Π G ϕ Jacquet-Langlands endoscopy endoscopic transfer LLC G, LLC J 5
6 endoscopic transfer [ ] L ϕ Π G ϕ, ΠJ ϕ L (endoscope) endoscopy Arthur ϕ S Sϕ Arthur ( := π ) 0 CentĜ(Im ϕ)/z(ĝ)w F L Arthur [A1] L ϕ: W F SL 2 (C) L G Im ϕ L G Ĝ L G CentĜ(Im ϕ) := g Ĝ x Im ϕ, gx = xg CentĜ(Im ϕ) Ĝ G L L G = Ĝ W F W F Ĝ W F Ĝ Z(Ĝ) Z(Ĝ)W F := g Z(Ĝ) w WF, wgw 1 = g L G Z(Ĝ)W F Ĝ H := Cent Ĝ (Im ϕ)/z(ĝ)w F C H H 0 Arthue S Sϕ Arthur π 0 (H) ) Arthur Π G ϕ SArthur = π 0 (H) = H/H 0 H H ϕ 1 1 : Sϕ Arthur 1:1 (?) Π G ϕ G = GL n Schur S ϕ L 1 [HT], [He2], [Scho] U 3 Arthur [ ] G Whittaker 1 [Vog], [GGP] Vogan J L G (pure inner form) Arthur S S 1 1 S Vogan ϕ / = S Vogan ϕ := π 0 ( CentĜ(Im ϕ) ) / = 1:1 (?) J G Vogan [Vog], [GGP] Kottwitz Kottwitz Hasse-Weil [Ko2] Kottwitz [Ra], [Kal] Langlands-Vogan Kottwitz G Z(G) Ĝ Ĝder := [Ĝ, Ĝ] Kottwitz 1 1 Z(Ĝder) W F C 1:1 H 1( Γ F, G ad (F ) ) = J J G 6 Π J ϕ / =
7 Kottwitz [Ko1, Proposition 6.4] G ad := G/Z(G) G χ: Z(Ĝ)W F C χ Z(Ĝder) W F Z(Ĝ)W F χ: Z(Ĝ)W F C H 1( Γ F, G ad (F ) ) = J J G χ G J χ Z(Ĝder) W F / = Z(Ĝ)W F χ J χ G = GL n 3.1 ( Langlands-Vogan Kottwitz ). G Z(G) CentĜ(Im ϕ)/z(ĝ)w F ϕ (elliptic) χ: Z(Ĝ)W F C LLC ϕ,χ : ρ : CentĜ(Im ϕ) ρ = χ 1:1 Π Jχ Z(Ĝ)W F ϕ χ ρ Sϕ Arthur 3.1 Arthur 3.1 Langlands 2.1 Arthur 3.1 CentĜ(Im ϕ) Sϕ Arthur, S Vogan ϕ 5, 6 Rapoport-Zink Langlands [HS] SL n L G Ĝder [A2] 3.3. ρ π Π Jχ ϕ Langlands-Vogan J χ (1) ϕ (2) L Π J χ ϕ (3) L Π J χ ϕ ϕ (1) L SL 2 (C) ϕ SL2 (C) (2) L Π J χ ϕ ϕ SL2 (C) ρ π Mœglin, Tadić G ρ π Langlands [MT], [Mœ] 7 / =
8 4. GL n (F q ) Langlands (toy model) GL n (F q ) q F q q GL n (F q ) GL 2 (F q ) SL 2 (F q ) 100 Frobenius, H. Jordan, Schur [Jo], [Schu] Green 60 GL n (F q ) [Gr1] Green p [ ] n = n n r (r 2) n M n1,...,n r := GL n1 GL nr GL n M n1,...,n r P n1,...,n r GL n P n1,...,n r GL n i = 1,..., r (ρ i, V i ) GL ni (F q ) P n1,...,n r (F q ) M n1,...,n r (F q ) = GL n1 (F q ) GL nr (F q ) ρ 1 ρ r GL(V1 V r ) ρ Ind GL n(f q ) P n1,...,nr (F q) ρ (ρ 1,..., ρ r ) GL n (F q ) GL n (F q ) θ : F q n C θ, θ q,..., θ qn 1 θ θ, θ θ = θ qi i 1 θ θ x F q n xqn = x θ qn = θ θ Green GL n (F q ) 4.1 (Green ([Gr1])). 1:1 GL n (F q ) θ : F qn C / = GL n (F q ) θ GL n (F q ) π θ Langlands 2.1 Harish-Chandra Lie θ L 4.1 GL n (F q ) Langlands θ π θ [ ] GL 2 (F q ) Green 4.1 Brauer π θ Deligne-Lusztig GL n (F q ) [DL] Deligne-Lusztig U n Tate-Thompson [Ta, pp. 102] SL 2 Drinfeld GL n F q G 8 /
9 G(F q ) GL n (F q ) X 1 DL := v = F n q det ( ) q 1 X qj 1 i = 1. X n det(x qj 1 i ) (i, j) X qj 1 i n DL F q (n 1) GL n Deligne-Lusztig DL F q - DL q = 2 n = 2 DL q 1 det(x qj 1 i ) = det ( X1 X q 1 X 2 X q 2 ) q 1 = ( X 1 X q 2 Xq 1 X 2) q 1 = 1 Drinfeld SL 2 Deligne-Lusztig XY q X q Y = 1 (q 1) GL n (F q ) F qn DL (g, α) GL n (F q ) F qn v DL (g, α) v = g αx 1. αx n v GL n (F q ) F q n (g, α) v DL l H i c( DL, Ql ) GLn (F q ) F q n Q l π θ Deligne-Lusztig 4.2 (Deligne-Lusztig ([DL])). l q Q l = C GL n (F q ) Hom F q n ( ) Hc i ( ) π θ i = n 1 DL, Ql, θ = 0 i n 1 F qn 1 θ θ DL L θ π θ DL GL n (F q ) 4.2 DL Green π θ 4.3 (Green ([Gr1])). 4.1 θ π θ g GL n (F q ) g GL n (F q ) P g (T ) = det(t g) F q [T ] P g (T ) = Q(T ) e Q(T ) F q [T ] Q(T ) α F q d m = dim Fq d Ker ( g α id ) deg Q(T ) 1 Tr π θ (g) = ( 1) n 1 i=0 9 θ qi (α) m 1 j=1 (1 q j deg Q(T ) )
10 P g (T ) = Q(T ) e Q(T ) F q [T ] Tr π θ (g) = 0. GL 2 (F q ) 4.1, 4.3 [ ] 4.3 g Jordan α Jordan m Tr π θ (g) L θ m 1 (1 S j deg Q(T ) ) j=1 g Jordan Green S = q F q Deligne-Lusztig [Lu1], [Lu2] Green 4.1, 4.3 Deligne- Lusztig [DL] 4.1, 4.3 [DL] Green [Ma] [Lu1] Green [DL] GL n (F q ) [Gr1] 44 [Gr2] Lusztig [DL] G G(F q ) Deligne-Lusztig [ ] Green Deligne-Lusztig 4 Langlands Green Deligne-Lusztig 2 [DR] 0 Langlands-Vogan Deligne-Lusztig Bruhat-Tits [Kal] Langlands Rapoport-Zink 5 Rapoport-Zink Deligne- Lusztig [Yo1], [Hara], [Vol], [VW] Rapoport-Zink [Yo1], [ImT] Deligne-Lusztig Langlands Deligne-Lusztig Rapoport-Zink Deligne-Lusztig p Rapoport-Zink Deligne-Lusztig Rapoport-Zink Lubin-Tate Drinfeld 1970 [De], [Dr3], [Dr1], [Dr2] Deligne-Lusztig Drinfeld SL 2 Drinfeld Tate-Thompson [Ta, pp. 102] Tate G(F q ) Langlands [DL] Langlands 10
11 5. Rapoport-Zink Rapoport-Zink [RZ] [Ra], [Far1], [SW] (1) Rapoport-Zink Rapoport-Zink M (2) Rapoport-Zink Q p G J E/Q p (3) Rapoport-Zink M Hc( i ) M, Q l G(Qp ) J(Q p ) W E Langlands-Vogan Jacquet-Langlands Kottwitz G G Rapoport-Zink G Rapoport-Zink Deligne-Lusztig [RZ] Rapoport-Zink p p Rapoport-Zink p p p [DOR] p Langlands Rapoport-Zink Rapoport-Zink 9 ( F, B,, OB, V,,, b, µ, L ) F p B F : B B (xy) = y x ( x, y B), 2 = id Q p - O B B O F - V B, : V V Q p b B, v, w V bv, w = v, b w b G ( Qur ) p basic isocrystal Q ur p Q p p µ: G m G Q p Hodge isocrystal Hodge [RZ] F p O B p X 0 L V O B - lattice chain Rapoport-Zink parahoric [RZ] Rapoport-Zink EL PEL 9 PEL EL, 7 ( E, G, J, χb, r µ, X 0, λ ) 11
12 E µ Q p E G Q p PEL G(Q p ) = (x, a) GL B (V ) Q p v, w V, gv, gw = a v, w EL G(Q p ) = GL B (V ) Rapoport-Zink G J G b basic Rapoport-Zink J G Levi χ b : Z(Ĝ)W F C r µ : Ĝ GL(V µ) G C Ĝ r µ µ: G m G Hodge V µ C- X 0 F p p B p X 0, X 0 f Hom ( X 0, X 0) Zp Q p (quasi-isogeny) g f = id g Hom ( X 0, X 0) Zp Q p Q p - B ( End X ) Zp Q p X B PEL λ 0 : X 0 X0 X 0 : X 0 p X0 λ 0 : X 0 X0 λ 0 : ( X0 ) = X0 X0 λ 0 = λ 0 λ 0 p X 0 (quasi-polarization) Rapoport-Zink M Langlands p 2 ( 1) EL ( 2) PEL ( 3) PEL F Q p Q p Q p B Q p Q p Q p 2 O B Z p Z p B V Q r p Q 2r p B r+s, ( ) V V L p i Z r p i Z p i Q 2r p i Z p i O r+s B i Z E Q p Q p B (r s ) Q p (r = s ) G GL r GSp 2r GU r,s Rapoport-Zink b, µ, L Rapoport-Zink ( 1) Lubin-Tate [HT] GL n Langlands Rapoport- Zink ( 2) Siegel 12
13 [KO], [LO], [Hara] ( 3) [Vol], [VW], [Zh] p X 0 X 0 ( 1): X 0 F p r 1 p r = 1 X 0 = Gm [p ] r = 2 X 0 = Ess [p ] E ss F p ( 2): X 0 = (E ss [p ]) r λ 0 : X 0 X 0 E ss ( 3): X 0 = (E ss [p ]) (r+s) λ 0 : X 0 X0 ( 2) B ι: B ( ) End X0 Zp Q p x O B ι(x) Lie LieX 0 diag ( ϕ(x),..., ϕ(x), ϕ(x) p,..., ϕ(x) p ) r s ϕ: O B F p 2 mod p diag ϕ(x) r ϕ(x) p s G J B p X 0 X 0 X 0 B λ 0 Q p J(Q p ) := q-isog OB, Q p λ 0 ( X0 ) J(Q p ) p isocrystal J(Q p ) G Q p - Rapoport-Zink M p Spf O E ur M O E ur E ur M p M 0 0 Rapoport-Zink p p n n Rapoport-Zink M n M n M 0 p Rapoport-Zink M := lim M n n M n n 0 Rapoport-Zink M Êur p p p Rapoport-Zink p Berkovich adic (pre-)perfectoid (pre-)perfectoid [SW] M C p - M (C p ) C p E p M (C p ) p M M (C p ) 5 ( ) M (C p ) = X, ι, λ, ρ, η X, ι, λ, ρ, η X O Cp p O Cp C p ι: B ( End X ) Zp Q p Q p - Lie Kottwitz Hodge µ 13 /
14 PEL λ: X X ( ) ( ) ρ: X 0 OCp Fp /po Cp X OC OCp /po p Cp B λ0 λ Q p η : V = Vp X X : X Tate V p X := ( lim X[p s ](O s Cp ) ) Zp Q p EL B PEL λ V p X V p X V p X V p ( Gm [p ] ) =: Q p (1) 1 Q p Q p (1) = Q p V p X V p X Q p η V, V p X λ V p X Q p η η Q p (1) = Q p 5 (X, ι, λ, ρ, η), (X, ι, λ, ρ, η ) f : X X ι = f ι, λ = f λ f 1, ρ = ( f (mod p) ) ρ, η = f η G(Q p ) J(Q p ) W E Rapooprt-Zink l l p Hc( i ) M, Q l := lim H i ( ) n c Mn ÊurC p, Q l G(Q p ) J(Q p ) M C p - G(Q p ) V J(Q p ) X 0 (g, j) G(Q p ) J(Q p ) (g, j) ( X, ι, λ, ρ, η ) = ( X, ι, λ, ρ j 1, η g 1 ) W E I E W E I E = ΓÊur := Gal ( Êur /Êur) ΓÊur C p I E H i c( M ÊurC p, Q l ) Galois Γ E Weil descent Weil W E [RZ, p.100, 3.48], [Far1, p.71, 4.4] 6. Langlands Kottwitz Rapoport-Zink H i c( M, Q l ) G(Qp ) J(Q p ) W E Langlands Kottwitz l p Q l = C 5 Rapoport-Zink (F, B,, O B, V,,, b, µ, L) (E, G, J, χ b, r µ, X 0, λ) Rapoport-Zink M G G Z(G) G J Langlands 2.1 Langlands-Vogan Kottwitz 3.1 ϕ: W F SL 2 (C) L G G L 14
15 ϕ 3.1 SL 2 (C) ϕ SL2 (C) Ĝ L L G = Ĝ W F E W E Ĝ W E = Ĝ W E r µ : Ĝ GL(V µ) W E W E ϕ WE L G pr 1 Ĝ r µ GL(V µ ) Im(pr 1 ϕ WE ) CentĜ(Im ϕ) W E CentĜ(Im ϕ) V µ : W E CentĜ(Im ϕ) GL(V µ ) CentĜ(Im ϕ) ρ ( ) W (ρ) := Hom CentĜ(Im ϕ) ρ, Vµ V µ W E CentĜ(Im ϕ) W (ρ) W E τ Π J ϕ L ϕ J L ρ τ 3.1 τ CentĜ(Im ϕ) ρ τ Z( Ĝ) W F = χ b χ b Rapoport-Zink χ b G J 6.1 (Kottwitz ([Ra], Conjecture 5.1)). G(Q p ) W E ( ) Hom J(Qp) Hc i ( ) π W (ρ τ ρ π ) i = dim M M, Q l, τ = π G(Q p )-smooth 0 i dim M G(Q p )-smooth G(Q p ) π G L π Π G ϕ 3.1 π Cent (Im ϕ) Ĝ ρ τ Z( = 1 Ĝ) W G F 6.1 J(Q p ) τ Π J ϕ M G(Q p ) π Π G ϕ W E L ϕ r µ M Langlands 4.2 Deligne-Lusztig G Drinfeld Rapoport-Zink [Ra, Conjecture 5.1] i Deligne- Lusztig Lubin-Tate Boyer [Boy] GSp 4 GU 1,2 [ItM1], [ItM2], [It] L π Hdim M c
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18 [Laf] Lafforgue, L., Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, [LRS] Laumon, G., Rapoport, M., Stuhler, U., D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), no. 2, [LO] Li, K.-Z., Oort, F., Moduli of supersingular abelian varieties, Lecture Notes in Mathematics, Springer-Verlag, Berlin, [Lu1] Lusztig, G., The discrete series of GL n over a finite field, Annals of Mathematics Studies, No. 81. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, [Lu2] Lusztig, G., On the discrete series representations of the classical groups over finite fields, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp Canad. Math. Congress, Montreal, Que., [Ma] Macdonald, I. G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, [Mi1] Mieda, Y., Lefschetz trace formula and l-adic cohomology of Rapoport-Zink tower for GSp(4), preprint, arxiv: [Mi2] Mieda, Y., Zelevinsky involution and l-adic cohomology of the Rapoport-Zink tower, preprint, arxiv: [Mœ] Mœglin, C., Classification et changement de base pour les séries discrètes des groupes unitaires p-adiques, Pacific J. Math. 233 (2007), no. 1, [MT] Mœglin, C., Tadić, M., Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc. 15 (2002), no. 3, [Mœ] Mœglin, C., Classification et changement de base pour les séries discrètes des groupes unitaires p-adiques, Pacific J. Math. 233 (2007), no. 1, [Mo] [Ra] [RV] [RZ] [Ro] Mok, C.-P., Endoscopic classification of representations of quasi-split unitary groups, to appear in Memoirs of the American Mathematical Society. Rapoport, M., Non-Archimedean period domains, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), , Birkhäuser, Basel, Rapoport, M., Viehmann, E., Towards a theory of local Shimura varieties, preprint, arxiv: Rapoport, M., Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, 141. Princeton University Press, Princeton, NJ, Rogawski, J. D., Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, 123. Princeton University Press, Princeton, NJ, [Schu] Schur, I., Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 132 (1907), [Scho] Scholze, P., The local Langlands correspondence for GL n over p-adic fields, Invent. Math. 192 (2013), no. 3, [SW] [Ta] Scholze P., Weinstein, J., Moduli of p-divisible groups, preprint, arxiv: Tate, J. T., Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), , Harper & Row, New York [Vog] Vogan, D. A., Jr., The local Langlands conjecture, Representation theory of groups and algebras, , Contemp. Math., 145, Amer. Math. Soc., Providence, RI, [Vol] Vollaard, I., The supersingular locus of the Shimura variety for GU(1, s), Canad. J. Math. 62 (2010), no. 3, [VW] Vollaard, I., Wedhorn, T., The supersingular locus of the Shimura variety of GU(1, n 1) II, Invent. Math. 184 (2011), no. 3, [Yo1] Yoshida, T., Local class field theory via Lubin-Tate theory, Ann. Fac. Sci. Toulouse Math. (6) 17 (2008), no. 2, [Yo2] Yoshida, T., On non-abelian Lubin-Tate theory via vanishing cycles, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), , Adv. Stud. Pure Math., 58, Math. Soc. Japan, Tokyo, [Zh] Zhang, W., On arithmetic fundamental lemmas, Invent. Math. 188 (2012), no. 1, [ ], ( ),, 2004 [ ], ( ),. [ ], U 2(F ), U 3(F ) endoscopic description,. [ ], p,. [ ], GL n (F ),. [ ], GL n (F ) Langlands,. [ ], GL n (F ),. [ ],,. [ ], GL 2,.,,, 18
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