(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec Z Q Spec Z 3 Z?? 4 5 Hausdorff : variety, scheme, algebraic space, algebraic stack,. 6 R 2 K C(x) R 2 g 2, g 3 K C(x, 4x 3 g 2 x g 3 ) genus 0 genus 1 genus Riemann Riemann genus 2 1
R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q 13 14 Riemann 15 16 17 K/Q Riemann 18 Riemann K/Q K ( 7 N P N (C) CP N (^_^;) 8 9 [ ] 10 compact Riemann 11 F (field F ) K (Körper K) (^_^;) f Körper K? 12 C/k C over k C/k C Spec k 13 F q q q p 14 15 Kähler 16 17 F q 18 L/K Riemann Hilbert Riemann Abel Riemann Abel Artin reciprocity map Frobenius map Riemann [Serre] 2
) K K p K Q p = 2, 3, 5, 7, 11, Q Q p Q R 19 p Q p Laurent C((ξ p )) = { } i a i ξ p a i C for any i Z and a i = 0 for i 0. i Z 20 : Q p = { a i p i ai = 0, 1,, p 1 for any i Z and a i = 0 for i 0. }. i Z Z Q p Z p Z p C[[ξ p ]] Q genus 0 Riemann P 1 (C) = C { } C(x) p C C(x) C((ξ p )) ξ p = x p Laurent Laurent Q p Q Q p C(x) C((ξ )) ξ = x 1 Laurent K/Q K K R C K o K p K p Riemann p O R,p ( ) m K = C(R) m K p O R,p m Ôp 21 Riemann 22 Riemann Spec Z? :? 19 p p Q p a a = p n b/c (n, b, c Z b,c p ) a p := p n a = 0 a p := 0 p p d p (a, a ) := a a p (a, a Q) Q Q d p Q Q Q p Q p p Q p p Q Q p p 20 ξ p 21 p ξ p ξ p (p) = 0 K p C((ξ p )), Ô p C[[ξ p ]] K p m C locally linearly compact K p K p ( Q p Q p ) R C locally compact [Lefschetz] 22 ( ) 3
Riemann Riemann Q 23 3 Riemann 24 G = SL 2 (C) 25 Riemann R Wess-Zumino-Witten model (WZW ) 26 G = SL 2 (C) 1 2 Lie Lie 27 g = sl 2 (C) = Lie G trace 0 2 [A, B] := AB BA Lie g (X Y ) := Tr(XY ) for X, Y g g basis E, F, H : (3.1) E := [ ] 0 1, F := 0 0 [ ] [ ] 0 0 1 0, H :=. 1 0 0 1 n + := CE, n := CF, h := CH h g Cartan subalgebra (maximal abelian subalgebra) n ± g maximal nilpotent subalgebra h dual space h g weight k P +, P k : (3.2) P + := { λ h λ(h) Z 0 }, P k := { λ P + λ(h) k }. P k level k dominant integral weight 28 23 [Ohio] 24 chiral theory non-chiral theory 2 chiral theory holomorphic part non-chiral theory holomorphic part anti-holomophic part chiral theory non-chiral theory mirror symmetry super conformal field theory non-chiral theory chiral theory 25 SL n (C) SL(n; C) SL(n, C) (^_^;) 26 G C Lie Riemann trivial group bundle R G R Riemann C Lie G C ( ) G/C WZW [TUY] r Riemann non-trivial group bundle 27 Lie the Virasoro algebra Virasoro Virasoro Lie? 28 Lie g (θ θ) = 2 (θ g highest root) P k := { λ P + (λ θ) k } 4
Lie G = SL 2 (C) WZW conformal block 29 conformal block WZW conformal block data : (3.3) (3.4) (3.5) Riemann R p 1,, p N R; k; level k dominant integtral weights λ 1,, λ N P k. Riemann Lie 30 affine Lie ĝ (3.6) ĝ := g C C((ξ)) C k Lie (3.7) [X f + a k, Y g + b k] := [X, Y ] fg + (X Y ) Res(df g) k ξ=0 31 X, Y g, f, g C((ξ)), a, b C, df(ξ) := f (ξ)dξ ( ξ ) Res ξ=0 : (3.8) Res ( a i ξ i dξ) := a 1 ξ=0 i Z (a i C). C((ξ)) Laurent C[ξ, ξ 1 ] ĝ A (1) 1 Kac-Moody Lie 32 Kac-Moody Lie 33 g C[[ξ]] ĝ Lie subalgebra X g X 1 ĝ g ĝ Lie subalgebra λ P k (3.9) (3.10) (g ξc[[ξ]] + n + )v k,λ = 0, Hv k,λ = λ(h)v k,λ ( ); (F ξ 1 ) k λ(h)+1 v k,λ = 0 ( ). 29 chiral theory non-chiral theory holomorphic part antiholomorphic part conformal blocks conformal block? 30 C ( 0 ) Lie 31 Res ξ=0 (df g) C((ξ)) inifinitesimal tame symbol [Garland] 12 32 g X n ĝ X (1) n [Kac] 33 highest weight integral representations supercuspidal Weil ŝp n boson 5
v k,λ 0 ĝ ( ) 34 L k,λ L k,λ ĝ 35 Riemann R p affine Lie ĝ p p Riemann ξ p ξ p (p) = 0 Riemann R f K = C(R) Laurent f(ξ) K p = C((ξ p )) K K p ĝ ξ ξ p Lie ĝ p = g K p C k g Ôp = g C[[ξ p ]] ĝ p Lie subalgebra affine Lie ĝ p p R affine Lie ĝ R (adelic affine Lie algebra) K = C(R) A R : (3.13) A R := p R K p := { (f p ) p R p R K p p fp Ôp }. A R 36 f K p R Laurent f p K p f (f p ) p R K A R 37 ĝ R vector space (3.14) ĝ R := g A R C k Lie (3.15) [X f + a k, Y g + b k] := [X, Y ] fg + (X Y ) Res R (df g) k X, Y g, f = (f p ), g = (g p ) A R, a, b C, df := (f p(ξ p ) dξ p ) 34 g Lie g lowest root θ root vector F θ : (3.11) (3.12) (g ξc[[ξ]] + n + )v k,λ = 0, Hv k,λ = λ(h)v k,λ for H h; (F θ ξ 1 ) k (λ θ)+1 v k,λ = 0. g C[[ξ]] g ξc[[ξ]] + h + n + p maximal compact subgroup Iwahori subgroup affine Lie 35 Affine Lie 2 : (1) loop algebra, (2) Kac-Moody Lie (2) Lie L k,λ Lie 36 A R C linearly locally comact AR Riemann A R R C 37 K A R Riemann R R C R 6
( ξ p ) Res R : (3.16) Res R ω := Res ω p for ω = (ω p ) Ω 1 A := K p dξ p. ξ p R p =0 p R well-defined affine Lie ĝ R ĝ p ĝ R Lie subalgebra g K := g(k) := g K 38 K A R g K ĝ R Lie conformal block data data ĝ R (3.3), (3.4), (3.5) λ = (λ(p)) p R : (3.17) λ(p i ) := λ i for i = 1,, N, λ(p) := 0 for p R {p 1,, p N }. L k,λ : (3.18) L k,λ := L k,λ(p) p R := { ϕ = (ϕ p ) p R L k,λ(p) p ϕp = v k,λ(p) }. L k,λ ĝ R 39 L k,λ ĝ R conformal block A R ĝ R Riemann R K A R g K ĝ R R g K ĝ R g K = g K L k,λ dual space L k,λ 40 Definition 3.1 L k,λ g K conformal block conformal blocks : (3.19) CB(R, k, λ) := CB(R, p 1,, p N ; k, λ 1,, λ N ) := [ L k,λ ] gk := { Φ L k,λ Φ(Aϕ) = 0 for ϕ L k,λ and A g K }. 38 Lie 39 L k,λ ĝ R 40 V = L k,λ dual space V := Hom C (V, C) 7
WZW R = P 1 (C), g = sl 2 (C) [TK] 41 [TUY] Riemann conformal blocks 42 (g Lie ) [TUY] CB(R, k, λ) N-pointed curve (R, p 1,, p N ) 43 N-pointed stable curves family vector bundle projectively flat regular connection conformal blocks conformal blocks 44 Example 3.2 ([Verlinde], [MS]) g = sl 2 (C) ĝ level k characters k + 1 modular τ 1/τ : (3.20) S λ,µ = ( ) 2 1/2 (λ(h) + 1)(µ(H) + 1) sin k + 2 k + 2 for λ, µ P k. Riemann R genus g conformal blocks R R N p 1,, p N : (3.21) dim CB(R, k, λ) = µ P k ( 1 S 0,µ ) 2g 2 N i=1 S λi,µ S 0,µ. Verlinde formula 45 46 G = GL 2 G Z (Z G m ) G, G 1 Q A := A Q := R Q p G A := G(A) = p: 41 [TK] conformal block vertex operator 42 [TUY] conformal block vacuum 43 stable curve 44 [TUY] 45 Verlinde conjecture N = 0 [Verlinde] (3.15) [MS] (A.7) [MS] Verlinde formula WZW [MS] [TUY] R parabolic bundles moduli line bundle global sections Verlinde formula global section conformal block conformal blocks non-trivial [Oxford] G. Segal Seminar 4 46 WZW 8
GL 2 (A) G G Q := G(Q) 47 G R := G(R), G Qp := G(Q p ) G A G A = G R G Qp p: diagonal embedding Q A G Q G A G A Haar 48 Z G m Q A = A Q (3.13) A R G A Lie (3.14), (3.15) ĝ R 49 π G A π, π p G R, G Qp π G A 50 π G R, G Qp π, π p 51 : (3.22) π π p: π p. π π, π p (3.18) L k,λ, L k,λ(p) 52 4 N SL 2 (Z) SL(Z/NZ) f N 53 SL 2 (Z/NZ) U N, B N ( ) (4.1) U N := { [ ] } 1 b 0 1 B N := { [ ] } a b 0 d SL 2 (Z/NZ) 47 C X C A X A-rational point set A X(A) 48 Haar A Q Riemann ( Feynman ) 49 G A ĝ R Lie ĝ R G A Lie C G A [ ], [Moore], [PR] 50 (admissible representation) 51 [Flath] 52 (3.18) 53 f N trivial f N Chevalley [Moore] Chapter IV 9
SL 2 (Z) Γ(N), Γ 1 (N), Γ 0 (N) : (4.2) Γ(N) := Ker f N Γ 1 (N) := f 1 N (U N ) Γ 0 (N) := f 1 N (B N ). Γ(N) SL 2 (Z) SL 2 (Z) (congruence subgroup) Γ i (N) Γ SL 2 (Z) G + R := { g G R det g > 0 } H := { τ C Im τ > 0 } : (4.3) gτ := aτ + b cτ + d [ ] a b ( H) for τ H, g = G + c d R. Γ H Y (Γ) := Γ\H Y (Γ) compact Riemann X(Γ) (Y (Γ) ) X(Γ) cusp Example 4.1 (Γ = SL 2 (Z)) : Y (1) = SL 2 (Z)\H X(1) (4.4) Y (1) P 1 (C) {pt}, X(1) P 1 (C). Y (1) τ H E τ := C/Z + τz Y (1) Example 4.2 (Γ = Γ 0 (N)) Y 0 (N) = Γ 0 (N)\H E N c (E, c) τ H E τ 1 Z ( C) N c τ E τ N τ (E τ, c τ ) Y 0 (N) (E, c) 54 (E, c) (E, c ) E E c c Y 0 (N) X 0 (N) Y (Γ) (+ α) Riemann (rank 2 ) Γ 0 (N) (Z/NZ) character ψ (Z/NZ) 0 Z Z/NZ ψ ψ ψ modulo N Dirichlet character k g = G + [ ] a b c d R, τ H j(g, τ) := (cτ + d)(det g) 1/2 54 Y (N) := Γ(N)\H Y 1 (N) := Γ 1 (N)\H + α [Silverman] Appendix C 10
Definition 4.3 H f level N, weight k, character ψ : [ ] a b (4.5) f(γτ)j(γ, τ) k ψ(a) 1 = f(τ) for γ = Γ 0 (N); c d (4.6) f X 0 (N) cusp ( ) f f : (4.7) f X 0 (N) cusp 0 M k (N, ψ), S k (N, ψ) (d(γτ)) k/2 = j(γ, τ) k (dτ) k/2 k ψ = 1 M k (N, 1) X 0 (N) k/2 3 Lemma 4.4 ( ) G Q G + R G A G A K 0 (N) : { [ ] } a b (4.8) K p := GL 2 (Z p ) c d c 0 mod N, (4.9) K 0(N) := p: K p, K 0 (N) := SO(2) K 0(N). G + RK 0(N) G A Lemma G A = G Q G + RK 0(N) G Q G + RK 0(N) = Γ 0 (N) : (4.10) Z A G Q \G A /K 0 (N) Γ 0 (N)\H = Y 0 (N). Ẑ := p: Z p A = Q R +Ẑ A Ẑ (Z/NZ) Dirichlet character ψ Q grossencharacter ψ K 0 (N) character χ : (4.11) (4.12) χ(k 0) := ψ(a ) [ a for k 0 b ] = K c d 0(N); χ(r θ ) := e ikθ [ ] cos θ sin θ for r θ = SO(2). sin θ cos θ 11
f S k (N, ψ) G A ϕ f : (4.13) ϕ f (γg k 0) := f(g i)j(g, i) k χ(k 0). for γ G Q, g G + R, k 0 K 0(N). sl 2 (R) Casimir operator Proposition 4.5 f ϕ f S k (N, ψ) G A ϕ 55 : (4.14) (4.15) (4.16) (4.17) (4.18) ϕ(zγg) = ϕ(g)ψ(z) for γ G Q, z Z A ; ϕ(gk 0 ) = χ(k 0 )ϕ(g) for k 0 K 0 (N); ϕ = k ( ) k 2 2 1 ϕ; ϕ slowly increasing ; ([ ] ) 1 x ϕ g dx = 0 ( ). Q\A 0 1 : (4.19) GL r (K)\ GL r (A R )/ GL r (O R ) { [E] E R rank r }. K Riemann R A R (3.13) R O R := p R C[[ξ p ]] A R [E] E GL r SL r c 1 (E) = 0 r = 2 Example 4.1 (+ (4.10)) 56 57 ( ) g A R ĝ R ĜR := SL 2 (A R ) C G K := SL 2 (K), G(O R ) := SL 2 (O R ) ĜR ĝ R L k,λ ĜR S := { p 1,, p N } R G(O R ) I : (4.20) (4.21) (4.22) { [ ] } a b I p := SL 2 (C[[ξ p ]]) c d c 0 mod (ξ p) for p S, I p := SL 2 (C[[ξ p ]]) for p / R S; I := p R I p. 55 [Gelbart] 56 (4.19) 57 (4.19) quotient stack [BL] 12
I (4.9) K 0 (N) Example 4.2 (+ (4.10)) : (4.23) M := SL 2 (K)\ SL 2 (A R )/I { [(E, c)] E c = (c p ) p S }, (4.24) (4.25) E R rank 2 c 1 (E) = 0 ; p S c p fiber E p 1 [(E, c)] (E, c) (E, c) quasi parabolic vector bundle Î := I C k λ Î 1 C k,λ C k,λ 58 F := SL 2 (A R )/I ĜR/Î M line bundles L k,λ, L k,λ L k,λ global sections H 0 (F, L k,λ ) L k,λ (4.26) CB(R, k, λ) [ L k,λ] gk [ L k,λ] GK [ H 0 (F, L k,λ ) ] G K H 0 (M, L k,λ ). conformal block quasi parabolic bundles line bundle L k,λ global section 59 [ ] :,, 1973, 380pp. [ ] : metaplectic, 14, 1968 12,, 103 126. [BL] Beauville, A. and Laszlo, Y.: Conformal blocks and generalized theta functions, preprint 1993, alg-geom.9309003. [Flath] Flath, D.: Decomposition of representations into tensor products, Proc. Symp. Pure Math. 33, part 1, 1979, 179 183. [Garland] Garland, H.: The arithmetic theory of loop groups, Publ. Math. IHES 52, 1980, 5-136. 58 N = 1 [BL] 59 Hecke Langlands program 13
[Gelbart] Gelbart, S. S.: Automorphic forms on adele groups, Ann. Math. Stud. 83, 1975, 267pp. [Kac] Kac, V.: Infinite dimensional Lie algebra, Third edition, Cambridge University Press, 1990, 400pp. [Moore] Moore, C. C.: Group extensions of p-adic and adelic linear groups, Publ. Math. IHES 35, 1968, 5 70. [Lefschetz] Lefschetz, L.: Algebraic topology, Amer. Math. Soc. Colloq. Publ., 1942. [MS] Moore, G. and Seiberg, N.: Classical and quantum conformal field theory, Commun. Math. Phys. 123, 1989, 177 254. [Ohio] The arithmetic of function fields, Proc. of the Workshop at The Ohio State Univ. June 17 26, 1991, edited by D. Goss, D. R. Hayes, and M. I. Rosen, Ohio State Univ. Math. Res. Inst. Publ. 2, Walter de Gruyter, 1992, 482pp. [Oxford] Oxford Seminar on Jones-Witten Theory, preprint 1988, 122pp. [PR] Prasad, G. and Raghunathan, M. S.: Topological central extension of semi-simple over local fields I, II, Ann. Math. 119, 1984, 143-201, 203-268. [Serre] Serre, J.-P.: Algebraic groups and class fields, Translation of the French edition, Graduate Texts in Mathematics 117, Springer-Verlag, 1975, 207pp. [Silverman] Silverman, J. H.: The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, 1986, 400pp. [TK] Tsuchiya, A. and Kanie, Y.: Vertex operators in conformal field theory on P 1 and monodromy representations of braid group, Adv. Stud. Pure Math. 16, 1988, 297 372. [TUY] Tsuchiya, A., Ueno, K., and Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure Math. 19, 1989, 459 566. [Verlinde] Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B300 [FS22], 1988, 360 376. 14