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1 2 2 MATHEMATICS.PDF (p n ), ( ) GL 2 (Z) SL 2 (Z) SL 2 (Z)

2

3 , a 0 + a + a b b 2 b 3 () + b n a n + b n a n., () b 2 b n a 0 + b. (2) a + a a n. a 0, a,..., a n, b, b 2,..., b n 0, (). n.,. a 0 + b a = a 0a + b a n = k, ()., b 2 b k a k + b k a k = a kb k a k a k + b k b k 2 b k b k a k a 0 + b a + a a k 2 + a k + = a 0 + b b 2 b k 2 a k b k. a + a a k 2 + a k a k + b k, n = k () n = k. n = k ()., n, (). 3

4 , a 0 + a + a a n + a n (3)., a 0 + a + a [a 0, a, a 2,..., a n ] a n ). a 0, a, a 2,..., a n (3)., [a 0, a, a 2,..., a n ] = [a 0, [a, a 2,..., a n ]] = [a 0, a, [a 2,..., a n ]] = = [a 0, a, a 2,..., a n 2, [a n, a n ]]. [a 0, a, a 2,..., a n ], 0., [a, a 2,..., a n ] 0, [a 2,..., a n ] 0,..., [a n, a n ] 0, a n 0 (4).,, [a 0, a, a 2,..., a n ]., a, a 2,..., a n, (4), [a 0, a, a 2,..., a n ]. a 0, a, a 2,..., a n, [a 0, a, a 2,..., a n ]. ) [] 2 [k 0, k,..., k n ], [a 0, a,..., a n ],. 4

5 a, a 2,..., a n 0,. n = 3, a = 0, a 0 + = a 0 + = a 0 + a 2 + a 3 a + a 2 + a 3 a 2 + a 3..2 [, 2, 3] = = = = [ 4, 3, 2] = = = = n 0. a 0, a, a 2,..., a n., [a 0, a, a 2,..., a n ], n = 0 n =, a =. n 2., s = a + a a n [a 0, a,..., a n ] = a 0 + s (5)., n 2, s >, (5).., n., n 0, n =, [a 0, a ] = a 0 + a. a >,.., n 0, n = a =..5 a, b, α, β, 0 < α <, 0 < β <., a + α = b + β, a = b α = β. 5

6 0 < α <, 0 < β <, < β < β α < α <. a + α = b + β, a b = β α, < a b <. a b, a b = 0., β α = 0..6 m, n, n m. a 0, b 0, a, a 2,..., a n, b, b 2,..., b m., [a 0, a,..., a n ] = [b 0, b,..., b m ],. (i) m = n, a 0 = b 0, a = b,..., a n = b n (ii) m = n +, a 0 = b 0, a = b,..., a n = b n +, b n+ = 0 i n i, s i = a i+ + a i t i = b i+ + a i i n 2, s i >, t i >,.5 a n b m.,.,. a i + s i = b i + t i = a i = b i a 0 = b 0, a = b, a 2 = b 2,..., a n 2 = b n 2 a n + a n = b n + t n (6) a n >, (6), t n >..5 a n = b n, a n = t n. a n,.4, m = n, a n = b n m = n +, a n = b n +, b n+ =. a n =, (6), t n =. m n + t n >, m = n, b n =. 6

7 .7 m, n, n m. a 0, b 0, a, a 2,..., a n, b, b 2,..., b m., s, t., [a 0, a,..., a n, s] = [b 0, b,..., b m, t], a 0 = b 0, a = b,..., a n = b n s = [b n+, b n+2,..., b m, t]., m = n s = t. 0 i n i,, s i >, t i >,.5 s i = a i+ + a i a n + s t i = b i+ + a i b m + t.,., a i + s i = b i + t i = a i = b i, s i = t i a 0 = b 0, a = b, a 2 = b 2,..., a n = b n a n + s = b n + b n+ + + b m + t. s >, t >,.5, a n = b n., m = n s = t. s = b n+ + b n b m + t 2 (p n ), ( ) (a n ), (p n ), ( ), p n = a n p n + p n 2, p =, p 2 = 0, = a n + 2, q = 0, q 2 = (7). n. a, a 2,..., a n, p, p 2,..., p n q, q 2,...,, p n,. 7

8 2. n a n., n (i) n (ii) < +. (i) n., q 2 =, q = 0, q 0 = a 0 q + q 2 =. a, a 2, q = a q 0 + q = a, q 2 = a 2 q + q 0 = a a n 3, k < n k k q k, a n = a n (n ) + (n 2) = 2n 3 n. (8), n n. (ii), q = a q 0 + q = a, q 2 = a 2 q + q 0 = a a 2 +. a, a 2, q < q 2. n 3, a n, (i) n 2 2, = a n (n 2) >. 2.2 = q 0 q, n 0 +., n 4 (8) 2n 3 > n, n 4 n <. 2.3 a 0, n a n., n 0 8

9 (i) n < p n (ii) p n < p n+. (i) n., p 2 = 0, p =, p 0 = a 0 p + p 2 = a 0. a, a 2, p = a p 0 + p = a a n 2, k < n k k < p k, a n p n = a n p n + p n 2 p n + p n 2 > (n ) + (n 2) = 2n 3 n., n 0 n < p n. (ii), p 0 = a 0, p = a a 0 +, a, p 0 < p. n 2, a n, (ii) n 2 < p n 2, p n = a n p n + p n 2 > p n + (n 2) > p n. 2.4 n a n., a 0, lim =. n lim p n =. n 2., n n. n. a 0, 2.3, n p n n. n p n. 2.5 n 2, (i) p n + p n+ = ( ) n 9

10 (ii) p n, gcd(p n, ) =. (i) n., p q 2 p 2 q =, n = 2 (i). n = k (i), p k+ q k p k q k+ = (a k+ p k + p k )q k p k (a k+ q k + q k ) = (p k q k p k q k ) = ( ) k = ( ) k. n = k (i)., n (i). (ii) g = gcd(p n, ) >, (i) g, g., g,. g =. 2.6 n a n > 0., p 0 q 0 < p 2 q 2 < < p 2k q 2k < p 2k+2 q 2k+2 < < p 2k+3 q 2k+3 < p 2k+ q 2k+ < < p 3 q 3 < p q., k. 2.5 p n+2 +2 p n = p n+2 p n p n+ p n + = ( ) n (9), p n+2 p n +2 = (a n+2 p n+ + p n ) p n (a n ) = (p n+ p n + )a n+2 = ( ) n a n+2. p n+2 +2 p n = ( )n a n

11 > 0, +2 > 0, a n+2 > 0, n = p n+2 +2 p n > 0 = p n+2 +2 > p n, (0) n = p n+2 +2 p n < 0 = p n+2 +2 < p n ()., n p n /, n p n /., (9) > 0, + > 0, p n+ p n = p n+ p n + = ( )n n = p n+ + p n > 0 = p n+ + > p n, (2) n = p n+ + p n < 0 = p n+ + < p n (3)., u 0 v, u < v, u < v +, (0) (3) p u q u < p v+ q v+ < p v q v, v < u, v < u +, () (2) p u q u < p u+ q u+ < p v q v.,,.. 3 (a n ), (p n ), ( ) (a n ) 2 (7). 3. n 0, 0 t, [a 0, a,..., a n, t] = tp n + p n 2 t + 2 (4)., n 2, a, a 2,..., a n. n. p =, p 2 = 0, q = 0, q 2 =, [t] = tp + p 2 tq + q 2.

12 , p 0 = a 0 p + p 2 = a 0, q 0 = a 0 q + q 2 =, [a 0, t] = a 0 + t = ta 0 + t, n = 0, (4). = tp 0 + p tq 0 + q. n 2, n (4), 3., [ [a 0, a,..., a n 2, a n, t] = a 0, a,..., a n 2, a n + ] t,, n (4). = (a n + /t)p n 2 + p n 3 (a n + /t) (a n + /t)p n 2 + p n 3 (a n + /t) = (ta n + )p n 2 + tp n 3 (ta n + ) 2 + t 3 = t(a n p n 2 + p n 3 ) + p n 2 t(a n ) + 2 = tp n + p n 2 t + 2., t a, a 2,..., a n, 0., n 0 (4). 3.2 n 0, 0 s, t, [a 0, a,..., a n, s] [a 0, a,..., a n, t] = ( ) n (s t) (s + 2 )(t + 2 )., n 2, a, a 2,..., a n. 2.5, p n 2 p n 2 = ( ) n 2 = ( ) n. 2

13 , 3., [a 0, a,..., a n, s] [a 0, a,..., a n, t] = sp n + p n 2 s + 2 tp n + p n 2 t + 2 = (sp n + p n 2 )(t + 2 ) (s + 2 )(tp n + p n 2 ) (s + 2 )(t + 2 ) = stp n + sp n 2 + tp n 2 + p n 2 2 (stp n + tp n 2 + sp n 2 + p n 2 2 ) (s + 2 )(t + 2 ) = (s t)(p n 2 p n 2 ) (s + 2 )(t + 2 ) ( ) n (s t) = (s + 2 )(t + 2 ). 3.3 n 0. n 2, a, a 2,..., a n., s, t. (a) [a 0, a,..., a n, s] = [a 0, a,..., a n, t] s = t. (b) n, [a 0, a,..., a n, s] < [a 0, a,..., a n, t] s < t. (c) n, [a 0, a,..., a n, s] < [a 0, a,..., a n, t] s > t n 0., n, a, a 2,..., a n.,. [a 0, a,..., a n ] = p n n, 3., t = a n, [a 0, a,..., a n ] = a np n + p n 2 a n + 2 = p n 3

14 ., p =, p 2 = 0, q = 0, q 2 =, p 0 = a 0 p + p 2 = a 0, q 0 = a 0 q + q 2 =,. n = 0. p 0 q 0 = a 0 = [a 0 ] 0 n m, ω = [a 0, a, a 2,..., a m ], [a 0, a, a 2,..., a n ] = p n ω n. gcd(p n, ) = > 0, p n /. 3.5 n a n. c n = [a 0, a, a 2,..., a n ], n = 0,, 2,... (c n ), ω. lim c n = ω n 3.4, n 0. c n = [a 0, a, a 2,..., a n ] = p n m, n, n m. 2.6, p m p n q m p n+ p n +., 2.5, 2., n, + n +, p n+ p n = p n+ p n + = ( )n p m p n q m ( )n. + ( ) n + n(n + ) n 2. 4

15 , p m p n q m n 2. ε > 0. δ = / ε, n > δ n. n 2 < δ 2 = ε m n > δ = p m p n q m < ε. (c n )., ω, lim n c n = ω. 3.6 f(x) [a, ), lim x f(x) = l. (i) f(a) < l, c f(a) < c < l, f(ξ) = c ξ > a ξ. (ii) l < f(a), c l < c < f(a), f(ξ) = c ξ > a ξ. (i) b a < b l f(b), [a, b], ξ f(ξ) = c a < ξ < b. x > a x f(x) < l, lim x f(x) = l, ε > 0, δ, x ε = (l c)/2, x > δ = f(x) l < ε. x > δ = l f(x) < l c 2 = f(x) > l + c 2 t = max{δ, a}, f(a) < c < f(t). [a, t], ξ f(ξ) = c a < ξ < t. (ii) (i). > c. 3.7 n 0. n, a, a 2,..., a n., ω. (i) [a 0, a,..., a n, a n+ ] < ω < [a 0, a,..., a n ], s, s > a n+. ω = [a 0, a,..., a n, s] 5

16 (ii) [a 0, a,..., a n ] < ω < [a 0, a,..., a n, a n+ ], s, s > a n+ ω = [a 0, a,..., a n, s]. f(x) = [a 0, a,..., a n, x]. f(x) a j + x (j = 0,,..., n) /x, (0, )., f(a n+ ) = [a 0, a,..., a n, a n+ ]., f(x) = [a 0, a,..., a n, a n + /x], lim f(x) = [a 0, a,..., a n ]. x, s s > a n+ ω = [a 0, a,..., a n, s], (i), (ii), 3.6 (i), (ii). 4 ω. ω, ω a 0, ω = a 0 + ω (5) ω. ω. ω = ω a 0 >, ω, ω a, ω = a + ω 2 (6) ω 2. ω 2. (6) (5), ω 2 = ω a > ω = a 0 + = [a 0, a, ω 2 ] a + ω 2 6

17 ., a, a 2,..., a n, ω n >, ω = a 0 + = [a 0, a, a 2,..., a n, ω n ] (7) a + a a n + ω n. n 0 (7), ω., ω n (7) n. ω, a 0 = q, q = a 0 = [a 0 ].. ω, m, n, ω = m/n, n > 0. ω,, m = a 0 n + r, 0 < r < n, n = a r + r 2, 0 < r 2 < r, r = a 2 r 2 + r 3, 0 < r 3 < r 2,, r n = a n r n + r n+, 0 < r n+ < r n, r n = a n+ r n+ a 0, a,..., a n, a n+, r, r 2,..., r n, r n+., n, r, r 2,..., r n, a, a 2,..., a n., m n = a 0 + r n = a 0 + n/r, n r = a + r 2 r = a + r /r 2, r r 2 = a 2 + r 3 r 2 = a 2 + r 2 /r 3,, r n = a n + r n+ = a n +, r n r n r n /r n+ r n = a n+ r n+.,, ω = m n = a 0 + = [a 0, a,..., a n+ ] a + a a n+.,. 7

18 4. q, a 0 a, a 2,..., a n, a n+ q = [a 0, a,..., a n, a n+ ]., ω = n/r, ω 2 = r /r 2, ω 3 = r 2 /r 3,..., ω n = r n /r n, ω n+ = r n /r n+., ω, = = = = [, 2, 3] = = + 7 = 4 + = [ 4, 3, 2] = = = = 3 + = [3, 7, 7] ω, n ω n+, ω = a 0 + a + a a n + ω n+,.., n ω n., ω, ω = + ( 2 ) = = + = 2 + ( 2 ) =

19 = + ( 3 ) = = + = + = = ( 3 ) = ω. (a n ), n a n, (c n ) c n = [a 0, a, a 2,..., a n ] (n = 0,, 2,...),., ω. lim c n = ω n ω, 4., b 0 b, b 2,..., b m ω = [b 0, b,..., b m ]. lim n c n = ω, (c 2k ) (c 2k+ ) (c n ), ω. 2.6, k 0 c 2k < ω < c 2k+. s > a n+ 3.7, n > m n, s, [b 0, b,..., b m ] = [a 0, a,..., a n, s]..7 b m = [a m, a m+,..., a n, s]., b m [a m, a m+,..., a n, s].. ω. 9

20 5, ω, ω = [a 0, a,..., a n, ω n ] (n = 0,, 2,...) ω. ω (a n )., n a n, ω n >. (a n ), (p n ), ( ),. 3.4, p n = a n p n + p n 2, p =, p 2 = 0, = a n + 2, q = 0, q 2 = [a 0, a,..., a n ] = p n. p n / ω n. 5. n 0 ω p n ( ) n = ( ω n+ + ) (8). 3.,. [a 0, a,..., a n, ω n+ ] = ω n+p n + p n ω n+ +, 2.5, ω p n = [a 0, a,..., a n, ω n+ ] p n = ω n+p n + p n ω n+ + p n = (ω n+ p n + p n ) p n (ω n+ + ). (ω n+ + ) (ω n+ p n + p n ) p n (ω n+ + ) = (p n p n ) = ( ) n = ( ) n. (8). 5.2 p 0 q 0 < p 2 q 2 < < p 2k q 2k < p 2k+2 q 2k+2 < < ω < < p 2k+3 q 2k+3 < p 2k+ q 2k+ < < p 3 q 3 < p q. 20

21 n 0. 5., > 0, > 0, ω n+ >, ω p n ( ) n = ( ω n+ + ). n = ω p n > 0 = ω > p n, n = ω p n < 0 = ω < p n., n p n / ω, n p n / ω. 2.6,. 5.3 n ω p n < ω p n (9). 3., ω n+ +, ω = ω n+p n + p n ω n+ +. (ω n+ + )ω = ω n+ p n + p n., ( ω n+ (ω p n ) = (ω p n ) = ω p ) n. ω n+,, ω p n = ω n+ ω p n. 0 < < < ω n+ (9). 0 < ω n+ <. 5.4 n 0 ω p n > (20)

22 5.3,, 2.5,, (20). ω p n+ < ω p n + p n+ p n + ω p n+ + ω p n < 2 ω p n. + p n+ p n = ( ) n p n+ p n + = p n+ p n = n 0 ω p n < +. q 2 n > 0, + > 0, ω n+ > a n+, ω n+ + > a n+ + = +., 5., ω p n = ( ) n ( ω n+ + ) = ( ω n+ + ) <. +, 2. ( 2.2), n 0 +,. 5.6 n 0, δ n,. p n = ω + δ n, δ n < δ n = (p n ω). ω., 5.5, ω p n <. q 2 n 22

23 ,,,, ω p n <. < p n ω <. < (p n ω) <., < δ n <.,. 5.7 p n lim = ω. n 5.5, n 0 ω p n <., 2.4 (n ), /qn 2 (n )., ω p n / 0 (n ). q 2 n 5.8 ω, p n / (n 0) ω. p, q, q > 0., qω p < ω p n = + q.. qω p < ω p n q < +., p n x + p n+ y = p, x + + y = q (2)., 2 p n,, 2.5 p n+ p n + = ( ) n, (p n+ p n + )y = p qp n. y = ( ) n (p qp n )., +, 2 p n+,, (p n+ p n + )x = qp n+ p+. 23

24 , x = ( ) n (qp n+ p+ ).., (2) (x, y)., x 0, y 0. x = 0, qp n+ = p gcd(p n, ) =, q k+ q. q +., x 0., x, x ω p n ω p n. y = 0, p n x = p, x = q, qω p = x( ω p n )., qω p = x ω p n ω p n.. y 0. x y., y < 0 = x = q + y > 0 = x > 0, y > 0 = + y + > q = x = q + y < 0 = x < 0. ω, 5.2 p n / < ω < p n+ /+ p n+ /+ < ω < p n /., ω p n + ω p n. x, y (2), qω p = ( x + + y)ω (p n x + p n+ y) = x( ω p n ) + y(+ ω p n+ )., x( ω p n ) y(+ ω p n ). x, qω p = x ω p n + y + ω p n+ x ω p n ω p n..,. 24

25 5.9 ω, p n / (n ) ω., p/q., p, q Z, q > 0, gcd(p, q) =., ω p q < ω p n = < q.. n, ω p/q < ω p n / q, qω p = q ω p q < ω p n = ω p n., 5.8, + q., +. n, ω, p/q., p, q Z, q > 0, gcd(p, q) =., ω p q < 2q 2, p/q ω.. ω p/q < /2q 2 p/q ω, n 0, p/q p n / qp n p, qp n p = qp n p q q q = p n p q ( ) ( = pn ω + ω p ) q ω p n + ω p q < ω p n + 2q 2. q 0 =, 2. ( ) n, k 0, q k q < q k+., 5.8, ω p k = q kω p k, qq k < q k q k qω p q k <. 2qq k = q q k ω p q ω p k + 2q 2 < + 2qq k 2q 2., q < q k, q k q. q k 25

26 5. ω. ω 2 p n /, p n+ /+, ω p n < ω p n+ < (22). 2q 2 n + 2q 2 n+ n , p n / < ω < p n+ /+ p n+ /+ < ω < p n /., p n+ p ( ) ( n pn+ = ω + ω p ) n + +, p n+ + p n > 0, p n+ + ω > 0, ω p n > 0 p n+ p n + = ω p n+ + ω p n.,., 2.5 p n+ p n + = ( ) n, p n+ p n + = p n+ p n + =. + +,, (22), + ω p n+ + ω p n = q 2 n+ + 2qn q 2 nq 2 n+, ( + ) 2 0., = +. n, 2. < +,., n = 0,, q = q 0 =., q = a q 0 + q = a a =. 5.2, p 2 q 2 < ω < p q p = a 0 + = a 0 +, q a p 2 = a 0 + q 2 a + = a 0 + a 2 + a 2 = a 0 + a 2 a 2 +, 0 < p q ω < p q p 2 q 2 = a 2 a , (22),. 26

27 5.2 ω > 0, x/y, x/y ω, y/x /ω., ω >, ω ω = a 0 + a + a a n +, /ω ω = 0 + a 0 + a + a a n +., ω > a 0. x/y ω n, x y = a 0 + a + a , y/x y x = 0 + a 0 + a + a 2 + +, /ω n +. 0 < ω <, ω > /ω., /ω ω = a 0 + a + a a n +, /(/ω) = ω ω = 0 + a 0 + a + a a n +. x/y ω n, y/x /ω n. a n a n 6 GL 2 (Z) SL 2 (Z) 2 ± GL 2 (Z) : GL 2 (Z) = P = p q p, q, r, s Z, det P = ps qr = ± r s. 6. GL 2 (Z). 2 P, Q GL 2 (Z), det P Q = det P det Q = ±, P Q GL 2 (Z). 27

28 GL 2 (Z) E = 0. 0 P = p q GL 2 (Z), P = r s det P P. s r q GL2 (Z) p 2 SL 2 (Z) : SL 2 (Z) = P = p q p, q, r, s Z, det P = ps qr = r s = {P GL 2 (Z) det P = }. 6.2 SL 2 (Z) GL 2 (Z) 2. P, Q GL 2 (Z) det P Q = det P det Q, GL 2 (Z) {±}, P det P. SL 2 (Z).,, GL 2 (Z)/SL 2 (Z) = {±}., [GL 2 (Z) : SL 2 (Z)] = SL 2 (Z), S =, T = det S = det T =, S, T SL 2 (Z). S, T SL 2 (Z) Γ. SL 2 (Z) Γ. p q SL 2 (Z), ps q 0 = p = s = ±., 0 s q = S q, q = S q T

29 , p q Γ., 0 s r 0 = min r p q SL 2 (Z) \ Γ r s, r 0. SL 2 (Z) \ Γ (2, )- r 0, P 0 = p 0 q 0 r 0 s 0., n, n Z, s 0 = r 0 n + n, 0 n < r 0.,, r 0, 0 s 0 r 0 n < r 0. P 0 S T = q 0 p 0 n p 0 Γ. s 0 r 0 n r 0, S T Γ, P 0 Γ.., SL 2 (Z) = Γ. 7 GL 2 (Z) C { }, P = p r, px + q rx + s, x, x s/r P x = p r, x =, x = s/r q GL 2 (Z), x C { }, r 0 s r = 0, px + q, x P x = s, x =. 7. ±q x = x ± q., ±. 0 29

30 P GL 2 (Z) P GL 2 (Z).,, x C { } P x = ( P ) x x = 0 x = /x GL 2 (Z) C { }. x C { }. E x = x. P = p q, Q = p q GL 2 (Z). r s r s x, r x + s 0, (rp + sr )x + (rq + ss ) 0, P Q x = pp + qr pq + qs x rp + sr rq + ss x =, = (pp + qr )x + (pq + qs ) (rp + sr )x + (rq + ss ), P (Q x) = P p x + q p p x + q r r x + s = x + s + q r p x + q r x + s + s = p(p x + q ) + q(r x + s ) r(p x + q ) + s(r x + s ) = (pp + qr )x + (pq + qs ) (rp + sr )x + (rq + ss ). pp + qr rp + sr, r 0 P Q = p r, r = 0 r 0, r = r = 0,,, p Q = r, r 0, r = 0 p P = r, r 0, r = 0 P p r = pp + qr rp + sr 30

31 . r x + s = 0, r = 0, s = 0, p s q r = det Q 0., x = s /r., P (Q x) = P = p/r., P Q x = p/r. (rp + sr )x + (rq + ss ) = 0, rp + sr = 0, rq + ss = 0, (pp + qr )(rq + ss ) (pq + qs )(rp + sr ) = det P Q 0., x = (rq + ss )/(rp + sr )., P Q x =., Q x = s/r, P (Q x) =., P Q x = P (Q x)., P x GL 2 (Z) C { }. x, y C { }, x y, P GL 2 (Z) x = P y. x, y C, x y, p, q, r, s. x = py + q, ps qr = ± ry + s, det P =, det P =. z, w C { }, P SL 2 (Z) w = P z , , 2 + = , 0 = 2 + = 2, 0 =. 7.5 C { }. 3

32 x, y, z C { }. ( ) x = E x, x x. ( ) x y, P GL 2 (Z), x = P y., y = E y = (P P ) y = P (P y) = P x., y x. ( ) x y y z, P, Q GL 2 (Z), x = P y y = Q z., x = P (Q z) = P Q z, x z.,. GL 2 (Z) SL 2 (Z), C { }. 7.6 (i). (ii). (iii)., P GL 2 (Z), x Q { }, P x Q { }.,.,. x, y C, P GL 2 (Z) x = P y. x, y,, x., x, y, y = P x. 7.7 Q { } 2., p/r, gcd(p, r) =, ps qr = q, s Z., P SL 2 (Z), P = p r,. q s p r = P., p /r, p/r, Q SL 2 (Z) p r = Q. 32

33 , p r = P Q p r, P Q SL 2 (Z)., p/r p /r., x 2 + bx + c = 0 2 θ = b + b 2 4c, θ = b b 2 4c 2 2., P = b, det P = θ = P θ x C { }, GL 2 (Z) x = {P GL 2 (Z) x = P x}, SL 2 (Z) x = {P SL 2 (Z) x = P x}., (i) GL 2 (Z) x GL 2 (Z). (ii) SL 2 (Z) x SL 2 (Z). (i) E, E GL 2 (Z) x., GL 2 (Z) x. P, Q GL 2 (Z) x, P Q x = P (Q x) = P x = x, P x = P (P x) = (P P ) x = E x = x., P Q, P GL 2 (Z) x. (ii) (i). 7.0 S =, T = 0., E., 0 0 (i) GL 2 (Z) S, T, E GL 2 (Z). (ii) SL 2 (Z) S, E SL 2 (Z). (i), P = S, T, E, P GL 2 (Z), P =. 33

34 P = p r, q GL 2 (Z), P =., r = 0. s ps = ps qr = det P = ±, p = ±, d = ± ( )., P = ± q 0 ± ( )., q = S q, q = T S q, 0 0 q = S q, q = T S q. 0 0, P S, T, E. (ii) det P = ± det P = (i), P SL 2 (Z) P = SL 2 (Z) S, E. 7. x C, x 2., P GL 2 (Z), x = P x P = ±E., GL 2 (Z) x = SL 2 (Z) x = {±E}., E. P = p r q GL 2 (Z), s., x = P x = px + q rx + s rx 2 + (s p)x q = 0. 34

35 x 2, r = s p = q = 0. ps qr = det P = ±, p = s = ±., P = ±E.,.,, GL 2 (Z) x, SL 2 (Z) x {±E}.,. 8 SL 2 (Z) z,, Re z, Im z : z = Re z + Im z. 8. z C, P = p q GL 2 (Z), r s. Im (P z) = det P Im z rz + s 2, z z. w = P z = pz + q rz + s.,, w = pz + q rz + s. w w = pz + q rz + s pz + q rz + s (pz + q)(rz + s) (pz + q)(rz + s) = (rz + s)(rz + s) (psz + qrz) (psz + qrz) = (rz + s)(rz + s) (ps qr)(z z) = rz + s 2. det P = ps qr, z z = 2 Im z, w w = 2 Im w, Im w = det P Im z rz + s 2. 35

36 8.2 z., S Z Z, (0, 0). R { rz + s (r, s) S \ {(0, 0)} } (23)., rz + s = 0 rz + s = 0 r = s = 0 (r, s) = (0, 0), 0 (23)., (23). x = Re z, y = Im z, rz + s = (s + rx) + ry, rz + s 2 = (s + rx) 2 + r 2 y 2. S \ {(0, 0)} =, S (r 0, s 0 ) (0, 0). R = r 0 z + s 0. R (23), (23) R., rz + s < R (r, s). rz + s < R, z, y 0., (24) 2,, (24), s + rx < R, ry < R. (24) r < R y. (25) s rx < s + rx < R., (25) ( s < R + rx < R + x ). (26) y (25), (26), rz + s < R (r, s). 8.3 z, rz + s., 8.2 (23). 36

37 , : z n, rz s < n, 0 < r n r, s Z., Dirichlet. C H = {z C Im z > 0}., C F = {z C z > /2 Re z < /2} {z C z = /2 Re z 0} SL 2 (Z). z F Im z > 0, F H. 8.4 =, Re = 0, F. ρ = ( + 3)/2 3., ρ =, Re z = /2, ρ F., ρ + = ( + 3)/2 6, ρ + =, Re z = /2, ρ + F. 8.5 z H, w F, z w. P = p r q SL 2 (Z), 8. z H, s Im (P z) = Im z rz + s 2 > 0. S = {(r, s) P SL 2 (Z)} 8.2, rz + s P. P, w 0 = P z., w 0 z H. a, w 0 + a = a w 0, w 0 + a w 0., /2 Re w 0 + a < /2 a, w = w 0 + a., Re (w 0 + a) = Re w 0 + a, Im w = Im w 0., w = w = 0 w 0 37

38 , w w. 8., Im w = Im w w 2 = Im w 0 w 2., w 2 z., Im w 0, Im w Im w 0., w 2 = Im w 0 Im w., w >, w F. w =, w ( = w =, Re ) = Re w w, w w F. 8.6 z, w F, P SL 2 (Z), w = P z., z = w., ±E, z, ρ P = ±E, ±T, z = ±E, ±T S, ±(T S) 2 z = ρ., S =, T = T S = 0, (T S) 2 = 0., ρ = ( + 3)/2 3. Im w < Im z P P, Im z Im w. P = p q, r s det P =, 8., w = P z = pz + q rz + s. (27) Im w = Im z rz + s 2. Im z Im w, rz + s = Im z. (28) Im w 38

39 Im (rz + s) = r Im z, r Im z = Im (rz + s) rz + s. F ρ, 3 = Im ρ Im z. 2, 3 r. 2, r Z, r = 0 ±. r 2 3. r = 0, ps qr =, p = s = ±. (27) z, w F, q = 0., z = w. w = z ± q. r =, P P r =., r =. r =, (28) z + s. (29) z F s Z, z > z + s > (29)., z =., s 0, (29) z = ρ s =., z =, s = 0 z = ρ, s =. z =, s = 0, ps qr =, r = q =. (27) w = p z. z =, z ( = z =, Re ) = Re z. z z, w F, p = 0, /z = p =, /z = ρ +., z = w =,, ρ + = ρ 2 = /ρ, z = w = ρ. z = ρ, s =, ps qr =, r = p q =. p = q +. (27) w =, (q + )ρ + q ρ + = q + ρ = q + (ρ + ). ρ + ρ ρ + = ρ ρ 2 = ρ = ρ2 = ρ +. w F, q =., p = 0, z = w = ρ. 39

40 8.7 z, w H., z w, z w z 0 F. ( ) 8.5, z 0 F z z 0., w 0 F w w 0. z w, z 0 w , z 0 = w 0. ( ) z z 0, z 0 w, z w. 9 2 θ 2, 2 ax 2 + bx + c = 0, gcd(a, b, c) =, a > 0 (30), θ Q., θ. 2 2, θ (30) 2 2, ax 2 + bx + c = 0, gcd(a, b, c) =, a > 0, a x 2 + b x + c = 0, gcd(a, b, c ) =, a > 0. a = au (u Q), 0 = a θ 2 + b θ + c u(aθ 2 + bθ + c) = (b bu)θ + (c cu). θ Q, b bu = c cu = 0., b = bu, c = cu. u, u = m/n, gcd(m, n) =, n > 0, a n = am, b n = bm, c n = cm., n a, b, c. gcd(a, b, c) =, n =., u., m a, b, c. gcd(a, b, c ) =, m = ±., u = ±., a > 0, a > 0 u =., a = a, b = b, c = c, θ (30) 2. 2 θ (30) 2 D θ., θ D 2. 40

41 (30), b + D, 2a b D 2a 2. 2., 2 θ, 2 θ, θ. θ 2, θ 2 (30), 0. D = b 2 4ac 2 2, 2 2., θ 2, θ., θ θ, θ θ., θ, 3 : (i) θ 2. (ii) θ 2 ax 2 + bx + c = 0, a 0, D = b 2 4ac 0. (iii) θ 2 x 2 + b x + c = 0, D = b 2 4c 0., 2 θ, θ, 3 : (i) θ, θ 2. (ii) θ, θ 2 ax 2 + bx + c = 0, a 0, D = b 2 4ac 0. (iii) θ, θ 2 x 2 + b x + c = 0, D = b 2 4c 0. (3),. (i) (ii). (ii) (iii) ax 2 + bx + c = 0 θ, aθ 2 + bθ + c = 0. a, θ 2 + b θ + c = 0, b, c Q., b = b/a, c = c/a., D = b 4a c = b2 4ac a 2 = D a 2 4

42 , D 0, D 0. (iii) (i) x 2 + b x + c = 0 θ, θ 2 + b θ + c = 0. b = b /b 2, c = c /c 2 (b, b 2, c, c 2 Z, b 2 > 0, c 2 > 0), b 2 c 2, 2 aθ 2 + bθ + c = 0, a = b 2 c 2 > 0, b = b c 2, c = b 2 c. g = gcd(a, b, c), g >, aθ 2 + bθ + c = 0 g., b 2 4ac g 2 = b2 c 2 2 4b 2 2c c 2 g ( 2 b 2 = b2 2c 2 2 g 2 b 2 2 = b2 2c 2 2 g 2 D ) 4 c c 2, D 0, (b 2 4ac)/g 2 0., (iii) (i),, θ, θ (30) 2 ( ) D 2 θ, D 0, D 0 (mod 4). D 2 θ, a, b, c, aθ 2 + bθ + c = 0, gcd(a, b, c) =, a > 0, θ = b ± D, D = b 2 4ac. 2a θ 2, D 0., 0 (mod 4), b D b 2 (mod 4), b., D 0 (mod 4), b, D b 2 4. a =, c = b2 D, θ = b + D

43 , b 2 4ac = D, aθ 2 + bθ + c = 0, gcd(a, b, c) =, a > 0. D 0, θ., θ D 2. D (mod 4), b. (30) 2 θ, θ ax 2 bx + c = 0., /θ cx 2 + bx + a = 0., θ /θ 2.,. 9.3 θ 2, P GL 2 (Z) ω = P θ., ω 2, ω = P θ. P = p q pθ + q, ω = P θ = r s rθ + s, ω = P θ = pθ + q rθ + s, ω.. θ ω + ω = pθ + q rθ + s + pθ + q rθ + s (pθ + q)(rθ + s) + (pθ + q)(rθ + s) = (rθ + s)(rθ + s) = 2prθθ + qr(θ + θ) + 2qs r 2 θθ + rs(θ + θ) + s 2., ωω (pθ + q)(pθ + q) = (rθ + s)(rθ + s) = p2 θθ + pq(θ + θ) + q 2 r 2 θθ + rs(θ + θ) + s 2. θ + θ, θθ Q, ω + ω, ωω Q., ω, ω 2 x 2 + bx + c = 0, b = (ω + ω ), c = ωω. ω, 2 0., 9., ω, ω 2. ω = ω, θ = P ω = P ω = θ., θ 2, θ θ., ω ω., ω, ω. 43

44 9.4 θ, θ 2, θ + θ θθ, 2., + 2, 3 x 2 2x = 0, x 2 3 = 0, D 0, D 0 (mod 4). θ, D 2 ax 2 + bx + c = 0, D = b 2 4ac, θ = b + e D, e = ± 2a., ω θ, P = p r θ = P ω = pω + q rω + s q GL 2 (Z) s (32)., a = ap 2 + bpr + cr 2, b = 2apq + b(ps + qr) + 2crs, (33) c = aq 2 + bqs + cs 2, ω D 2 a x 2 + b x + c = 0, D = b 2 4a c, ω = b + e D 2a, e = e det P., gcd(a, b, c) = gcd(a, b, c ) =. D, D ( 9.2). A = a b/2, b/2 c [ ] θ A θ = aθ 2 + bθ + c = 0., (32), P ω = pω + q = (rω + s) θ. rω + s 44

45 , A = a b /2, (33), b /2 c t P AP = A. (34), [ ] a ω 2 + b ω + c = ω A ω [ = t ω ] P AP ω = t P ω A P ω [ ] = (rω + s) 2 θ A θ = 0., ω 2 a x 2 + b x + c = 0.. 2, b 2 4a c = 4 det A = 4 det t P AP = 4(det P ) 2 det A = 4 det A = b 2 4ac P = s det P r q = ± s p r = D. q (32), p ω = P θ = sθ q rθ + p. θ, ω θ, ω 2, 9.3, ω ω = sθ q rθ + p sθ q rθ + p (ps qr)(θ θ) = (p rθ)(p rθ) = det P a(θ θ) a. 45

46 , (p rθ)(p rθ) = p 2 pr(θ + θ) + r 2 θθ ( = p 2 pr b ) + r 2 c a a = ap2 + bpr + cr 2 a = a a., e = e det P, a (ω ω) = det P a(θ θ) = e D.,, (34), ω = b + e D 2a, ω = b e D 2a. A = t (P )A P, P = s det P r q. p, a = a s 2 b rs + c r 2, b = 2a qs + b (ps + qr) 2c pr, c = a q 2 b pq + c p 2. g = gcd(a, b, c ), g a, b, c., g > gcd(a, b, c) >.,, gcd(a, b, c) = g = , 2 2, 2 2., θ 2, θ θ, < θ < 0, < θ. 46

47 0. D, D 0 (mod 4). (i) D 0 (mod 4), r D/4, θ = r + D/4 D 2. (ii) D (mod 4), r D/4, θ = (r + D)/2 D 2. (i) θ 2 x 2 2rx + r 2 D/4 = 0, ( 2r) 2 4(r 2 D/4) = D. 2 θ = r D/4, r < θ < 0 < θ. (ii) θ 2 x 2 rx + (r 2 D)/4 = 0, ( r) 2 4(r 2 D)/4 = D. 2 θ = (r D)/2, r ω θ > 0., 0 < D r < 2, < θ < m. m 2 x 2 m = 0, 4m. m, m m 2., m + m 2 (x m ) 2 m = 0, 4m. m m, 2 2. m m + m 2., m + m < m m < 0, < m + m, D > 0, 2 ax 2 + bx + c = 0, D = b 2 4ac (35) 2. θ 2, 2 (35). a < 0, θ, ( b) 2 4( a)( c) = b 2 4ac = D., a > 0. θ θ 2, < θ < 0, < θ. (36) 47

48 , a > 0, ( b D)/2a < ( b + D)/2a., (37), θ = b + D 2a b a = θ + θ > 0,, θ = b D. (37) 2a c = θθ < 0. a a > 0, b < 0, c < 0., c = c, D = b 2 + 4a c > b 2., b < D. (38), b = b, (37), θ = b + D 2a, θ = b D. 2a, (36),, a > 0,, (38), θ < < θ aθ < a < aθ. D b D + b < a < < a < D. (39) D, (38), (39) b a., θ = ( b + D)/2a.. A = a c b GL 2 (Z), 0 < d < c., a/c d a c = [a 0, a,..., a n ], n det A = ad bc = ( ) n 48

49 ., a/c p k /q k (0 k n).,. A = p n p n,, n., a n =, [a 0, a,..., a n, a n ] = [a 0, a,..., a n + ], a n 2,., [a 0, a,..., a n ] = [a 0, a,..., a n, ] ad bc = ( ) n n. p n /, > 0., ad bc = ± gcd(a, c) =, c > 0, a = p n, c =., p n d b = ad bc = ( ) n = p n p n., p n (d ) = (b p n ). (40) gcd(p n, ) =, (d )., d d = 0., 0 < d < c, > 0, d < d < c =., d = 0., d =., > 0, (40) b = p n..2 θ >, a, b, c, d, 0 < d < c, ad bc = ± ω = aθ + b cθ + d., θ ω. 49

50 . 3., ω = a c b θ = p n d p n θ = p nθ + p n θ + = [a 0, a,..., a n, θ]. θ >, θ ω..3 2 θ, ω, θ ω,,, ζ > n 0, m 0. θ = [a 0, a,..., a n, ζ], ω = [b 0, b,..., b m, ζ] (4) ( ) θ (4). p n / θ n, θ = [a 0, a,..., a n, ζ] = p nζ + p n ζ + p n p n = ±, θ ζ., ω ζ., θ ω. ( ) θ ω, a, b, c, d, ω = aθ + b, ad bc = ±. (42) cθ + d a, b, c, d, cθ + d > 0. θ, θ = [a 0, a,..., a n, θ n ] = p n ζ + p n 2 ζ + 2., θ n >, p n / θ n. (42),, ω = a θ n + b c θ n + d. a = ap n + b, b = ap n 2 + b 2, c = cp n + d, d = cp n 2 + d 2. 50

51 a, b, c, d, a d b c = (ad bc)(p n 2 p n 2 ) = ±. 5.6, δ, δ,, p n = θ + δ, δ <, p n 2 = 2 θ + δ 2, δ <. c = (cθ + d) + d = (cθ + d) 2 + cδ, cδ 2. cθ + d > 0 0 < 2 <, n 0 < c < d. n ζ = θ n, ζ, θ,.2 ω. 2 2.,.,.. ω ω = [a 0, a, a 2,..., a n, ω n ] (n = 0,, 2,...) (43), n 0 m, ω n0 = ω n0 +m = = ω n0 +jm = (j = 0,, 2,...) (44), ω., ω. (44) m ω., n 0 = 0, ω., ω. 5

52 ω (43), (44) ( m ), ω n0., ω = [a 0, a, a 2,..., a n0, ω n0 ] = [a 0, a, a 2,..., a n0, a n0,..., a n0 +m, ω n0 ] = [a 0, a, a 2,..., a n0, a n0,..., a n0 +m, a n0,..., a n0 +m, ω n0 ] =, a n0,..., a n0 +m.., ω, ω = [a 0, a,..., a n0, ȧ n0,..., ȧ n0 +m ] ω = [ȧ 0,..., ȧ m ] = = [2,,,, 4] = [2,,,, 4,,, ]. 7, = = [ 4,,, ] ω, ω = [ȧ 0, a,..., ȧ n ], n. a 0, a 0., ω >., ω = [a 0, a,..., a n, ω] = p n ω + p n 2 ω

53 , ω 2 + ( 2 p n )ω p n 2 = 0., ω 2., f(x) = x 2 + ( 2 p n )x p n 2, n >, 2., < p n 2 < p n, 0 < 2, n =, p =, p 0 = a 0, q = 0, q 0 =, f(0) = p n 2 < 0, f( ) = 2 + p n p n 2 > 0., f(x) = 0 < x < 0., f(x) = 0 ω ω 2. ω >, < ω < 0., ω ω, ω = [a 0, a,..., a n, ȧ n, a n+,..., ȧ n+k ]. ω, 4.7 ω., ω = [ȧ n, a n+,..., ȧ n+k ], ω, , 3. ω = [a 0, a,..., a n, ω ] = p n ω + p n 2 ω + 2., ω ω., 9.6, ω 2. ω ω = [a 0, a, a 2,..., a n, ω n ] (n = 0,, 2,...), n, n 2 ω n = ω n2, n < n 2 53

54 , ω., ω n = [a n, a n+, a n+2,..., a n2, ω n2 ] = [a n, a n+, a n+2,..., a n2, ω n ] = [ȧ n, a n+, a n+2,..., ȧ n2 ], ω = [a 0, a, a 2,..., a n, ω n ] = [a 0, a, a 2,..., a n, ȧ n, a n +, a n +2,..., ȧ n2 ] θ 2, 2 ax 2 + bx + c = 0, gcd(a, b, c) =, D = b 2 4ac > 0., θ θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...), p n / θ, 9.5, θ = p n θ n + p n 2 θ n + 2. A n = ap 2 n + bp n + cqn, 2 B n = 2ap n p n 2 + b(p n 2 + p n 2 ) + 2c 2, C n = ap 2 n 2 + bp n cqn 2 2, θ n 2 A n x 2 + B n x + C n = 0, B 2 n 4A n C n = D. n, 5.6, δ n,, p n = θ + δ n, δ n <. ( A n = a θ + δ ) 2 ( n + b θ + δ ) n + cqn 2 = (aθ 2 + bθ + c)qn 2 + 2aθδ n + a δ2 n qn 2 + bδ n = 2aθδ n + a δ2 n qn 2 + bδ n. 54

55 ,, C n = A n, A n < 2 aθ + a + b. C n < 2 aθ + a + b., B 2 n 4A n C n = D > 0, B 2 n 4 A n C n + D < 4(2 aθ + a + b ) 2 + D., B n < 4(2 aθ + a + b ) 2 + D. A n, B n, C n n, (A n, B n, C n )., n, n 2, n 3, n < n 2 < n 3 (A n, B n, C n ) = (A n2, B n2, C n2 ) = (A n3, B n3, C n3 ). (A, B, C)., θ n, θ n2, θ n3 2 Ax 2 + Bx + C = 0, 2., θ. 2.5 θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...), θ n 2. θ 2 f(x) = ax 2 + bx + c = 0, θ. θ 2, < θ < 0, < θ. (f n (x)) ( f 0 (x) = f(x), f n (x) = x 2 f n a n + ) x (n =, 2,...), n 0, f n (x) 2, f n (θ n ) = 0 55

56 ., (θ n ) θ 0 = θ, θ n =, n 0. θ n a n (n =, 2,...) f n (θ n ) = 0, n. n 0, θ n > θ n, < θ n < 0. n = 0 θ 0 = θ. θ n, < θ n < 0. a n, θ n a n <.,, < θ n a n < 0. < θ n < 0., θ n 2., n θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...). 2.4, 2, n, m θ n = θ m, n < m. n, θ n = a n + θ n = a n θ n + θ n, θ m = a m + θ m = a m θ m + θ m. θ n θ n 2,, 9.3 θ n = a n + θ n, θ m = a m + θ m. 56

57 θ n = θ m θ n = θ m, θ n θ m = a n a m. 2.5, θ n, θ m 2, θ n, θ m 0., < θ n < θ n θ m < θ m <., < a n a m <. a n a m, 0., a n = a m., θ n = θ m.,., θ. θ 0 = θ m n 2.7 2, 2. θ 2, θ = [a 0, a, a 2,..., a n, θ n ] (n = 0,, 2,...). 2.4, 2, n, m θ n = θ m, n < m., θ = [a 0, a, a 2,..., a n, θ n ] = [a 0, a, a 2,..., a n, a n,..., a m, θ m ] = [a 0, a, a 2,..., a n, a n,..., a m, θ n ],.7, θ n = [a n,..., a m, θ n ]., θ n. 2.2, θ n 2. n, 3., θ = p n θ n + p n 2 θ n

58 , 2.5, p n 2 p n 2 = ( ) n 2 =., θ θ n. n, 2.5, 2.2, θ n+ 2., θ n+ n, θ θ n a, b, c, x, y f(x, y) = ax 2 + bxy + cy 2 (45) 2 2.,, 2. gcd(a, b, c) =, f(x, y). 2,, 2. (45), f(x, y) = x (ax + b2 ) y = = [ x [ x ( b + y ] ax + b y 2 y b 2 x + cy ) 2 x + cy ] y a b/2 x b/2 c y, 2 a b/2 (46) b/2 c 2 f. D = b 2 4ac 2 f. f (46) A, D = (4ac b 2 2a b ) = = 4 det A (47) b 2c. 3. f(x, y) = ax 2 + bxy + cy 2 2, D., 4af(x, y) = (2ax + by) 2 Dy 2. 58

59 D = b 2 4ac, 4af(x, y) = 4a 2 x 2 + 4abxy + 4acy 2 = (4a 2 x 2 + 4abxy + b 2 y 2 ) (b 2 y 2 4acy 2 ) = (2ax + by 2 ) 2 (b 2 4ac)y 2 = (2ax + by) 2 Dy 2. 2 f(x, y), (i) f f(x, y). (ii) f x, y Z f(x, y) 0. (iii) f x, y Z f(x, y) 0. (iv) f f., x, y Z, f(x, y) = 0 x = y = 0. (v) f f., x, y Z, f(x, y) = 0 x = y = 0.., f,. 3.2 f(x, y) = ax 2 + bxy + cy 2 2, D = b 2 4ac f. a 0, (i) D > 0 f. (ii) D = 0 a > 0 f. (iii) D = 0 a < 0 f. (iv) D < 0 a > 0 f. (v) D < 0 a < 0 f. a = 0, (i) D > 0 f. (ii) D = 0 c > 0 f, 0. (iii) D = 0 c < 0 f, 0. (iv) D = 0 c = 0 f 0. a 0 : (i) ( ) 3., f(x, y). (ii) ( ) a > 0, D = 0 3., 4af(x, y) = (2ax + by) 2. a > 0, f(x, y)., 2ax + by = 0 f(x, y) = 0, f. 59

60 (iii) ( ) (iv) ( ), (ii). 3., D < 0 a > 0, f(x, y)., x, y Z f(x, y) = 0 = 2ax + by = y = 0 = x = y = 0., f. (v) ( ) (iv). (i) (v),, ( ). a = 0 : (i) ( ) D = b 2, b 0., f(x, y) = bxy + cy 2. (ii) ( ) D = b 2, b = 0., f(x, y) = cy 2., y = 0 x f(x, y) = 0, f., y 0 f(x, y) 0, f 0. (iii) ( ) (ii). (iv) ( ) D = b 2, b = 0., a = b = c = 0., f 0. (i) (iv)., a = 0 D = b 2 0.,., ( ). 3.3 f(x, y) = ax 2 + bxy + cy 2 2, D = b 2 4ac f.,. (i) f D > 0. (ii) f, 0 D = 0 a > 0 c > 0. (iii) f, 0 D = 0 a < 0 c < 0. (iv) f 0 D = 0 a = c = 0. (v) f D < 0 a > 0. (vi) f D < 0 a < 0. (ii) ( ), D = 0 c > 0, 2 f(x, y) x, y, D = 0 a > 0. (iii) ( )., f(x, y) = ax 2 + bxy + cy 2, y =, x 2 g(x) = f(x, ) = ax 2 + bx + c (48) 60

61 ., g(x) = ax 2 + bx + c, ( ) x f(x, y) = y 2 g = ax 2 + bxy + cy 2 y., g(x) f(x, y) 2, f(x, y) g(x) 2., f(x, y) g(x) [ ] f(x, y) = ax 2 + bxy + cy 2 = x y A x, A = a b/2 y b/2 c, P = p q GL 2 (Z) r s x = P y x y, (49),, f(x, y) = x = px + qy, y = rx + sy [ x y ] t P AP x y = a x 2 + b x y + c y 2., a = ap 2 + bpr + cr 2 (= f(p, r)), b = 2apq + b(ps + qr) + 2crs, c = aq 2 + bqs + cs 2 (= f(q, s)). (50), a, b, c., (49), a x 2 + b x y + cy 2, f(x, y) P [ ] f(x, y) = ax 2 + bxy + cy 2 = x y A x, A = a b/2, (5) y b/2 c ] f(x, y ) = a x 2 + b x y + c y 2 = [x y A x, A = a b /2 (52) b /2 c y 6

62 f = f, f = f a = a, b = b, c = c., f = f A = A., f f, P GL 2 (Z) A = t P AP (53)., t P P. f(x, y) f (x, y ) f f., (49) f(x, y) f (x, y ). P = p q, 2 (53) (50) r s., det P =, det P = f(x, y) = ax 2 + bxy + cy 2, f (x, y ) = a x 2 + b x y + c y 2, f = f, n, n 2 Z f(n, n 2 ) = f (n, n 2 ). ( ) f = f, a = a, b = b, c = c, n, n 2 Z f(n, n 2 ) = an 2 + bn n 2 + cn 2 2 = a n 2 + b n n 2 + c n 2 2 = f (n, n 2 ). ( ), a = f(, 0) = f (, 0) = a, c = f(0, ) = f (0, ) = c., a + b + c = f(, ) = f (, ) = a + b + c., b = b. 4.2 f, f 2. f f f, f. 62

63 f, f A, A (5), (52), f f, P = p q GL 2 (Z), (50). r s g = gcd(a, b, c). (50) g a, b, c. g >, f., g = f, f 2, D, D f, f., A, A f, f. f f, P GL 2 (Z) A = t P AP., (47), D = 4 det A = 4 det t P AP = 4 det t P det A det P = 4(det P ) 2 det A = 4 det A = D f(x, y), f (x, y ), f (x, y ) 2, A, A, A. ( ) E = 0, A = t EAE. 0 ( ) f f, P GL 2 (Z), A = t P AP., A = t P A P, f f. ( ) f f f f., P GL 2 (Z), A = t P AP.,, P GL 2 (Z), A = t P A P., A = t (P P )A(P P ), f f. D 0, D 0 (mod 4). D 2 ax 2 + bx + c, D = b 2 4ac, 2 ax 2 + bx + c = 0 b + D, 2a b D 2a,, 2. 9., 2. 63

64 4.5 f(x, y) = ax 2 + bxy + cy 2, f (x, y ) = a x 2 + b x y + c y 2 D 2., θ f 2 ax 2 + bx + c, θ f 2 a x 2 + b x + c. (i) f f θ = θ. (ii) f f θ θ. (iii) f f θ θ., θ = ( b + D)/2a, θ = ( b + D)/2a. (i) ( ) f f, 2, θ = θ. ( ) θ = θ,, a, b, a, b Z D Q, b + D 2a = b + D 2a. (a b ab ) + (a a ) D = 0. a b a b = a a = 0., a = a, b = b., f f, b 2 4ac = b 2 4a c., c = c., f = f. (ii) A, A f, f, A = a b/2, A = a b /2. b/2 c b /2 c ( ) f f., P = p q SL 2 (Z), r s A = t P AP. (33). ω = P θ, 9.5 ω = θ., θ = P θ. ( ) θ θ., P = p r, a = ap 2 + bpr + cr 2,, 9.5, θ = ( b + D)/2a., b + D 2a = b + D 2a. q SL 2 (Z), θ = P θ. s b = 2apq + b(ps + qr) + 2crs, (54) c = aq 2 + bqs + cs 2 64

65 , (a b a b ) + (a a ) D = 0. a, b, a, b Z D Q, a b a b = a a = 0., a = a, b = b., (54) A = t P AP., f f. (iii) (ii) , f ( ), a > 0 ( a < 0)., 2., 2 2, ( ) ( )., 4 (50), P GL 2 (Z), ( ) 2 P 2 ( )., f(x, y) 2, f(x, y) 2. f(x, y) P f (x, y ), P f(x, y) f (x, y )., f, f A, A, A = t P AP A = t P ( A)P., 2. 2 f(x, y) = ax 2 + bxy + cy 2, a > 0, c 0 b 2 4ac 0, c > 0. 2 f(x, y) = ax 2 + bxy + cy 2 2, a < b a < c 0 b a = c (55) f(x, y) 2 g(x) θ., f 2, θ >, 2 Re θ < 2 θ =, 2 Re θ 0., F SL 2 (Z),. f 2 θ F 65

66 θ θ, g(x) 2., 2 Re θ = θ + θ = b a, θ = θθ = c a. (56) f, (55) 0 b a = c a < b a < c a. (57), (56) θ F., θ F, (56) (57)., (55), f. 5.2 D < 0 2. f(x, y) = ax 2 + bxy + cy 2 D 2, (55) b a c., D = 4ac b 2 4b 2 b 2 = 3b 2., b D 3.., b., b, 4ac = b 2 D (a, c)., a, c., D, D f, f 2. f(x, y) = ax 2 + bxy + cy 2., D = b 2 4ac f, g(x) = ax 2 + bx + c f 2, θ = ( b + D)/2a g(x). D < 0 a > 0, θ H. F SL 2 (Z), θ θ 0 F ( 8.5). 9.5, θ 0 2 g 0 (x ). g 0 (x ) 2 f 0 (x, y ), 5., f 0., θ θ 0, 4.5, f f ,. 66

67 f(x, y) = ax 2 + bxy + cy 2 f (x, y ) = a x 2 + b x y + c y 2 2., g(x), g (x ) f, f 2, θ g(x), θ g (x ). f f, 4.5, θ θ., F SL 2 (Z), 5., θ, θ F., 8.6, θ = θ., 4.5, f = f. 5.5 f, f 2., f f, f f 2 f 0. ( ) 5.3, 2 f 0 f f 0., 2 f 0 f f , f 0 = f 0. ( ) f f 0, f 0 f, f f. 6 2 D, 0. D 2 f(x, y) = ax 2 + bxy + cy 2 2, f a > 0, a b + c > 0, a + b + c < 0, c < 0 (58). f(x, y) 2 g(x) = ax 2 + bx + c, g(0) = c, g() = a + b + c, g( ) = a b + c, (58) a > 0, g( ) > 0, g(0) < 0, g() < 0 (59)., 2b = g() g( ), f b < 0. f, 0, g. 6. f(x, y) 2, g(x) f 2, θ g(x)., f(x, y) 2, θ 2,, < θ < 0, < θ (60)., θ θ, g(x) 2. 67

68 f(x, y) = ax 2 +bxy+cy 2, D = b 2 4ac, g(x) = ax 2 +bx+c, θ = ( b+ D)/2a, θ = ( b D)/2a., g(x), g(x) = a(x θ)(x θ). (6) f, (59). θ < θ a > 0, (6), x, < 0, θ < x < θ, g(x) = = 0, x = θ x = θ, > 0, x < θ θ < x. g( ) > 0, g(0) < 0, g() < 0, < θ < 0 < θ., (60)., (60), D a = θ θ > 0 a > 0., (6), g( ) > 0, g(0) < 0, g() < 0., (59). 6.2 D 0., D 2. f D 2, f(x, y) = ax 2 + bxy + cy 2, 6. 0 < b + D 2a < < b + D 2a,, 0 < b + D < 2a < b + D. b < D., b., b, 4ac = b 2 D (a, c)., a, c., D, D D > 0 2 f, f 2. 68

69 f(x, y) = ax 2 + bxy + cy 2., D = b 2 4ac f, g(x) = ax 2 + bx + c f 2, θ = ( b + D)/2a g(x). D 0, θ , θ 2 θ , θ 0 2 g 0 (x ). g 0 (x ) 2 f 0 (x, y ), 6., f 0., θ θ 0, 4.5, f f n, f(x, y) = ax 2 + bxy + cy 2 2. x, y ax 2 + bxy + cy 2 = n (62) (x, y), n 2 f., gcd(x, y) =,, n f. (62) g = gcd(x, y) > (x, y), g 2 n, (62) g 2, a ( ) 2 x + b x g g y ( ) 2 y g + c = n g g 2, n/g 2 f(x, y). P = p q GL 2 (Z), r s x = P y, (50) f(x, y) = ax 2 + bxy + cy x y f (x, y ) = a x 2 + b x y + c y 2., x = P x, P = s y y det P r q, det P = ± p, f(x, y) = n (x, y) f (x, y ) = n (x, y )., n 2 f, n f P 2 f., n, f(x, y), f (x, y ) 2, n f, f. 69

70 7. n D 2, z z 2 D (mod 4n) (63). (x 0, y 0 ) (62). gcd(x 0, y 0 ) =, z 0, w 0 Z x 0 z 0 + y 0 w 0 =., x = P y x y, P = x 0 w 0 y 0 z 0, P GL 2 (Z), 4 (50) f(x, y) = ax 2 + bxy + cy f (x, y ) = nx 2 + b x y + c y 2. f f, 4.3 f, f., b 2 4nc = D., z = b (63)., (63) z = b, c Z b 2 4nc = D., f(x, y) = nx 2 + bxy + cy 2, f D, (62) (x, y) = (, 0). 70

71 [] : 2,, 97. [2] :,, 972. [3] :,, 98. [4] :,, 992. [5], : UBASIC,, 994. [6] G. M., E. M. : I,,

72 , , , , ,

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x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy

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