) Binary Cubic Forms / 25
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1 ) Binary Cubic Forms / 25
2 ) Binary Cubic Forms / 25
3 1.1 ( ) = 5+7, 16 = 5+11, 36 = 7+29, 1.2 ( ) p p , ) Binary Cubic Forms / 25
4 2 ) Binary Cubic Forms / 25
5 a,b,c (a 0) x ax 2 +bx+c = 0 2 D = b 2 4ac 2 2 2x 2 2x 3 = 0, x 2 4x 3 = 0 ( 2) ( 3) = 4+24 = 28, ( 3) = = 28. ) Binary Cubic Forms / 25
6 2 x = b± D 2a (D = b 2 4ac) 2 x = 2± x 2 2x 3 = 0, = 1± 7, α = x 2 4x 3 = 0, x = 4± 28 = 2± 7, β = β +1 β +1 = = (5+2 7)(3 7) (3+ 7)(3 7) = 1+ 7 = α. 2 ) Binary Cubic Forms / 25
7 2 2X +1 1 x = X +1 2 ( ) 2 2X +1 2X = 0, X +1 X +1 2(2X +1) 2 2(2X +1)(X +1) (X +1) 2 = 0, 8X 2 +8X +2 4X 2 6X 2 X 2 2X 1 = 0, X 2 4X 3 = 0. (2 ) 1 2 x y ) Binary Cubic Forms / 25
8 a,b,c x, y 2 f(x,y) = ax 2 +bxy +cy D(f) = b 2 4ac D(f) < 0 a > 0 ) Binary Cubic Forms / 25
9 f(x,y) = 2x 2 2xy 3y 2, g(x,y) = x 2 4xy 3y 2 f(x,y) x 2x+y, y x+y f(2x+y,x+y) = 2(2x+y) 2 2(2x+y)(x+y) 3(x+y) 2 = 2(4x 2 +4xy +y 2 ) 2(2x 2 +3xy +y 2 ) 3(x 2 +2xy +y 2 ) = x 2 4xy 3y 2 = g(x,y). ) Binary Cubic Forms / 25
10 2 2 a,b,c f(x,y) = ax 2 +bxy +cy 2, g(x,y) = a x 2 +b xy +c y 2 p,q,r,s ps qr = 1 (x,y) (px+ry,qx+sy) g(x,y) = f(px+ry,qx+sy) f(x,y) g(x,y) f(x,y) g(x,y) ) Binary Cubic Forms / 25
11 f(x,y), g(x,y) D(f) = D(g) ) Binary Cubic Forms / 25
12 f(x,y) = 34x 2 42xy +13y 2 D(f) = = 4 g 1 (x,y) = f(y, x) = 13x 2 +42xy +34y 2 g 2 (x,y) = g 1 (x 2y,y) = 13(x 2y) 2 +42(x 2y)y +34y 2 = 13x 2 10xy +2y 2, g 3 (x,y) = g 2 (y, x) = 2x 2 +10xy +13y 2, g 4 (x,y) = g 3 (x 2y,y) = 2(x 2y) 2 +10(x 2y)y +13y 2 = 2x 2 +2xy +y 2, g 5 (x,y) = g 4 (y, x) = x 2 2xy +2y 2, g 6 (x,y) = g 5 (x+y,y) = (x+y) 2 2(x+y)y +2y 2 = x 2 +y 2. ) Binary Cubic Forms / 25
13 34x 2 42xy +13y 2 x 2 +y x 2 +y 2 ) Binary Cubic Forms / 25
14 2 2 m 0 m 2 2 h f 1 (x,y),...,f h (x,y) m 2 2 f i (x,y) h h = h(m) m x 2 +y 2 h( 4) = 1 ) Binary Cubic Forms / 25
15 f(x,y) = x 2 2y 2 D(f) = 8 p = 3, q = 2, r = 4, s = 3 ps qr = = 9 8 = 1 f(3x+4y,2x+3y) = (3x+4y) 2 2(2x+3y) 2 = 9x 2 +24xy +16y 2 2(4x 2 +12xy +9y 2 ) = x 2 2y 2 = f(x,y). f(x,y) D(f) > 0 D(f) < ) Binary Cubic Forms / 25
16 2 2 ( ) 2.4 (Baker, Stark 1966) h(m) = 1 m 9 3, 4, 7, 8, 11, 19, 43, 67, h(m) = 1 m ) Binary Cubic Forms / 25
17 a,b,c,d x, y 3 f(x,y) = ax 3 +bx 2 y +cxy 2 +dy D(f) = b 2 c 2 +18abcd 4ac 3 4b 3 d 27a 2 d 2 ) Binary Cubic Forms / 25
18 f(x,y) = ax 3 +bx 2 y+cxy 2 +dy 3, g(x,y) = a x 3 +b x 2 y+c xy 2 +d y 3 p,q,r,s ps qr = 1 (x,y) (px+ry,qx+sy) g(x,y) = f(px+ry,qx+sy) f(x,y) g(x,y) f(x,y) g(x,y) ) Binary Cubic Forms / 25
19 f(x,y) g(x,y) D(f) = D(g) m 0 m 2 3 h f 1 (x,y),...,f h (x,y) m 2 3 f i (x,y) h h = h(m) m 2 3 ) Binary Cubic Forms / 25
20 2 3 f(x,y) = ax 3 +bx 2 y +cxy 2 +dy 3 b,c 3 m ĥ(m) 2 3 f(x,y) m h 1 (m), h 2 (m) ĥ 1 (m), ĥ2(m) ) Binary Cubic Forms / 25
21 2 3 f(x,y) = x 3 +3x 2 y y 3 D(f) = ( 1) 27( 1) 2 = = 81. f( x y,x) = ( x y) 3 +3( x y) 2 x x 3 = x 3 3x 2 y 3xy 2 y 3 +3x 3 +6x 2 y +3xy 2 x 3 = x 3 +3x 2 y y 3 = f(x,y). ) Binary Cubic Forms / 25
22 3.2 (Nakagawa 1998) m ĥ 1 (27m)+ 1 3ĥ2(27m) = h( m), 3h 1 (m)+h 2 (m) = ĥ( 27m). ) Binary Cubic Forms / 25
23 1995 m 2 ) Binary Cubic Forms / 25
24 [3] ) Binary Cubic Forms / 25
25 [1] [2] J. Nakagawa, On the relations among the class numbers of binary cubic forms, Invent. math. 134, (1998) [3] [4], [5] F ) Binary Cubic Forms / 25
> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3
13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >
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[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
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[ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =
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(u(x)v(x)) = u (x)v(x) + u(x)v (x) ( ) u(x) = u (x)v(x) u(x)v (x) v(x) v(x) 2 y = g(t), t = f(x) y = g(f(x)) dy dx dy dx = dy dt dt dx., y, f, g y = f (g(x))g (x). ( (f(g(x)). ). [ ] y = e ax+b (a, b )
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