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 Riemann 15

Size: px
Start display at page:

Download "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 Riemann 15"

Transcription

1 (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 ( 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

2 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 Riemann 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 Kähler F q 18 L/K Riemann Hilbert Riemann Abel Riemann Abel Artin reciprocity map Frobenius map Riemann [Serre] 2

3 ) 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

4 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 [ ] [ ] , H := 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 [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

5 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] g X n ĝ X (1) n [Kac] 33 highest weight integral representations supercuspidal Weil ŝp n boson 5

6 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

7 ( 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

8 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 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

9 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

10 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

11 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

12 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 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)) ( ) 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

13 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, ,, [BL] Beauville, A. and Laszlo, Y.: Conformal blocks and generalized theta functions, preprint 1993, alg-geom [Flath] Flath, D.: Decomposition of representations into tensor products, Proc. Symp. Pure Math. 33, part 1, 1979, [Garland] Garland, H.: The arithmetic theory of loop groups, Publ. Math. IHES 52, 1980, N = 1 [BL] 59 Hecke Langlands program 13

14 [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, [Lefschetz] Lefschetz, L.: Algebraic topology, Amer. Math. Soc. Colloq. Publ., [MS] Moore, G. and Seiberg, N.: Classical and quantum conformal field theory, Commun. Math. Phys. 123, 1989, [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, , [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, [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, [Verlinde] Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory, Nuclear Physics B300 [FS22], 1988,

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,

More information

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half Wess-Zumino-Witten 1999 3 18 Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie 2 1.1 Affine Lie.............................. 2 1.2..................................... 3 2 WZW 4 3 Knizhnik-Zamolodchikov-Bernard

More information

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+ 1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)

More information

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

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t e-mail: koyama@math.keio.ac.jp 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

More information

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T

SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary

More information

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1 014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β

More information

2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W

2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W SGC -83 2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ 1 3 4 Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W [14] c = 1 CFT [8] Rational CFT [15], [56]

More information

I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x

I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]

More information

Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo

Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,

More information

平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (

平成 15 年度 ( 第 25 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = ( 1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i

More information

1

1 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

More information

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler

More information

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe 3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)

More information

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X 2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ

More information

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) = 2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I

More information

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h 2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h

More information

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P

Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo

More information

平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,

平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, B 2, B 3 A i 1 B i+1 A i+1 B i 1 P i i = 1, 2, 3 3 3 P 1, P 2, P 3 1 *1 19 3 27 B 2 P m l (*) l P P l m m 1 P l m + m *1 A N

More information

1 Part I (warming up lecture). (,,...) 1.1 ( ) M = G/K :. M,. : : R-space. R-space..

1 Part I (warming up lecture). (,,...) 1.1 ( ) M = G/K :. M,. : : R-space. R-space.. ( ) ( ) 2012/07/14 1 Part I (warming up lecture). (,,...) 1.1 ( ) M = G/K :. M,. : : R-space. R-space.. 1.2 ( ) ( ): M,. : (Part II). 1 (Part III). : :,, austere,. :, Einstein, Ricci soliton,. 1.3 : (S,

More information

n ( (

n ( ( 1 2 27 6 1 1 m-mat@mathscihiroshima-uacjp 2 http://wwwmathscihiroshima-uacjp/~m-mat/teach/teachhtml 2 1 3 11 3 111 3 112 4 113 n 4 114 5 115 5 12 7 121 7 122 9 123 11 124 11 125 12 126 2 2 13 127 15 128

More information

main.dvi

main.dvi SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1

More information

2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,

2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,, 15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ

More information

非可換Lubin-Tate理論の一般化に向けて

非可換Lubin-Tate理論の一般化に向けて Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012

More information

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................

More information

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2 2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp

More information

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T

λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T 0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >

More information

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2 On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,

More information

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]).

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]). 3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S 3 1 1 (CMC 1), 1 ( [AA]) 3 H 3 CMC 1 Bryant ([B, UY1]) H 3 CMC 1, Bryant ([CHR, RUY1, RUY2, UY1, UY2, UY3,

More information

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib ( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson

More information

Dynkin Serre Weyl

Dynkin Serre Weyl Dynkin Naoya Enomoto 2003.3. paper Dynkin Introduction Dynkin Lie Lie paper 1 0 Introduction 3 I ( ) Lie Dynkin 4 1 ( ) Lie 4 1.1 Lie ( )................................ 4 1.2 Killing form...........................................

More information

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1 1998 1998 7 20 26, 44. 400,,., (KEK), ( ) ( )..,.,,,. 1998 1 '98 7 23, 24 :,,,,, ( ) 1 3 2 Cech 6 3 13 4 Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing

More information

1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,

1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0, 2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)

More information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2

2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2 1 vs. 90 mescoscopic physics 1 2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2 (: ) [1]: 1. Newton =[( ) vs. ] (a) =0 x v ( p = mv) [ a), b), c)] (b) = :

More information

2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12

More information

( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv

( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv ( ) 1 ([SU] ): F K k Z p - (cf [Iw2] [Iw3] [Iw6]) K F F/K Z p - k /k Weil K K F F p- ( 41) Z p - Weil Weil F F projective smooth C C Jac(C)/F ( ) : 2 3 4 5 Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K

More information

note4.dvi

note4.dvi 10 016 6 0 4 (quantum wire) 4.1 4.1.1.6.1, 4.1(a) V Q N dep ( ) 4.1(b) w σ E z (d) E z (d) = σ [ ( ) ( )] x w/ x+w/ π+arctan arctan πǫǫ 0 d d (4.1) à ƒq [ƒg w ó R w d V( x) QŽŸŒ³ džq x (a) (b) 4.1 (a)

More information

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1) φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x

More information

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ

Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R

More information

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys

More information

SUSY DWs

SUSY DWs @ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding

More information

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n

More information

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

More information

Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ,

Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ, A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X

More information

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q

Chern-Simons   Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0

More information

Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2

Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 C II,,,,,,,,,,, 0.2. 1 (Connectivity) 3 2 (Compactness)

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3

1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}

More information

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i

2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i [ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk

More information

D 24 D D D

D 24 D D D 5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6

More information

untitled

untitled Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker

More information

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 + 2016 12 16 1 1 2 2 2.1 C s................. 2 2.2 C 3v................ 9 3 11 3.1.............. 11 3.2 32............... 12 3.3.............. 13 4 14 4.1........... 14 4.2................ 15 4.3................

More information

0. Intro ( K CohFT etc CohFT 5.IKKT 6.

0. Intro ( K CohFT etc CohFT 5.IKKT 6. E-mail: sako@math.keio.ac.jp 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y

More information

( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt

( ) Lemma 2.2. X ultra filter (1) X = X 1 X 2 X 1 X 2 (2) X = X 1 X 2 X 3... X N X 1, X 2,..., X N (3) disjoint union X j Definition 2.3. X ultra filt NON COMMTATIVE ALGEBRAIC SPACE OF FINITE ARITHMETIC TYPE ( ) 1. Introduction (1) (2) universality C ( ) R (1) (2) ultra filter 0 (1) (1) ( ) (2) (2) (3) 2. ultra filter Definition 2.1. X F filter (1) F

More information

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple

More information

2

2 III ( Dirac ) ( ) ( ) 2001. 9.22 2 1 2 1.1... 3 1.2... 3 1.3 G P... 5 2 5 2.1... 6 2.2... 6 2.3 G P... 7 2.4... 7 3 8 3.1... 8 3.2... 9 3.3... 10 3.4... 11 3.5... 12 4 Dirac 13 4.1 Spin... 13 4.2 Spin

More information

62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P

62 Serre Abel-Jacob Serre Jacob Jacob Jacob k Jacob Jac(X) X g X (g) X (g) Zarsk [Wel] [Ml] [BLR] [Ser] Jacob ( ) 2 Jacob Pcard 2.1 X g ( C ) X n P P 15, pp.61-80 Abel-Jacob I 1 Introducton Remann Abel-Jacob X g Remann X ω 1,..., ω g Λ = {( γ ω 1,..., γ ω g) C g γ H 1 (X, Z)} Λ C g lattce Jac(X) = C g /Λ Le Abel-Jacob (Theorem 2.2, 4.2) Jac(X) Pcard

More information

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2

1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2 1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac

More information

Q p G Qp Q G Q p Ramanujan 12 q- (q) : (q) = q n=1 (1 qn ) 24 S 12 (SL 2 (Z))., p (ordinary) (, q- p a p ( ) p ). p = 11 a p ( ) p. p 11 p a p

Q p G Qp Q G Q p Ramanujan 12 q- (q) : (q) = q n=1 (1 qn ) 24 S 12 (SL 2 (Z))., p (ordinary) (, q- p a p ( ) p ). p = 11 a p ( ) p. p 11 p a p .,.,.,..,, 1.. Contents 1. 1 1.1. 2 1.2. 3 1.3. 4 1.4. Eisenstein 5 1.5. 7 2. 9 2.1. e p 9 2.2. p 11 2.3. 15 2.4. 16 2.5. 18 3. 19 3.1. ( ) 19 3.2. 22 4. 23 1. p., Q Q p Q Q p Q C.,. 1. 1 Q p G Qp Q G

More information

On a branched Zp-cover of Q-homology 3-spheres

On a branched Zp-cover of Q-homology 3-spheres Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

More information

日本数学会・2011年度年会(早稲田大学)・総合講演

日本数学会・2011年度年会(早稲田大学)・総合講演 日本数学会 2011 年度年会 ( 早稲田大学 ) 総合講演 2011 年度日本数学会春季賞受賞記念講演 MSJMEETING-2011-0 ( ) 1. p>0 p C ( ) p p 0 smooth l (l p ) p p André, Christol, Mebkhout, Kedlaya K 0 O K K k O K k p>0 K K : K R 0 p = p 1 Γ := K k

More information

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull

Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and

More information

The Physics of Atmospheres CAPTER :

The Physics of Atmospheres CAPTER : The Physics of Atmospheres CAPTER 4 1 4 2 41 : 2 42 14 43 17 44 25 45 27 46 3 47 31 48 32 49 34 41 35 411 36 maintex 23/11/28 The Physics of Atmospheres CAPTER 4 2 4 41 : 2 1 σ 2 (21) (22) k I = I exp(

More information

Perturbation method for determining the group of invariance of hierarchical models

Perturbation method for determining the group of invariance of hierarchical models Perturbation method for determining the group of invariance of hierarchical models 1 2 1 1 2 2009/11/27 ( ) 2009/11/27 1 / 31 2 3 p 11 p 12 p 13 p 21 p 22 p 23 (p ij 0, i;j p ij = 1). p ij = a i b j log

More information

1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC

1   nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC 1 http://www.gem.aoyama.ac.jp/ nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC r 1 A B B C C A (1),(2),, (8) A, B, C A,B,C 2 1 ABC

More information

1 [BPZ] model Wess-Zumino-Witten model,, compact Riemann, principal G-bunlde quasi parabolic structure) family, family base space twisted D-module) 11

1 [BPZ] model Wess-Zumino-Witten model,, compact Riemann, principal G-bunlde quasi parabolic structure) family, family base space twisted D-module) 11 2003 12 26 ) 71 1995 11 2 ) 1 2 2 Twisted diffrential operator tdo) 6 21 compact Riemann quasi parabolic G-bundle 6 22 compact Riemann quasi parabolic G-bundle family 6 23 Lie algebroid dg Lie algebroid

More information

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n ( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally

More information

( ) (, ) ( )

( ) (, ) ( ) ( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................

More information

SO(2)

SO(2) TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6

More information

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x

More information

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2 7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)

More information

1. Γ, R 2,, M R. M R. M M Map(M, M) 3, Aut R (M). ρ : Γ Aut R (M) Γ. M R n, R, R ρ : Γ Aut R (M) GL n (R) := {g M n (R) det(g) R } 4. ρ Γ R R M.,,.,,

1. Γ, R 2,, M R. M R. M M Map(M, M) 3, Aut R (M). ρ : Γ Aut R (M) Γ. M R n, R, R ρ : Γ Aut R (M) GL n (R) := {g M n (R) det(g) R } 4. ρ Γ R R M.,,.,, I ( ) (i) l, l, l (ii) (Q p ) l, l, l (iii) Artin (iv). (i),(ii). (iii) 1. (iv),.. [9]. [4] L-,.. Contents 1. 2 2. 4 2.1. 4 2.2. l 5 2.3. l 9 2.4. l 10 2.5. 12 2.6. Artin 13 3. 15 3.1. l, l, l 15 3.2.

More information

数学メモアール 第4巻, (2004)

数学メモアール 第4巻, (2004) 6 9 7 i 2002 5 6 0, [TUY] Lie, P ŝl 2, 3 ) OPE) 2) 3) factorization property ), 2, 4 2) 4, 3) 7, 2, OPE 3, P n 3, 4, 5 6, 7 6, 7 factorization property,,, Lie Lie 2, [K] 5.2, 7.9 5.2 3.4, 7.9, 6.2, 6,

More information

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like () 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)

More information

コホモロジー的AGT対応とK群類似

コホモロジー的AGT対応とK群類似 AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)

More information

Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4

Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4 Part 4 2001 6 20 1 2 2 generator 3 3 L 7 4 Manin triple 8 5 KP Hamiltonian 10 6 n-component KP 12 7 nonlinear Schrödinger Hamiltonian 13 http//wwwmathtohokuacjp/ kuroki/hyogen/soliton-4txt TEX 2002 1 17

More information

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()

More information

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2 hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................

More information

December 28, 2018

December 28, 2018 e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................

More information

Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n

Part y mx + n mt + n m 1 mt n + n t m 2 t + mn 0 t m 0 n 18 y n n a 7 3 ; x α α 1 7α +t t 3 4α + 3t t x α x α y mx + n Part2 47 Example 161 93 1 T a a 2 M 1 a 1 T a 2 a Point 1 T L L L T T L L T L L L T T L L T detm a 1 aa 2 a 1 2 + 1 > 0 11 T T x x M λ 12 y y x y λ 2 a + 1λ + a 2 2a + 2 0 13 D D a + 1 2 4a 2 2a + 2 a

More information

16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P

16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P 15, pp.15-60 Riemann-Roch 1 1.1 (a) k A n = A n (k) = {(x 1,, x n ) x 1,, x n k} (affine space) A 1, A 2 k[x] = k[x 1,, X n ] n k- X = (X 1,, X n ) A n f(x) k[x], x A n, f x f(x) k[x] I, V = V (I) = {x

More information

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,. 1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i 1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,

More information

3 exotica

3 exotica ( / ) 2013 2 23 embedding tensors (non)geometric fluxes exotic branes + D U-duality G 0 R-symmetry H dim(g 0 /H) T-duality 11 1 1 0 1 IIA R + 1 1 1 IIB SL(2, R) SO(2) 2 1 9 GL(2, R) SO(2) 3 SO(1, 1) 8

More information

, 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U

, 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U W ( ) 1. ( )W Kac-Moody Virasoro,,,,, 4, Langlands.,, W., W, W ([A1, A2, A3, A7]). Premet[Pre] W ( )W, Kostant[Kos]. W Slodowy, primitive ideal. Premet Losev[Los2]. primitive ideal. W. ( )W Losev. Kac-Moody

More information

untitled

untitled 18 18 8 17 18 8 19 3. II 3-8 18 9:00~10:30? 3 30 3 a b a x n nx n-1 x n n+1 x / n+1 log log = logos + arithmos n+1 x / n+1 incompleteness theorem log b = = rosário Euclid Maya-glyph quipe 9 number digits

More information

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α

SO(3) 49 u = Ru (6.9), i u iv i = i u iv i (C ) π π : G Hom(V, V ) : g D(g). π : R 3 V : i 1. : u u = u 1 u 2 u 3 (6.10) 6.2 i R α (1) = 0 cos α SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij

More information

Introduction 2 / 43

Introduction 2 / 43 Batalin-Vilkoviski ( ) 2016 2 22 at SFT16 based on arxiv:1511.04187 BV Analysis of Tachyon Fluctuation around Multi-brane Solutions in Cubic String Field Theory 1 / 43 Introduction 2 / 43 in Cubic open

More information

2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald Weyl Demazure. g, h Cartan., Q := i I Zα i h root lattice, Q + := i I Z 0α

2018 : msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald Weyl Demazure. g, h Cartan., Q := i I Zα i h root lattice, Q + := i I Z 0α 2018 : 2018 21 msjmeeting-2018mar-02i003 : Demazure ( ) 1. Macdonald 1.1. Weyl Demazure. g, h Cartan, Q := i I Zα i h root lattice, Q + := i I Z 0α i Q, P := i I Zϖ i h g weight lattice ;, ϖ i h, i I,

More information

YITP50.dvi

YITP50.dvi 1 70 80 90 50 2 3 3 84 first revolution 4 94 second revolution 5 6 2 1: 1 3 consistent 1-loop Feynman 1-loop Feynman loop loop loop Feynman 2 3 2: 1-loop Feynman loop 3 cycle 4 = 3: 4: 4 cycle loop Feynman

More information

Kaluza-Klein(KK) SO(11) KK 1 2 1

Kaluza-Klein(KK) SO(11) KK 1 2 1 Maskawa Institute, Kyoto Sangyo University Naoki Yamatsu 2016 4 12 ( ) @ Kaluza-Klein(KK) SO(11) KK 1 2 1 1. 2. 3. 4. 2 1. 標準理論 物質場 ( フェルミオン ) スカラー ゲージ場 クォーク ヒッグス u d s b ν c レプトン ν t ν e μ τ e μ τ e h

More information

G H J(g, τ G g G J(g, τ τ J(g 1 g, τ = J(g 1, g τj(g, τ J J(1, τ = 1 k g = ( a b c d J(g, τ = (cτ + dk G = SL (R SL (R G G α, β C α = α iθ (θ R

G H J(g, τ G g G J(g, τ τ J(g 1 g, τ = J(g 1, g τj(g, τ J J(1, τ = 1 k g = ( a b c d J(g, τ = (cτ + dk G = SL (R SL (R G G α, β C α = α iθ (θ R 1 1.1 SL (R 1.1.1 SL (R H SL (R SL (R H H H = {z = x + iy C; x, y R, y > 0}, SL (R = {g M (R; dt(g = 1}, gτ = aτ + b a b g = SL (R cτ + d c d 1.1. Γ H H SL (R f(τ f(gτ G SL (R G H J(g, τ τ g G Hol f(τ

More information

2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)

2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8) (Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s)

More information

DVIOUT-HYOU

DVIOUT-HYOU () P. () AB () AB ³ ³, BA, BA ³ ³ P. A B B A IA (B B)A B (BA) B A ³, A ³ ³ B ³ ³ x z ³ A AA w ³ AA ³ x z ³ x + z +w ³ w x + z +w ½ x + ½ z +w x + z +w x,,z,w ³ A ³ AA I x,, z, w ³ A ³ ³ + + A ³ A A P.

More information

xia2.dvi

xia2.dvi Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,

More information