Computing Differential Modular Forms

Transcription

1 Comuting Differential Modular Forms Chris Hurlburt Northern Illinois University July 28, 2003

2 Let > 3 be a rime number. Let R be a comlete, discrete valuation ring whose maximal ideal is generated by and whose residue field is algebraically closed. Let φ : R R be the unique lifting of the Frobenius morhism to R. Let δ : R R be the ma δ(x) = φ(x) x. Then δ is the unique -derivation on R of the identity. In general Definition 1 A -derivation is a set theoretic ma, δ : A B, from a ring A to an A-algebra B such that δ(x + y) = δx + δy + C (x, y) δ(xy) = y δx + x δy + δxδy for all x, y A where C (X, Y ) = X +Y (X+Y ).

3 Very Comutable with Calculus tye Rules! First we must define an extension of C (X, Y ). Definition 2 For any q, let C ext ( q ( q) q) = Note that C ext (X + Y ) = X +Y (X+Y ) a very natural definition. = C (X, Y ), and so this is Lemma 1 Let δ : A B be a -derivation, let g = q, x, y A, and let n > 0 be an integer. Then the following are true. 1. δ( q) = δq + C ext ( q) 2. δ( 1) = 0 3. δ( x) = δx n ( n 4. δ(x n ) = ) 1 x (n ) (δx) = xn + (x + δx) n =1 ( ) 1 δx 5. δ = x x (x + δx) ( y 6. δ = x) x δy y δx x (x + δx)

4 Consider the following sequence of R-algebras where R f,δ M f 1,δ M 1 f 2,δ M 2... M = R[a 4, a 6, 1 ]ˆ M 1 = R[a 4, a 6, δ(a 4 ), δ(a 6 ), 1 ]ˆ M 2 = R[a 4, a 6, δ(a 4 ), δ(a 6 ), δ 2 (a 4 ), δ 2 (a 6 ), 1 ]ˆ each f i is the natural embedding, = 2 4 (4a a 2 6), and δ mas a 4 δ(a 4 ) a 6 δ(a 6 ) δ(a 4 ) δ 2 (a 4 ) δ(a 6 ) δ 2 (a 6 ) Then if X = Sec M, the first jet sace of X would be Sec M 1.

5 Let E be the ellitic curve over M 0 defined by the homogeneous equation f(x, Y, W ) = W Y 2 X 3 a 4 XW 2 a 6 W 3. Let U and V be the affine oen subsets of E given by the equations f(x, y, 1) and f(z, 1, w) resectively. So and on U V U = Sec M 0 [X, Y]/(f(X, Y, 1)) = Sec M 0 [x, y] V = Sec M 0 [Z, W]/(f(Z, 1, W)) = Sec M 0 [z, w] z = x/y w = 1/y. whence E = U V Next we define the first jets of U and V to be the sets U 1 = Sec M 1 [X, Y, δx, δy]/(f(x, Y, 1), δf(x, Y, 1)) = Sec M 1 [x, y, δx, δy] V 1 = Sec M 1, [Z, W, δz, δw]/(f(z, 1, W), δf(z, 1, W)) = Sec M 1 [z, w, δz, δw] Then E 1, the first jet sace of E, is the gluing of U 1 and V 1 by the mas z = x/y w = 1/y δz = x δy y δx y (y + δy) δy δw = y (y + δy)

6 Just as E is a grou scheme, E 1 is also a grou scheme. We can exress this grou law in terms of coordinates on either U or V. On V, the origin is (0, 0, 0, 0). Let P i = (z i, w i, δz i, δw i ). Then the inverse of P 0 is P 0 = ( z 0, w 0, δz 0, δw 0 ). The coordinates P 3 of the sum P 1 P 2 are given by z 3 = α µ w 3 = β µ where δz 3 = µ δα α δµ µ (µ + δµ) δw 3 = µ δβ β δµ µ (µ + δµ) α = 2w 2 z 1 w 1 z 1 + 2w 1 z 2 3a 6 w 2 w 2 1z 2 a 4 w 2 1z w 2 z 2 + 3w 2 2z 1 a 6 w 1 + w 2 2z 2 1a 4 β = 3w 2 z 2 z z 1 w 2 2w 1 a 4 3z 1 w 1 z w 2 1 w 2 w 2 1z 2 a 4 w 2 2 µ = 3a 6 w 2 w 1 (w 2 w 1 ) + 3z 2 z 1 (z 2 z 1 ) + a 4 (w 2 2z 1 w 2 1z 2 ) + w 1 w 2 + 2a 4 w 2 w 1 (z 2 z 1 ). Note: If z 1 = z 2, it is necessary to tae the limits of the grou law.

7 Let M(R) = {(a, b) R 2 4a b 2 R }. Then the set M(R) is in one-to-one corresondence with the set of airs consisting of an ellitic curve over R and an invertible 1-form. For any f M 1 if we substitute a, b, δa, δb in for a 4, a 6, δa 4, δa 6, then f defines a ma (still denoted by f) from M(R) to R. This element in M 1 is in fact uniquely determined by the ma from M(R) to R. Therefore we call f a δ modular function of order 1. We define a δ-character of order 1 to be a grou homomorhism χ : R R of the form χ = χ m,n where ( ) n φ(λ) χ m,n (λ) = λ m. λ A δ modular function has weight χ if for any λ R f(λ 4 a, λ 6 b) = χ(λ)f(a, b) for all (a, b) M(R). A δ modular form is a δ modular function with a weight.

8 A δ modular form is isogeny-covariant if for any two airs (a, b) and (ã, b) with an etale isogeny of degree N between the corresonding ellitic curves that ulls bac dx y to dx y f(a, b) = N /2 f(ã, b) where is a constant that deends solely on the weight. Note that for χ = χ m,n the constant is = m + n(1 ). Theorem 2 (Buium) For all δ-characters, χ, of order 1 there is u to multilication by a constant in Z a unique isogeny covariant δ modular form of weight χ.

9 We describe from the construction of the unique isogeny covariant differential modular form of weight χ 1, 1, called f jet now to comute f jet. We must Find two sections s U : U 0 M 1 U 1 and s V : V 0 M 1 V 1. Find the difference of sections under the grou law, s V s U, which determines a ma from (U V ) 0 M 1 E 1. Select the δz = ζ coordinate from this difference. Aly log φ 1 (ξ), the formal logarithm of the Frobenius twist of F 1 the formal grou of the ellitic curve to ζ = δz. Tae the residue class of log φ 1 (ζ) F 1 Then f jet =< log F φ 1 1 (ζ), ω >, the residue class of log φ 1 (ζ). F 1

10 Proosition 3 The section s U modulo 2 is where (x, y, A (P U,0 + P U,1 ), B (P U,0 + P U,1 )) A = 24 (4a x 2 a 4 9xa 6 ) B = 23 (9y)(2xa 4 3a 6 ) P U,0 = δa 4 x δa 6 + C ext (y 2 x 3 a 4 x a 6 ) R U,0 = C ext (Af x + Bf y ) + A ( δ(3)x 2 + C ext ( 3x 2 a 4 )) + B δ(2)y P U,1 = 3x PU,0A δa 4 P U,0 A + PU,0B 2 2 P U,0 R U,0. Proosition 4 The section s V modulo 2 is (z, w, C (P V,0 + P V,1 ), D (P V,0 + P V,1 )) where C = z( 3 2 a 6w a 4 z) D = 3 2 a 6w 2 wa 4 z + 1 P V,0 = δa 6 w 3 δa 4 z w 2 + C ext (w z 3 a 4 zw 2 a 6 w 3 ) R V,0 = C ext (Cg z + Dg w ) + C ( δ(3)z 2 + C ext ( 3z 2 a 4 w 2 )) + D ( δ(3)a 6 w2 δ(2)a 4 z w + C ext (1 3a 6 w 2 2a 4 zw)) P V,1 = 3z (P V,0 C ) 2 + (2a 4 w ( P V,0 D ) + δa 4 w 2 )(P V,0 C ) (a 4 z + 3a 6 w )(P V,0 D ) 2 (3δa 6 w 2 + 2δa 4 z w )( P V,0 D ) P V,0 R V,0

11 Proosition 5 The ζ = δz term modulo 2 of s U s V where δα modulo 2 is ζ = δα (8w) + δαδµ (8w) 2 is δα = ( ( 2(3 )w 5 a 6 B A 2w 3 B A 3w 3 B 2 z + 2 w 3 B A + 4w 4 a 4 z B A + w 4 a 4 A2) PU,0 2 ( (2 + w D z B + 4w 2 D a 4 z2 B 2(3 )w 4 C a 6 B + 4w 2 D a 4 z A + 2(3 )w 3 D a 6 A + 2(3 )w 3 D a 6 z B 4w 3 B a 4 z C + 2 w D A 2 C B w 2) P V, 0 3 w 4 δ(a 6 )A δ(2)w 2 A 2w 3 δ(a 4 )z A δ(3)w 4 a 6 A) P U, 0 ( + 3 D C a 6 w2 + a 4 z2 D 2 + D C w 2 a 4 C2 + 3 z a 6 w D 2) PV,0 2 ( + 2w D δ(a 4 )z 2 δ(3)z a 6 w2 D 3 D z δ(a 6 )w 2 + 2w 2 δ(a 4 )z C ) + δ(2)c w δ(2)z D + 3 w 3 C δ(a 6 ) + δ(3)w 3 C a 6 P V, 0 ( + w 2 A 2w 3 a 4 z A + 2w 2 z B 3 w 4 a 6 A 2 w 2 A ) P U, 1 ( + 2 C w D z + 2w 2 a 4 z C w C 2a 4 z2 w D 2 z D + 3 w 3 C a 6 3 D z a 6 w2) P V,1 ) ( + w 2 A 2w 3 a 4 z A + 2w 2 z B 3 w 4 a 6 A 2 w 2 A ) ( P U,0 + 2 C w D z + 2w 2 a 4 z C w C 2a 4 z2 w D 2 z D + 3 w 3 C a 6 3 D z a 6 w2) P V,0 and δµ is modulo δµ = ( B w 2 3(3 )z 2 w A 3w 3 a 4 z B 2 w 3 a 4 A 2(2 )w 3 a 4 z B 2 w 3 B a 4 z 3(3 )z 3 w B w 3 a 4 A 3(3 )a 6 w4 B ) P U, 0 ( + 2(2 )D a 4 z w 2 w 2 a 4 C 3(3 )C z 2 2a 4 z w D D w 2 a 4 C 3(3 )a 6 D w 2) P V, 0 + 2δ(3)z 3 + 2δ(2)w 2 a 4 z + 2(2 )w 2 δ(a 4 )z + 2w 2 δ(a 4 )z + 2δ(3)a 6 w3 + 2(3 )δ(a 6 )w 3 C ext (3a 6 w 3 + 3a 6 w 3 + 3z 3 + 3z 3 + a 4 w 2 z + a 4 w 2 z + w + w2a 4 w 2 z + 2a 4 w 2 z)

12 Proosition 6 Let log φ 1 (ξ) be the formal logarithm of the Frobenius twist of the formal grou of the ellitic curve. F 1 Then log φ 1 (ζ) = ζ modulo 2 F 1 Proof: Recall log φ 1 (ξ) := ξ + φ(c 1) ξ φ(c 2 ) ξ F where the c i are the coefficients of the ower series exansion of the invariant differential. We also now that the invariant differential ω(z) = (1 + 2a 4 z )dz and so c 1 = 0, c 2 = 0, c 3 = 0, c 4 = 2a Hence modulo 2 the ower series log φ 1 (ξ) is the identity. F 1

13 We rovide an exlicit formula for the residue of xa y b call γ a,b. which we will We let ( n ) ) denote the binomial coefficient with the convention that = 0 if > n. ( n Proosition 7 Let a and b be ositive integers. Let m and n {0, 1, 2} be integers such that a = 3m + n. Then the residue of is x a y b 0 if b is even; γ a,b = ( )( ) m + m n b 1 m+ ( 1) 2 (a4 ) 3+2 n b 1 m 2 2+n (a 6 ) 2 if b is odd n =0 b 1 2 Using this formula, we define some more notation. Definition 3 Define µ a,b to be the residue class of xa Υ y b Υ = C ext (y 2 x 3 a 4 x a 6 ). Definition 4 Define τ a,b to be the residue class of xa Υ 2 y b Υ = C ext (y 2 x 3 a 4 x a 6 ). where where

14 Theorem 8 The reduction modulo 2 of f jet the unique, isogeny covariant δ-modular form of weight χ 1, 1 and order 1 is [ 72 (2 4 2(3 ))a 6 δ(a 4) + 16 ( (9 ))a 4 δ(a ] 6) γ 2, [ (1 2 )4 a 4 µ 0, + ( 18 (1 2 ) + 2(27 ))a 6 µ, ] +(12 (1 2 ) 2(18 )(1 + a 6 ))a 4 µ 2, 36 a 4 a 6 µ 2,3 2(12 )a 2 4 µ 3,3 ) + (H 0 + H 1 + H 2 + H 3 + H 4 + H 5 + H 6 + H 7 where H 0 is ( [ 1 (1 2 ) 24 δ(a 4 )6 a 4 γ 3, 1 + =1 ( ) (4a 4 ) 2 23 (9 [ ) (3 a 6 )24 (2a 4 )24 i=1 1 + =1 ( ( ) (6a 4 ) i ( 9a 6 ) i ( δ(a 4 )γ ++i, δ(a 6 )γ +i, + µ +i, ) i ) ] (6a 4 ) ( 9a 6 ) ( δ(a 4 )γ 2+, δ(a 6 )γ +, + µ +, ) 1 δ(a 4 )2 a 4 γ 3, + [ 1 [ 1 =1 =1 =1 ( ) (4a 4 ) i=1 ( ] )(2a 4 ) ( 3a 6 ) ( δ(a 4 )γ +, + µ, ( ) (6a 4 ) i ( 9a 6 ) i µ +i,3 i 1 ( ] + )(6a 4 ) ( 9a 6 ) µ +,3 =1 =1 ( ) ( ) (4a 4 ) (6a 4 ) i ( 9a 6 ) i µ ++i,3 i i=1 1 ( ] + )(6a ) 4 ) ( 9a 6 ) µ 2+,3

15 H 1 is ( 9 ( 2 δ(a 4) a 4 2(3 )γ 3, a 4 (2(3a 6 ) µ 2,3 + 2(3 )µ 5,3 + (2 + 4 )a 4 µ 3,3 + (2 8 )µ 2, ) +( 3 ) 2 a a 6 ) (2(3 )µ 4,3 + (2 + 4 )a 4 µ 2,3 + (2 8 )µ, ) a 2 4 (2(3 )µ 3,3 + (2 8 )µ 0, ) + ( 9 δ(a 4 ) a a 4 δ(a 6 ) δ(a 4) a 4 ) (2(3 )µ 3,3 + (2 8 )µ 0, ) +(( 3 a a 6 a 4 ) (2(3 )µ 5,5 + (2 8 )µ 2,3 )/, H 2 is 1 1 =1 i i=0 j=0 ( )( )( i i j ) ( 1) i ( 9(2 4 ) ) i (2a 4 ) j ( 3a 6 ) i j [ ( ( 36 a 6 a a 4 2 ) µ 2+j,3+2( i) + (36 a 6 a a a 6 2 ) µ +j,3+2( i) + 24 a 6 a 4 2 µ j,3+2( i) +36 a 4 µ 2+j,+2( i) + ( 72 a a 4 ) µ +j,+2( i) + 8 a 4 2 µ j,+2( i) )/ +((36 a 4 δ(a 4 )γ 3+j,+2( i) + (72 δ(a 4 ) a 6 12 δ(a 4 ) a 4 36 a 4 δ(a 6 ))γ 2+j,+2( i) + (28 a 4 2 δ(a 4 ) + 72 δ(a 6 ) a 6 12 a 4 δ(a 6 )) γ +j,+2( i) + (36 δ(a 4 ) a 4 a δ(a 4 ) a 4 a 6 8 a 4 2 δ(a 6 ) 24 a 4 2 δ(a 4 ))γ j,+2( i) ) +(( 16 δ(a 4 ) a δ(a 4 ) a 4 a δ(a 4 ) a a 2 4 δ(a 6 ) + 36 a 4 δ(a 6 ) a 6 )γ 2+j,3+2( i) ] +(24 δ(a 4 ) a δ(a 4 ) a 6 a δ(a 6 ) a a 4 δ(a 6 ) a δ(a 6 ) a 6 2 )γ 2+j,3+2( i) )/ H 3 is 1 1 =1 ( ) ( (( a 4 3 a a 6 a a a 2 4 ) µ 2+2, + ( 576 a 6 a a 4 3 ) µ +2, +(2304 a a a 6 a a 6 2 a 4 )µ 2, ) / 2 +(((576 δ(a 4 ) a 6 a a 4 2 δ(a 6 ) δ(a 4 ) a δ(a 6 ) a δ(a 6 ) a a 4 δ(a 6 ) a 6 )γ 2+2, +(576 a 6 a 4 2 δ(a 6 ) 1152 δ(a 4 ) a δ(a 4 ) a δ(a 4 ) a 6 a δ(a 6 ) a 4 3 )γ +2, ( 2304 δ(a 6 ) a δ(a 4 ) a 4 3 a δ(a 4 ) a 6 a δ(a 6 ) a 6 2 a a 6 a 4 2 δ(a 6 ) 3456 δ(a 4 ) a 6 a δ(a 4 ) a 4 2 a δ(a 4 ) a δ(a 4 ) a δ(a 6 ) a δ(a 4 ) a (1+) 4 a 6 ) / δ(a 4 ) a 2 6 a ( δ(a 4 ) a ) 6 )γ 2, )) 2

16 H 4 is ( 1 2 ( a 6 a a a 6 a 4 (3 ) a 6 a a a 6 2 a a 6 3 a a 4 4 )τ 2, / ( a 6 2 a a a a a 6 a a 6 3 a a a 6 a a a a 6 a a 6 2 a 4 2 )τ, / ( 3456 a a 4 5 a a a a 6 3 a a 6 a a 6 2 a a 4 2 a 6 )τ 0, / 3 ) + ( 1 2 ( a 6 a a 4 5 a a a 6 2 a a 6 4 a a a 6 2 a a a 4 (7 ) a 4 2 a a 6 3 a a 6 2 a 4 )τ 2,3 / ( a 4 5 a a 4 2 a a a 4 3 a a 6 2 a a 4 (7 ) a 6 2 a a a 6 a a a 6 2 a a 6 3 a a 6 3 a a 6 4 a a 6 5 )τ,3 / ( a 6 3 a a a 6 2 a a 6 4 a a 6 2 a a 6 a a 6 2 a a 4 3 a a a 4 (5 ) a 6 )τ 0,3 / 3 ) + ( 1 2 (23328 a 6 3 a a 6 2 a a 6 a a a a 4 3 a 6 )τ 2,5 / 3

17 H 5 is ( 1 2 ( δ(a 4) a 6 3 a δ(a 4 ) a 6 2 a δ(a 4 ) a 6 a δ(a 4 ) a 6 2 a δ(a 4 ) a a 4 4 δ(3) δ(a 6 ) a 6 2 a δ(a 6 ) a 6 a δ(a 6 ) a 6 a δ(a 6 ) a 6 3 a δ(a 4 ) a a 4 3 δ(3) + 18 δ(3) a δ(2) a a 4 2 δ(a 6 ) δ(a 4 ) a δ(3) a 4 2 a δ(a 4 ) a δ(a 4 ) a δ(a 6 ) a δ(a 4 ) a a 4 δ(a 6 ) a δ(a 4 ) a 6 a δ(a 4 ) a 6 a δ(a 6 ) a 4 4 )µ 2, / ( δ(a 4) a 6 a a 4 5 δ(a 4 ) a a 4 2 δ(a 4 ) a δ(a 6 ) a 6 2 a δ(a 6 ) a δ(a 4 ) a 6 2 a δ(a 4 ) a a 4 4 δ(3) δ(a 6 ) a 6 2 a δ(a 6 ) a 6 a δ(a 6 ) a 6 a δ(a 6 ) a 6 3 a δ(3) a δ(2) a a 4 2 δ(a 6 ) δ(2) a δ(a 6 ) a δ(2) a δ(3) a δ(2) a δ(3) a δ(3) a 4 2 a δ(a 4 ) a δ(a 6 ) a a 4 2 δ(a 4 ) a a 4 3 δ(3) a δ(a 6 ) a δ(a 6 ) a δ(2) a 4 3 a δ(a 4 ) a δ(2) a 6 2 a δ(3) a 6 2 a a 4 δ(a 6 ) a 6 )µ, / ( δ(a 4) a 6 a a 4 δ(a 4 ) a a 4 4 δ(a 4 ) a 6 (1+) a 4 5 δ(a 4 ) a a 4 2 δ(a 4 ) a δ(a 6 ) a 6 2 a δ(a 6 ) a δ(2) a δ(a 4 ) a 6 2 a a 4 5 δ(a 6 ) a a 4 4 δ(a 4 ) a 4 4 δ(a 4 ) a a 4 5 δ(a 4 ) a a 4 δ(a 4 ) a 4 δ(a 4 ) a 6 (3 +1) 2304 δ(a 4 ) a a 4 2 δ(a 6 ) a a 4 4 δ(3) 14 δ(3) a a 4 2 δ(a 4 ) a 6 (1+2 ) δ(a 4 ) a 4 4 a δ(a 6 ) a 6 a δ(a 4 ) a 4 3 a 6 (1+) δ(a 4 ) a a 4 2 δ(a 4 ) a a 4 5 δ(3) δ(3) a 6 2 a a 4 2 δ(a 6 ) a δ(a 6 ) a a 4 (7 ) δ(a 4 ) a 4 2 δ(a 4 ) a a 4 3 δ(3) a a 4 δ(a 4 ) a δ(a 6 ) a a 4 δ(a 4 ) a 6 (1+) 6912 δ(2) a 4 3 a 6 )µ 0, / 3 )

18 H 6 is ( 1 2 ( a 4 δ(a 4 ) a a 4 5 δ(a 4 ) a a 4 2 δ(a 4 ) a δ(a 6 ) a 6 2 a δ(a 6 ) a δ(2) a δ(a 4 ) a 6 2 a a 4 5 δ(a 6 ) a a 4 4 δ(a 4 ) a 4 4 δ(a 4 ) a a 4 δ(a 4 ) a 4 2 δ(a 6 ) a δ(3) a δ(a 6 ) a 6 a a 4 δ(a 6 ) δ(a 4 ) a 6 a a 4 4 δ(a 6 ) a δ(a 4 ) a 6 3 a a 4 4 δ(a 6 ) 36 δ(a 4 ) a δ(a 4 ) a a 4 δ(a 6 ) a δ(a 4 ) a a 4 5 δ(3) δ(3) a 6 2 a a 4 2 δ(a 6 ) a δ(a 6 ) a a 4 (7 ) δ(a 4 ) a 4 2 δ(a 4 ) a a 4 3 δ(3) a a 4 δ(a 4 ) a δ(a 4 ) a 6 5 and H 7 is a 4 (7 ) δ(a 6 ) 36 δ(2) a 4 a δ(3) a 6 a a 4 δ(a 6 ) a δ(a 4 ) a 4 3 a a 4 4 δ(3) a a 6 3 δ(3) a 4 )µ 2,3 / 3 ( 9 a 4 δ(a 6 ) 2 δ(3) + 12 δ(a 4 ) δ(2) a δ(a 4 ) δ(3) a δ(a 4 ) δ(2) a δ(a 4 ) a 4 δ(a 6 ) a δ(a 4 ) δ(3) a a 4 δ(a 6 ) 2 a a 4 5 δ(a 6 ) δ(a 4 ) δ(2) a 4 3 a a 4 2 δ(a 6 ) δ(3) a δ(a 4 ) δ(2) a 6 2 a a 4 3 δ(a 6 ) δ(3) δ(a 4 ) a 4 δ(a 6 ) a δ(a 4 ) δ(2) a δ(a 4 ) δ(3) a a 4 4 δ(a 6 ) δ(3) 2304 a 4 4 δ(a 6 ) δ(2) δ(a 4 ) δ(a 6 ) a 6 2 a δ(a 4 ) δ(a 6 ) a 4 (3 ) 9072 δ(a 4 ) δ(a 6 ) a δ(a 4 ) a 4 2 δ(a 6 ) a δ(a 4 ) a 4 (4 ) δ(a 6 ) a a 4 4 δ(a 6 ) a 4 2 δ(a 4 ) 2 a δ(a 4 ) a 4 3 δ(3) a δ(a 4 ) a 4 4 δ(3) 1440 δ(a 4 ) a 4 2 δ(a 6 ) a 4 δ(a 6 ) 2 a a 4 2 δ(a 4 ) 2 a a 4 5 δ(a 4 ) 2 a δ(a 4 ) δ(2) a δ(a 4 ) δ(3) a 6 2 a δ(a 4 ) δ(3) a 4 2 a δ(a 4 ) a 4 3 δ(a 6 ) a a 4 3 δ(a 6 ) 2 a a 4 6 δ(a 4 ) a 4 5 δ(a 4 ) δ(a 4 ) δ(a 6 ) a 4 (6 ) δ(a 4 ) a 4 5 δ(a 6 ) a 4 3 δ(a 4 ) 2 a δ(a 4 ) δ(a 6 ) a a 4 4 δ(a 4 ) 2 a a 4 3 δ(a 4 ) a 4 2 δ(a 6 ) a 4 2 δ(a 6 ) 2 a a 4 4 δ(a 6 ) 2 a 6 )γ 2, /( 3 )