Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer Zagier Gangl- -Zagi

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1 Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer Zagier Gangl- -Zagier GKZ Brown Brown Lie Brown ,, Broadhurst-Kreimer ( 97). Broadhurst-Kreimer, Zagier ( 94),., SL 2 (Z), Broadhurst-Kreimer.,., Gangl,, Zagier GKZ,, Brown (exceptional element)., Lie 1

2 ., Brown. Brown, Q (totally odd MZV), Lie, Lie,., Brown. Brown,, ( ). Brown, - -Zagier( 06) (linearized double shuffle space) Lie., 1 Broadhurst-Kreimer,. 2,, Brown, - -Zagier. Brown, 3 Brown, Gangl- -Zagier ( GKZ ). 4, Lie, Lie Brown., Brown, 2, 3. 1 Broadhurst-Kreimer, Zagier [23]., Broadhurst- Kreimer[5]., Broadhurst-Kreimer. 1. n = ( 1,..., n ), ζ() = ζ( 1,..., n ) = m 1 > >m n >0 1. m 1 1 m n n wt() = n, dep() = n., 1 > 1., Zagier. 2

3 Zagier Q Z, {d } 0 1/(1 t 2 t 3 ) = d t., Z (n) dim Q Z? = d. Zagier, d ([21, 8] ), {ζ( 1,..., n ) i {2, 3}} Z ([2, 3], Hoffman ).,, Q Galois Gronthendiec-Teichmüller. (, [6].) Broadhurst-Kreimer,., n ( 1) Q Z (n) : := ζ() wt() =, dep() n Q, Z (n) 0 := Q. n,. Broadhurst-Kreimer, Z (n) /Z(n 1) ( ). E(s), O(s), S(s) : E(s) = s2 s3, O(s) = 1 s2 1 s, S(s) = s 12 2 (1 s 4 )(1 s 6 ). {d,n },n 0.,n 0 d,n s t n = 1 + E(s)t 1 O(s)t + S(s)t 2 S(s)t (Broadhurst-Kreimer vector space ver. ) > n > 0, d,n =? dim Q Z (n) /Z(n 1). d,n n\

4 2, (bigraded algebra) (algebra generator). (graded algebra) Z, augumentation I. Z := Z (n), Z := Z, I = Z. >n>0 0 1 Z (filtered algebra) {0} = Z (0) Z (1)... Z (n) Z ( Z (n) := 0 Z (n) )., T := I/I 2, (cotangent space) T,., M ( :[15, p.14]): M :=,n>0,n>0 M (n), M(n) := T (n) /T (n 1) (n) = Z /(Z(n 1) + Z (n) I 2 ). {D,n },n>0. ( ) D,n 1 1 = 1 s t n 1 O(s)t + S(s)t 2 S(s)t (Broadhurst-Kreimer algebra generator ver. ) > n > 0 (, (, n) (2, 1)), D?,n = dim Q M (n). D,n n\ : D,n log, D,n n = 2, 3, 4,. ( :[12]) M (n) D,2 s s 8 = (1 s 2 )(1 s 6 ), D,3 s s 11 (1 + s 2 s 4 ) = (1 s 2 )(1 s 4 )(1 s 6 ), >0 >0 D,4 s = s12 (1 + 2s 4 + s 6 + s 8 + 2s 10 + s 14 s 16 ). (1 s 2 )(1 s 6 )(1 s 8 )(1 s 12 ) >0 ( 4 ) 4

5 n\ ζ(2) ζ(3) ζ(5) ζ(7) ζ(9) ζ(11) ζ(13) 2 ζ(3, 5) ζ(3, 7) ζ(3, 9) 3 ζ(3, 3, 5) 4 ζ(4, 4, 2, 2) ζ(15) ζ(17) ζ(19) ζ(3,11) ζ(5,9) ζ(3, 3, 3, 5) ζ(3,3,9) ζ(3,5,7) ζ(3,13) ζ(5,11) ζ(3,3,3,7) ζ(3,3,5,5) ζ(4,4,2,6) ζ(3,3,11),ζ(3,5,9) ζ(3,7,7),ζ(5,5,7) ζ(3,15) ζ(5,13) ζ(3,3,3,9),ζ(3,3,5,7) ζ(3,5,5,5),ζ(3,5,3,7) ζ(4,8,4,2) ζ(3,3,13),ζ(3,5,11) ζ(3,7,9),ζ(5,5,9) ζ(5,7,7) ζ(3,5,5) ζ(3,3,7) ζ(3,17) ζ(5,15) ζ(7,13) 2 - -Zagier, (double shuffle relations) Broadhurst-Kreimer - -Zagier [13].,., dim M (n) (linearized double shuffle space),. 2.1,.,. [1] [13] ( [1] ).,,.,, : r + s = (r > s 1), ζ(r)ζ(s) = ζ(r, s) + ζ(s, r) + ζ(), (1) ζ(r)ζ(s) = (( ) ( )) i 1 i 1 + ζ(i, j). r 1 s 1 i+j= i,j 1 ( a,b>0 = a>b>0 + b>a>0 + a=b>0 ) ( ). 5

6 , s = 1,,., ζ(1) := 0 ζ(1, r) := ζ(r, 1) ζ(r + 1)., ( ) (Hoffman [11], [1, 1.4] ). x, y Q ( ) H, H 1, H 0 : H := Q x, y H 1 := Q + Hy H 0 := Q + xhy. H 0. Q- Z : H 0 R Z(x 1 1 y x n 1 y) = ζ( 1,..., n ) 1., Z(1) = 1. H 0. : Q- : H 1 H 1 H 1 z := x 1 y, w, w H 1, z w z l w = z (w z l w ) + z l (z w w ) + z +l (w w ),., 1. H 1 H 1 = Q z 1, z 2,.... H 0., Hoffman[11] (i.e. Z(w w ) = Z(w)Z(w )). : Q- x : H H H w, w H u, v {x, y}, uw x vw = u(w x vw ) + v(uw x w ),., 1 x., x H 0, (i.e. Z(w x w ) = Z(w)Z(w )). : Q Z H 1 R[T ]. {, x}, H 1, H 0., Q Z : H 1 R[T ], ([1, 1.4.3]): Z H 0 = Z, Z (y) = T., H 1 = H 0 [y], w H 1 w = w 0 + w 1 y + w n y n (w 0, w 1,..., w n H 0 )., Z (w) = Z(w 0 ) + Z(w 1 )T + + Z(w n )T n. 1 x = dt/t, y = dt/(1 t) 0 1 ( ). 6

7 4. ( [13]) w 0 H 0, w 1 H 1 Z (w 0 w 1 w 0 x w 1 ) = 0., T Z reg = Z T =0. Z reg(w 0 w 1 w 0 x w 1 ) = 0. Proof.. R ρ : R[T ] R[T ]( [1, 1.4.6]). ρ(exp(t u)) = exp ( ( 1) n n ζ(n)un) exp(t u). n>1, Z x = ρ Z ( ). w 1 H 1 w 0 H 0, ρ R.. 0 = Z(w 0 )Z x (w 1 ) Z(w 0 )ρ(z (w 1 )) = Z x (w 0 x w 1 ) ρ(z (w 0 w 1 )) = Z x (w 0 w 1 w 0 x w 1 ). 5. y n w 1 H 1, Z reg(w 1 ) Z x reg(w 1 ) (mod Z (n 1) + Z (n) I 2 ). Proof. w 1, : w 1 = v 0 + v 1 y + + v n y n (v i H 0 ). v i y n i., ρ(t i ) Q[ζ() 2] T i, i, ρ(z (w 1 )) = Z reg(w 1 ) + Z(v 1 )ρ(t ) + + Z(v n )ρ(t n ) Z reg(w 1 ) mod Z (n 1) + Z (n) I 2 : Z x reg(w 1 ) = Z x (w 1 ) T =0 = ρ(z (w 1 )) T =0 Z reg(w 1 ) (mod Z (n 1) + Z (n) I 2 ). ( : i 2, ρ(t i ), v i 0 i 2 ρ(z (w 1 )) T =0 Z reg(w 1 )., 2 (ρ(t i ) (i 2) ), ρ(z (w 1 )) T =0 = Z reg(w 1 ).) 7

8 2.2, M Q., M n n.,,,,., M (n). Z (n) /(Z(n 1) +Z (n) I 2 ) :, (1), M (2).. F2 (x 1, x 2 ) := Zreg(z 1 z 2 )x x 2 1 2,, F x 2 (x 1, x 2 ) := 1, 2 >0 1, 2 >0 Z x reg(z 1 z 2 )x x F 2 (x 1, x 2 ) = ζ(2) 2 +ζ(2, 1)x 1+( ζ(2, 1) ζ(3))x 2 +ζ(2, 2)x 1 x 2 +( ζ(3, 1) ζ(4))x (i) F 2 (x 1, x 2 ) = F x 2 (x 1, x 2 ), (ii) 0 F 2 (x 1, x 2 ) + F 2 (x 2, x 1 ) 1, 2 >0 F x 2 (x 1 + x 2, x 2 ) + F x 2 (x 1 + x 2, x 1 ) (mod Z (1) + Z (2) I 2 ). Proof. 5., (1)., 0 Zreg(z 1 z 2 )x x Zreg(z 1 z 2 + z 2 z 1 )x x 2 1 2, 0 1, 2 >0 = 1, 2 >0 = i,j>0 Z x reg(z 1 i+j= i,j>0 x z 2 )x x , 2 >0 (( ) ( )) i 1 i 1 + Z x reg(z i z j )x x Z x reg(z i z j ) ( (x 1 + x 2 ) i 1 x j (x 1 + x 2 ) i 1 x j 1 1 ). 8

9 6, mod Z (1) + Z (2) I 2, Q f(x 1, x 2 ) : 0 = f(x 1, x 2 ) + f(x 2, x 1 ) = f(x 1 + x 2, x 2 ) + f(x 1 + x 2, x 1 ). (2) (2), Q[x 1, x 2 ] DSh 2, d Q DSh 2 (d). 7. 2, DSh 2 := {f(x 1, x 2 ) Q[x 1, x 2 ] f satisfy (2)}. dim M (2) dim DSh 2 ( 2). Proof. DSh 2 ( 2), q 1,..., q g (g = dim DSh 2 ( 2)). 6, F2 (x 1, x 2 ) M (2) DSh 2, ζ i M (2), (F 2 (x 1, x 2 ) 2 ) = ζ 1 q ζ g q g. ζ 1,..., ζ g,. Remar. - -Zagier, dim DSh 2 ( 2) = D,2., Broadhurst-Kreimer n = 2 2.,, DSh n Broadhurst-Kreimer. n :. n 2, Q[x 1,..., x n ] sh (n) l Z[GL n (Q)] (1 l n 1) : f(x 1,..., x n ) (n) sh l := f(x σ 1 (1),..., x σ 1 (n)). σ S n σ(1)<...<σ(l) σ(l+1)<...<σ(n) ( ),., w 1,..., w n, ([18] ): w 1 w l x w l+1 w n = w σ 1 (1) w σ 1 (n). (3) σ S n σ(1)<...<σ(l) σ(l+1)<...<σ(n), f (x 1,..., x n ) := f(x x n,..., x n 1 + x n, x n ). n ( {, x}): Fn(x 1,..., x n ) := Zreg(z 1 z n )x x n 1 n. ( 1,..., n) N n 2, D,n Broadhurst- Kreimer. 9

10 8. n > 1 l {1,..., n 1},. (i) Fn Fn x (mod Z (n 1) + Z (n) I 2 ), (ii) 0 Fn (n) sh l (Fn x ) (n) sh l (mod Z (n 1) + Z (n) I 2 ). Proof. 5., 0 Z reg(z 1 z l z l+1 z n ) (mod Z (n) I 2 ) Z reg(z 1 z l x z l+1 z n ) (mod Z (n 1) + Z (n) I 2 ), (3) Fn (n) sh l 0.,, z 1 z l x z l+1 z n.,, 0 = (Fl x ) (x 1,..., x l ) (Fn l) x (x l+1,..., x n ) = (Fn x ) (x 1,..., x n ) (n) sh l ( :[13], [12, Proposition 9]). 9. n n > 1, DSh n := {f Q[x 1,..., x n ] f sh (n) l, d DSh n (d). 7,. 10. ( - -Zagier [13]) > n > 1, dim M (n) = f sh (n) l = 0 (1 l n 1)}. dim DSh n ( n). Remar. dim DSh n ( n) D,n, Broadhurst-Kreimer., DSh n (d). DSh n (d), Goncharov [10] GL n (Z), - [14] B d > 0 n > 0, DSh n (d) = {0} ([13, Prop. 17]). d > 0,. [ ] d dim DSh 2 (d) = = D d+2,2 ([10],[13, Prop. 18]), 6 [ ] d 2 1 dim DSh 3 (d) = = D d+3,3 ([10, 14])

11 3 Gangl- -Zagier Broadhurst-Kreimer (Zagier [23, 8], [24, 3] ), dim Q Z (2)? = 2 1 dim S (SL 2 (Z)) ( : even)., (GKZ ).,.,, GKZ. 3.1,, ( Gangl- -Zagier[7] ). F (X, Y ) GL 2 (Z), Z[GL 2 (Z)] : F (X, Y ) ( a b c d ) = F (ax + by, cx + dy )., GL 2 (Z). ( ) ( ) S =, T =, ε = ( ) ( 0 1, δ = , F (X, Y ) ε = F (Y, X), F (X, Y ) δ = F ( X, Y )., W : W := er(1 T t T ) (= {P (X, Y ) Q[X, Y ] P (X, Y ) (1 T t T ) = 0})., d W (d). PGL 2 (Z), (1 T t T )S = (1 + S) (1 + T S + (T S) 2 ). PSL 2 (Z) S T, P W G = P (1 + S) = P (1 + T S + (T S) 2 ), G SL 2 (Z), G 0., P (1 + S) = 0.,, W = er(1 + S) er(1 + T S + (T S) 2 )., ε(t S)ε = (T S) 2 P ε = P δ W. W ε ±1 W ± ( t T = εt ε): W ± = er(1 T T ε). : P W P δ = P,. ). 11

12 , W = er(1 + S) er(1 + T S + (T S) 2 )., f(z), n-th r n (f) r n (f) = 0 f(it)t n dt ( : f L- r n (f) = n!/(2π) n+1 L(n + 1, f)), : P f (X, Y ) = i 0 f(z)(x zy ) 2 dz. (f ), P f (X, Y ) γ = γ 1 (i ) f(z)(x zy ) 2 dz, SL γ 1 (0) 2 (Z) S, T S, P f (X, Y ) (1 + S) = Pf (X, Y ) (1 + T S + (T S) 2 ) = 0. P f (X, Y ), Q., P f (X, Y ) (1 + S) = 0 rn (f) = ( 1) n r 2 n (f). P f (X, Y ), P f (X, Y ) ( ) HP 1 (SL 2(Z), V Q C), SL 2 (Z)., V 2 ; V = Q[X, Y ] ( 2). f S (SL 2 (Z)), Z[SL 2 (Z)]- φ f : SL 2 (Z) V Q C φ f (γ) = γ 1 (i ) f(z)(x zy ) 2 dz, φ i f HP 1 (SL 2(Z), V Q C). ( φ f (γ 1 γ 2 ) = φ f (γ 1 ) γ2 + φ f (γ 2 ).) φ f, φ f (S) = P f (X, Y ) W ( 2), S 2 = (T S) 3 = 1 (PSL 2 (Z) ) φ f (S) W. ( [17, 19].) 12. (Eichler- -Manin [16]), r + : S (SL 2 (Z)) W + ( 2) Q C, f P f (X, Y ) (1 + δ), r : S (SL 2 (Z)) W ( 2) Q C, f P f (X, Y ) (1 δ) 1. Remar. W (d) Q(X d Y d )., W (d) = 12

13 W,0 (d) Q(X d Y d ), W,0 (d) d W,0. W,0 = d>0 W,0 (d). 3.2 GKZ Gangl- -Zagier[7] GKZ.,. 13. (Gangl- -Zagier [7]) P (X, Y ) W ( 2), q r,s. P (X + Y, Y ) = ( ) 2 q r,s X r 1 Y s 1. r 1,. r+s= r,s odd r+s= r,s 1 q r,s ζ(r, s) 0 (mod Qζ()). 1. X 2 Y 8 3X 4 Y 6 + 3X 6 Y 4 X 8 Y 2 W (10). GKZ-, 14ζ(9, 3) + 75ζ(7, 5) + 84ζ(5, 7) 0 (mod Qζ(12)). 13. D, Z, Z r,s (r+s =, r, s 1) Q,. Z r,s + Z s,r + Z = (( ) ( )) i 1 i 1 + Z i,j. (4) r 1 s 1 i+j= ( i+j= i, j 1.), D = Z, Z r,s r + s = Q. (relation (4)) 14. a r,s,. (i) D,. a r,s Z r,s 0 (mod QZ ). r+s= (ii) 2 H(X, Y ) er(1 ε, V ),. H(X, Y ) ( t T 1) = ( ) 2 a r,s X r 1 Y s 1. r 1 r+s= 13

14 Proof. er(1 ε, V ) H r,s (X, Y ) = ( 2 r 1) (X r 1 Y s 1 +X s 1 Y r 1 ) (r+s = ), H r,s ( t T 1)., ( 2 r 1) X r 1 (X+Y ) s 1 = ( 2 )( i 1 ) i+j= i 1 r 1 X i 1 Y j 1, Z r,s + Z s,r (( ) ( )) i 1 i 1 + Z i,j (mod QZ ) r 1 s 1 i+j=. D (4),. 15. q r,s q r,s = q s,r (r, s : even).,. (i) D,. q r,s Z r,s 3 q r,s Z r,s (mod QZ ). r+s= r,s even r+s= r,s odd (ii) P (X, Y ) W ( 2),. P (X, Y ) ( ) 2 T = q r,s X r 1 Y s 1. r 1 r+s= Proof. (ii) (i). P (X, Y ) = ( 2 r+s= r 1) pr,s X r 1 Y s 1 W ( 2), Q(X, Y ) = P (X, Y ) T. Q (1 ε) = P (T T ε) = P, q r,s q s,r = { pr,s r, s : odd, 0 r, s : even. (5), Q δ Q = Q ev + Q od, Q ev er(1 δ, V ), Q od er(1 + δ, V ). Q od (5), (i.e. Q od er(1 ε, V )). Q ev (5), Q ev = ( ) 2 (p r,s + q s,r )X r 1 Y s 1 = P + Q ev ε (6) r 1 r+s=., Q ev ε Q ev = Q ev,+ + Q ev,, Q ev,± er(1 ε), (6) 2Q ev, = P., 14, (i) F, F 2Q ev,+ 2Q od ( )., 2Q ev, t T = P t T = P T ε = Q ε = Q ev,+ + Q ev, Q od, (Q ev,+ Q ev, Q od ) t T = Q S t T = P T S t T = P S = P = 2Q ev,, (2Q ev,+ 2Q od ) t T = Q ev,+ 3Q ev, Q od, (2Q ev,+ 2Q od ) ( t T 1) = Q od 3Q ev. 14

15 15, Z r,s ζ(r, s), 15 (i), q r,s = q s,r ζ(r)ζ(s) (r + s =, r, s even) ζ() ( )., GKZ ( 13). 4 Brown DSh DSh := n>0 DSh n = DSh n (d)., DSh 1 := d>0 Qx2d 1. 2, Broadhurst-Kreimer DSh n ( n) D,n. Brown [4], DSh Lie.,. d,n>0 4.1, DSh Lie., DSh., Brown ( [4] ). Q-. : Q[y 0,..., y r ] Q Q[y 0,..., y s ] Q[y 0,..., y r+s ] f(y 0,..., y r ) g(y 0,..., y s ) f g(y 0,..., y r+s ), f g(y 0,..., y r+s ) = s deg f+r ( 1) s f(y i, y i+1,..., y i+r )g(y 0,..., y }{{} i, y i+r+1,..., y r+s )+ }{{} i+1 s i i=0 i=1, : f(y i+r,..., y i+1, y i )g(y 0,..., y }{{ i 1, y } i+r,..., y r+s ). }{{} i s i+1 {f, g} = f g(0, x 1,..., x r+s ) g f(0, x 1,..., x r+s ) 16. (Goncharov-Racinet-Brown) DSh {, }, Lie (bigraded Lie algebra). 15

16 , {, }, Jacobi, f i DSh ni (d i ), {f 1, f 2 } DSh n1 +n 2 (d 1 + d 2 )., {x 2n 1 1, x 2n 2 1 } = (x 2n 1 = x 2n x 2n 2 2 x 2n 2 1 x 2n 1 ( x 2n 2 2 (x 2 x 1 ) ) 2n 2 + (x 2 x 1 ) ( 2n 1 x 2n 2 1 x 2n ) (1 T + T ε), (7) ) + x 2n 1 ( 2 (x2 x 1 ) 2n 2 ) x 2n Brown Lie DSh, Lie. DSh (DSh/{DSh, DSh} = DSh ab ), DSh Lie A. A := Lie Q [x 2 1, x 4 1, x 6 1,...]. Lie A n ( x 2m 1 (m = 1, 2,...) n 1 ), d A A n (d).. A n (d) DSh n (d). (8) 2. n = 2 : A 2 (4) = {0}, A 2 (6) = Q{x 2 1, x 4 1}, A 2 (8) = Q{x 2 1, x 6 1}, A 2 (10) = Q{x 2 1, x 8 1} + Q{x 4 1, x 6 1}, A 2 (12) = Q{x 2 1, x 10 1 } + Q{x 4 1, x 8 1}. Lie A Lie., A ( -, Schneps[20]) d > 0, a n1,n 2 a n1,n 2 {x 2n 1 1, x 2n 2 1 } = 0 2n 1 +2n 2 =d n 1 >n 2 >0, a n1,n 2 (X 2n 1 Y 2n 2 X 2n 2 Y 2n 1 ) W,0 (d). 2n 1 +2n 2 =d n 1 >n 2 >0 Proof. (7). 0 W,0 DSh 1 DSh 1 A 2 0., DSh 1 DSh 1 n 1,n 2 >0 Q(x2n 1 1 x 2n 2 2 x 2n 2 1 x 2n 1 2 ). 3. 1, (X 2 Y 8 X 8 Y 2 ) 3(X 4 Y 6 X 6 Y 4 ) W (10)., (7) {x 2 1, x 8 1} 3{x 4 1, x 6 1} = 0. 16

17 dim A 4 (8) = 0, dim DSh 4 (8) = 1., A DSh, Brown [4], A DSh Lie,.. p(x, Y ) W,0. p(x, Y ) XY (X Y ). p 0 (X, Y ), p 1 (X, Y ) : p 0 (X, Y ) = p(x, Y )/XY (X Y ), p 1 (X, Y ) = p(x, Y )/XY. 18. p(x, Y ) W,0, e p (y 0,..., y 4 ) := ( p1 (y σ(4) y σ(3), y σ(2) y σ(1) ) + (y σ(0) y σ(1) )p 0 (y σ(2) y σ(3), y σ(4) y σ(3) ) ), σ Z/5Z e p (x 1,..., x 4 ) := e p (0, x 1,..., x 4 ). σ. 19. (Brown[4]) d > 0,. e : W,0 (d) DSh 4 (d 2) p(x, Y ) e p (x 1,..., x 4 ). DSh DSh 1 Lie, 19 Brown (Brown exceptional element, ), DSh Brown., Brown DSh Lie,. 20. DSh. H 1 (DSh, Q) = DSh 1 e(w,0 ) H 2 (DSh, Q) = W,0 H i (DSh, Q) = 0 (i 3). Remar. Brown 20, 19 DSh 4 DSh Lie W, DSh? = A W ( W := Lie Q (e(w,0 )) ),, DSh 17 A 2., 20, DSh n ( n), D,n ( [4] ). 17

18 4.3 Brown, , A,. 21. >n>0 ( 1 ) dim An( n)? = 1 s t n 1 1 O(s)t + S(s)t 2. (9), ζ( 1,..., n ) x x n 1 n, A 3 ζ(2l 1 + 1,..., 2l n + 1) (l i 1)( ) Z A 3., (9). Brown.,., n Z O (n). O (n) n (mod 2),. O (n) O (n 1) := ζ() Z (n) i 3 : odd Q. = O (n 2) O (n 3) = O (n 4). ζ(2) Z (ζ(2))., Z,n odd,, n, n 1, (ζ(2)). Z odd,n = O (n) /(Z(n 1) O (n) + (ζ(2)) O (n) ). 22. (Brown ). 1 + >n>0 dim Z odd,n s t n? = 1 1 O(s)t + S(s)t 2., 21, 22,. n = 2 ( ) : 21, [ ] 2 dim A 2 ( 2) = ( : even) 6 3, 1 M (n)., ( 19). 18

19 . A Lie, Poincaré-Birhoff-Witt ([22, p226]) ( ) dim An ( n) 1 1 s = 1 s t n 1 s(st + 1) >n>0, t 2 dim A 2 ( 2) = [( 4)/4] ( : even)., 17, A 2 ( 2) dim W,0 ( 2) = dim S (SL 2 (Z)) = [( 4)/4] [( 2)/6], 12,17 (8), A 2 = DSh 2 : [ ] [ ] 4 2 dim A 2 ( 2) = dim S (SL 2 (Z)) = = dim DSh 2 ( 2). 4 6 n = 3 ( ) : Brown [4], Goncharov A 3 = DSh 3., A 3 (12), (Jacobi ){x 2 1, {x 2 1, x 8 1} 3{x 4 1, x 6 1}} = 0. n = 2 ( ) : 22,. dim Q Z odd,2? = 2 2 dim S (SL 2 (Z))., Z odd,2 = ζ(2i + 1, 2i 1) 1 i /2 1 Q/ π Q, GKZ ( 13) dim S (SL 2 (Z)),. dim Q Z odd,2 2 2 dim S (SL 2 (Z)). n = 3 ( ) : 22, dim Q Z,3 odd s =? O(s) 3 2O(s)S(s). >0, 15 ( )., 1 GKZ ζ(3)., 14(ζ(3, 9, 3) + 2ζ(9, 3, 3)) + 75(ζ(3, 7, 5) + ζ(7, 3, 5) + ζ(7, 5, 3)) + 84(ζ(3, 5, 7) + ζ(5, 3, 7) + ζ(5, 7, 3)) 0 (mod Z (2) 15 ). 36ζ(5, 5, 5) + 6ζ(5, 7, 3) + 15ζ(7, 5, 3) 14ζ(9, 3, 3) 0 (mod Z (2) 15 ),., ( GKZ ), 3 d=12 dim S d(sl 2 (Z)),. 19

20 [1] T. Araawa, M. Kaneo,, MI 23. [2] F. Brown, Mixed Tate motives over Z, Ann. of Math. 175(2) (2012), [3] F. Brown, On the decomposition of motivic multiple zeta values, Galois-Teichmüller theory and Arithmetic Geometry, Advanced Studies in Pure Mathematics. [4] F. Brown, Depth-graded motivic multiple zeta values, arxiv: [5] D. Broadhurst, D. Kreimer, Association of multiple zeta values with positive nots via Feynman diagrams up to 9 loops, Phys. Lett. B 393, no. 3-4 (1997), [6] H. Furusho, The Multiple Zeta Value Algebra And The Stable Derivation Algebra, RIMS-oyurou 1200, Algebraic number theory and related topics, (2001), [7] H. Gangle, M. Kaneo, D. Zagier, Double zeta values and modular forms, Automorphic forms and Zeta functions, Proceedings of the conference in memory of Tsuneo Araawa, World Scientific, (2006), [8] P. Deligne, A.B. Goncharov, Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ećole Norm. Sup. 38 (2005), 1 56 [9] A. B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5 (1998), [10] A. B. Goncharov, The dihedral Lie algebras and Galois symmetries of π (l) 1 (P 1 ({0, } µ N )), Due Math. J. 110(3) (2001), [11] M. Hoffman, The algebra of multiple harmonic series, J. of Algebra, 194 (1997), [12] K. Ihara, Derivation and double shuffle relations for multiple zeta values, RIMS 1549 (2007), [13] K. Ihara, M. Kaneo, D. Zagier, Derivation and double shuffle relations for multiple zeta values, Compositio Math. 142 (2006),

21 [14] K. Ihara, H. Ochiai, Symmetry on linear relations for multiple zeta values, Nagoya Math. J. 189 (2008), [15] C. Kassel, Quantum groups, Graduate Texts in Mathematics, 155. Springer-Verlag, New Yor, [16] W. Kohnen, D. Zagier, Modular forms with rational periods, Modular forms (Durham, 1983), , Ellis Horwood [17] S. Lang, Introduction to modular forms. Grundlehren der mathematischen Wissenschaften, No Springer-Verlag, Berlin-New Yor, [18] C. Reutenauer, Free Lie algebras (Oxford Science Publications, Oxford, 1993). [19] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Kanô Memorial Lectures, No. 1. Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Toyo; Princeton University Press, Princeton, N.J., [20] L. Schneps, On the Poisson Bracet on the Free Lie Algebra in two Generators, Journal of Lie Theory 16(1) (2006), [21] T. Terasoma, Mixed Tate motives and multiple zeta values, Invent. Math. 149(2) (2002), [22] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, [23] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., 120, Birhäuser, Basel (1994), [24] D. Zagier, Periods of modular forms, traces of Hece operators, and multiple zeta values, in Hoei-eishii to L-ansuu no enyuu (= Research on Automorphic Forms and L-Functions), RIMS Koyurou 843 (1993),

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