0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t

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1 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl type Euler, Jacobi η- (1 q n ) = n=1 (1 q n ) 3 = n=1 m= m= ( 1) m q 3m2 +m 2 ( 1) m (2m + 1)q m(m+1) 2. Typeset by AMS-TEX 1

2 3,8,10,14,15,21,24,... d MacDonald([M]) (1 q n ) d Weyl ; Weyl ( ) α + (1 e( α)) dim g α = w W( 1) l(w) e(wρ ρ) A l generalized Cartan g = g(a) Kac-Moody Lie e i, f i, g i W Weyl ( + ) e = l i=1 Zα i Z[[ξ 1,...ξ l ]] α i ξ i l(w) := #( + w ), ρ + ρ(h i ) = 1 (i = 1,.., l) Macdonald affine Kac-Moody Lie affine Kac-Moody Euler, Jacobi hyperbolic Kac-Moody Feingold-Frenkel ([FnF]) A = hyperbolic Euler, Jacobi, Macdonald 2πitr(P Z) e 0 N S 2 (Z) = (1 e 2πitr(NZ) ) mult(n) g P GL(2,Z) det(g)e 2πitr(gP t gz) N ν( R ) (1 e 2πitr(NZ) ) 2

3 Z P = ( ) 3 1/2, S 1/2 2 2 (Z) ν Cartan subalgebra h dual h S 2 (C) h {h i } h {α i } α i (h j ) = A ij h {γ i } γ 1 = α 1 γ 2 = α 1 α 2 α 3, γ 3 = α 1 α 2 h z i γ i ( ) i=1 z3 z ν 1 /2 z 1 /2 z 2 [FnF] Weyl - lifting 3 2, I-. type (3) G = Gal( Q/Q) Belyi([B])-Ihara-Drigne-Drinfel d([d]) G Grothendieck-Teichmuller modular type (3) I-. Langlands Drinfel d type (3) Langlands type (3) Drinfel d Feigin-Frenkel ([FgF]) I-. Hasse-Weil type (2) Hasse-Weil K Hasse-Weil 3

4 ([Sa1],[Sa2]) II. II-. type (2) R McKean-Trubowitz ([MT]) Hill Sp( ) θ(τ, z) = n Z e 2πinz+πin2 τ z θ(τ, z + 1) = θ(τ, z) θ(τ, z + τ) = e πi( 2z τ) θ(τ, z) τ θ(τ, z) = i 2 τ e πiz /τ θ( τ 1, τ 1 z) g = 1 n, z, τ τ Sp(1) H τ g Sp(g) Hill K, H L K τ Q Q 4

5 Hill n z τ τ modular form g = 1 Z C h Sp(1) g < Z g C g h g Sp(g) g = L K + 1H h Hill Sp( )? Sp( ) Hill q(ξ) Q = d2 q(ξ) Hill Q λ 0 < λ 1 < (, λ 0 ), (λ 1, λ 2 ),... Hill S dξ 2 + hyperelliptic type y 2 = R S (x) x R S l n = λ 2n λ 2n 1 l n n tied spectrum µ n [λ 2n 1, λ 2n ] S [MT] A n, B n (n = 1, 2, 3...) K φ φ(µ n ) 2 l 2 n=1 n 5 <

6 K K dual K { } K x 2 n = x n A n x n R, (nl n ) 2 < n=1 n=1 A n dual space K x n Z L L = m n A n n = 1, 2,... Z m n S winding number L dual lattice L L K m n A n dual basis 1 n { } L = n i 1 i ( ) n i Z i=1 C φ Φ dφ = φ R S dλ H = {dφ; 1 dφ dφ < } S K K H H S dφ dφ = 4 1 = 2 1 φ C i,j=1 = 2 1Q[n] R S 2 d(area) n i Q i,j n j (Q ij := B i (1 j )) Hill Θ(Q, z) := φ L e 2π 1z(φ)+πQ[n] 6

7 z = x + 1y x K, y H L φ Z Q S τ 1τ z Θ(Q, z + A n ) = Θ(Q, z) Θ(Q, z + B n ) = Θ(Q, z)e π 1( 2z(1 n ) B n (1 n )) θ(τ, x) = 1 t n e π(x n)2 t (τ = 1t) Θ(Q, x) = vol(k /L 1 ) det Q n e πq 1 [x+n] vol =, det = [0,1) E E θ(τ, x)dx = n P (E + n) dp (x) = πx 2 e t dx t 7

8 K /L E Θ(Q, x)dx = P (I 1 (E)) E I R (x i ) x i A i K /L dp (x) R τ Q S Hill Q Hill Q 1 Hill McKean-Trubowitz II- [K] type (2) S 1 P SL(2, C) 8

9 Γ = P SL(2, Z[T ]) T S 1 Γ ζ Γ (s) = p Prim(Γ) (1 N(p) s ) 1 Prim(Γ) Γ N ([Se1],[Se2]) P SL(2, Z) Q Q(T ) Q Q(T ) Prim(Γ) Q(T ) P SL(2, Z) K K 9

10 N(p) := exp(vol(coker(reg p ))) Reg p p C p K K 2 (C p ) regulator map P SL(2, Z) K regulator map log Γ = P SL(2, Z[T ]) ζ Γ (s) := (1 N(p) s ) 1 p Prim(Γ)/ g (s) ζ (g) Γ ζ Γ (s) = g=1 ζ (g) Γ (s). -Weil Beilinson-Bloch (s) s ζ (1) Γ higher regulator -Weil modular curve modular curve Beilinson-Bloch higher regulator L 10

11 [K] Γ [K] Diff(S 1 ) type (3) type (3) Γ Diff(S 1 ) Ghys-Sergiescu[GS] 11

12 R K F Ghys-Sergiescu R = Z, K = Q, F = R R = F[T ], K = F(T ), F = F((T 1 )) F R = Z[ 1], K = Q( 1), F = C π T R = R[1/π] general affine GA(R ) := {x π n x + p } π q ; p R, q, n Z Γ F/R piecewise linear h h piecewise linear F/R I n I n h GA(R ) Γ. F/R h h 1 Γh Diff(F/R) diffeo 12

13 References [B] G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Math. USSR Izv. 14 (1980), [D] V. G. Drinfel d, On quasitriangular quasi-hopf algebras and on a group that is closely connected with Gal( Q/Q), Leningrad Math. J. 2 (1991), [FgF] B. Feigin and E. Frenkel, Duality in W -algebras, International Math. Research Notes 6 (1991), Duke Math. J., [FnF] J. Feingold and I. B. Frenkel, A Hyperbolic Kac-Moody Algebra and the Theory of Siegel Modular Forms of Genus 2, Math. Ann. 263 (1983). [GS] E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes de cercle, Comment. Math. Helvetici 62 (1987), [K] S. Koyama, Zeta functions of loop groups, Advanced Studies in Pure Math. 21 (1992), [M] I. G. Macdonald, Affine root systems and Dedekind s η-function, Inv. Math. 15 (1972), [MT] H. P. McKean and E. Trubowitz, Hill s surfaces and their theta functions, Bull. Amer. Math. Soc. 84 (1978), [Sa1], Wiener, 792 (1992), [Sa2], Wiener, this volume (1992). 13

14 [Se1] A. Selberg, Harmonic analysis and discontinuous groups on weakly symmetric Riemannian surfaces with applications to Dirichlet series, J. Ind. Math. Soc. 20 (1956), [Se2], Harmonic Analysis, Collected Papers 39, Current address: Department of Mathematics, Princeton University, Princeton, NJ , U.S.A. 14

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1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification