2016 5 2 ) Binary Cubic Forms 2016 5 2 1 / 25
1 2 2 2 3 2 3 ) Binary Cubic Forms 2016 5 2 2 / 25
1.1 ( ) 4 2 12 = 5+7, 16 = 5+11, 36 = 7+29, 1.2 ( ) p p+2 3 5 5 7 11 13 17 19, 29 31 41 43 ) Binary Cubic Forms 2016 5 2 3 / 25
2 ) Binary Cubic Forms 2016 5 2 4 / 25
2 2.1 3 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) 2 4 2 ( 3) = 4+24 = 28, 4 2 4 1 ( 3) = 16+12 = 28. ) Binary Cubic Forms 2016 5 2 5 / 25
2 x = b± D 2a (D = b 2 4ac) 2 x = 2± 28 4 2x 2 2x 3 = 0, = 1± 7, α = 1+ 7 2 2 x 2 4x 3 = 0, x = 4± 28 = 2± 7, β = 2+ 7 2 2β +1 β +1 = 5+2 7 3+ 7 = (5+2 7)(3 7) (3+ 7)(3 7) = 1+ 7 = α. 2 ) Binary Cubic Forms 2016 5 2 6 / 25
2 2X +1 1 x = X +1 2 ( ) 2 2X +1 2X +1 2 3 = 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 2016 5 2 7 / 25
2 2 2.2 3 a,b,c x, y 2 f(x,y) = ax 2 +bxy +cy 2 2 2 D(f) = b 2 4ac D(f) < 0 a > 0 ) Binary Cubic Forms 2016 5 2 8 / 25
2 2 28 2 2 2 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 2016 5 2 9 / 25
2 2 a,b,c 2 2 2 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 2016 5 2 10 / 25
2.3 2 2 2 f(x,y), g(x,y) D(f) = D(g) ) Binary Cubic Forms 2016 5 2 11 / 25
2 2 2 2 f(x,y) = 34x 2 42xy +13y 2 D(f) = 42 2 4 34 13 = 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 2016 5 2 12 / 25
34x 2 42xy +13y 2 x 2 +y 2. 4 2 2 x 2 +y 2 ) Binary Cubic Forms 2016 5 2 13 / 25
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 2 2 4 2 2 x 2 +y 2 h( 4) = 1 ) Binary Cubic Forms 2016 5 2 14 / 25
f(x,y) = x 2 2y 2 D(f) = 8 p = 3, q = 2, r = 4, s = 3 ps qr = 3 3 2 4 = 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) < 0 2 4 6 ) Binary Cubic Forms 2016 5 2 15 / 25
2 2 (1777 1855) 2.4 (Baker, Stark 1966) h(m) = 1 m 9 3, 4, 7, 8, 11, 19, 43, 67, 163. 2.5 h(m) = 1 m ) Binary Cubic Forms 2016 5 2 16 / 25
2 3 3.1 4 a,b,c,d x, y 3 f(x,y) = ax 3 +bx 2 y +cxy 2 +dy 3 2 3 D(f) = b 2 c 2 +18abcd 4ac 3 4b 3 d 27a 2 d 2 ) Binary Cubic Forms 2016 5 2 17 / 25
2 3 2 2 3 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 2016 5 2 18 / 25
2 3 2 2 2 2 3 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 2016 5 2 19 / 25
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) 1 3 2 3 m h 1 (m), h 2 (m) ĥ 1 (m), ĥ2(m) ) Binary Cubic Forms 2016 5 2 20 / 25
2 3 f(x,y) = x 3 +3x 2 y y 3 D(f) = 4 3 3 ( 1) 27( 1) 2 = 4 27 27 = 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 2016 5 2 21 / 25
3.2 (Nakagawa 1998) m ĥ 1 (27m)+ 1 3ĥ2(27m) = h( m), 3h 1 (m)+h 2 (m) = ĥ( 27m). ) Binary Cubic Forms 2016 5 2 22 / 25
1995 m 2 ) Binary Cubic Forms 2016 5 2 23 / 25
[3] ) Binary Cubic Forms 2016 5 2 24 / 25
[1] 2 2012 [2] J. Nakagawa, On the relations among the class numbers of binary cubic forms, Invent. math. 134, 101-138 (1998) [3] http://www.math.tohoku.ac.jp/people/ohno.html [4], http://www.shokabo.co.jp/column-math/list.html [5] F 3 2014. ) Binary Cubic Forms 2016 5 2 25 / 25