(,)

,

, Riemann

, 2001

2001. . .

: , ,

3

5

1.

5

2.

7

3. 9

4. (inflections flexes)

15

5.

19

23

1.

23

2.

27

3. Riemann

37

4. Manin Hasse

41

5.

51

61

1.

61

2.

71

3.

75

79

81

, .

, . Q .

Fq (Fq) . Q, mod p p (Fp). (Fp) . Fq Riemann g Fq Manin Hasse Riemann. , Riemann - . Roquette Lemmermeyer Manin Hasse , , .

. , .. , Reed-Solomon . , . .

. . .

. Peter Roquette (Heidelberg) . Ruud Pellikaan (Eindhoven) .

. [1] [S1].

1.

(x, y)- . , . , .

.

() .

.

(w, x, y). (0, 0, 0) (w, x, y) (0, 0, 0). (w, x, y) (w, x, y) (0, 0, 0) , 0,

w = w, x = x y = y

(1.1.1)

2 (0, 0, 0). 2 (w, x, y) (1.1.1) ( ). (w, x, y) [w, x, y].

:

() w, x, y;

() R .

() Q .

() C .

w, x, y k

2(k):={[w, x, y] | w, x, y ( k }.

(x, y)- k2 2(k) :

(x, y) ([1, x, y].

[w, x, y] ( 2(k) w (0 k2

[w, x, y] = [1,] 2(k).

2(k)

= {[w, x, y] |aw + bx + cy = 0, (a, b, c) ( (0, 0, 0)}.

a = 1, b = 0, c = 0 w = 0 , [0, x, y].

w = 1 a + bx +cy = 0 bc ( 0.

a, b, c a(, b(, c( u ( k, u ( 0 a( = ua, b( = ub, c( = uc.

2.

l(w, x, y) = 0 l(w, x, y) .

Cf d [w, x, y] ( 2(k) f(w, x, y) = 0 f d k.

d f(w, x, y) = d(w, x, y), (w, x, y) [w, x, y] f(w, x, y) = 0, . ( [S1], , 2, 225)

w = 1. g(x, y) = f(1, x, y). deg g(x, y) ( d.

Cgaff = {(x, y) ( k2 | g(x, y) = 0}.

Cgaff = Cf ( k2.

g(x, y) d f(w, x, y) = wd g d f(1, x, y) = g(x, y).

f (w, x, y) = 0 f (w, x, y)2 = 0 Cf f (w, x, y) = 0. f f = f1 f2 fr :

.

f | f( Cf ( Cf(.

f k, k. , , k.

1.2.1 Cf d k f(w, x, y) (k[w, x, y] d , k

Cf() = {[w, x, y] ( 2() | f(w, x, y) = 0}.

f = f1a(1) f2a(2) fra(r) f d k[w, x, y] :

Cf() = .

Cfi () Cf a(i) Cfi.

.

.

.

, .

( K k P2(K) ( P2(K) Cf(K) ( Cf(K).

, :

1.2.2 f(X1, X2, , Xn) ( k[X1, X2, , Xn] k .

f(a1, a2, , an) = 0 a1, a2, , an k. f(X1, X2, , Xn) 0.

n .

n = 1. f(X) ( k[X] degF(X) = m. f(X) m k . , n =1.

n 1 . :

f(X1, X2, , Xn) = f0 + f1Xn + fmX

m 0 f1, f2, , fm ( k[X1, X2, , Xn 1]. f - fm ( 0. a1, a2, , an fm (a1, a2, , an 1) ( 0. , degf = m, m an f(a1, a2, , an) = 0, . f(X1, X2, , Xn) .

3.

k . k . Cf f(W, X, Y) ( k[W, X, Y] W f(W, X, Y) ( Cf ) g(X, Y) = f(1, X, Y) g(X, Y) = 0 k Cf().

. , 2(), Cf(K). .

1 = [w1, x1, y1] 2 = [w2, x2, y2] 2(). L : 1 + 2 , ( . L Cf()

f(w1 + w2, x1 + x2, y1 + y2) = 0.

:

(i) f(w1 + w2, x1 + x2, y1 + y2) = 0 , ( . , L W = 0. :

f(0, x, y) = 0, x, y ( .

1.2.2 f(0, , ) = 0. , W f(W, X, Y) L (W = 0) Cf().

(ii) f(1 + 2) . f(1 + 2) d .

f(0, 1) ( K[0, 1] d d ai, bi ( K f(0, 1) = ( 0. ( [1], 11 22).

f(1 + 2) = 0 d / r r-. L ( Cf(). :

:

1.3.1 L P2(K) Cf() d (deg f = d). ( [1], 23, 33).

1.3.2 Cf ( 2(), ( Cf(), Cf() d . ( [1], 24, 33). 1.3.3 Cf k . Cf ( k) Cf L.

Cf(K) P ( Cf().

P (a, b) f g(, ) g(a, b) = 0.

L P = (a, b)

.

g (a + t, b + t) = 0.

Taylor t.

t2 + = 0

: . t = 0 . P Cf P. + = 0.

1.3.4 Cf P.

: , .

2 2 0 2. Cf ( ).

r- : g, (r1)- , , r- . r r r . Cf

= 0

.

1.3.5 r Cf r.

1.3.6 g(, ) r r 1 ( r ( d. 1, .

2, .

...

1.3.7 Cf 1 (singular).

, P = (a, b) ,

g(a, b) = (a, b) = (a, b) = 0.

1.3.8 Cf - (non-singular) -. Cf .

1.3.9 Cf - m (m 1)(m 2).

1.3.10 g(X, Y) r r. r g(X, Y) = 0 gr (X, Y) = 0, ( gr (X, Y) g r), g .

1.3.11 r f(W, X, Y) = 0 (r 1) f P r. ( [1], 32, 36).

1.3.12 P=[1, a, b]

f(w, x, y) = 0

(P) = (P) = (P) = 0.

1.3.13 Cf m n , k. K k Cf(K) ((K) mn Cf . ( [1], 34, 37).

1.3.14 Cf k m n K k , Pi (i= 1, 2, ), ri si ,

( mn

( [1], 36, 40).

1.3.15 , .. , = 4 > 3(1. . . 1.3.16 Cf k n. K( k

# = n2

mn m. n(nm) nm. ( [1], 38, 41).

1.3.17 C0 C1 4 , ( P1, P2, P3, P4), . P1, P2, P3, P4

b0C0 + b1C1 b0, b1 ( K.( [1], 39, 42).

:

1.3.18 0 1 , 9 P2(K), . 0 1,

= b0 0 + b1 1 b0, b1 ( K.

( [1], 40, 42).

, ,

1.3.19 ( Bezout) Cf , , m n , , , , mn . ( [W])

4. (inflections flexes)

1.4.1 P = [w, x, y] Cf Cf

(i) P -

(ii) P 3.

1.4.2 - .

, ( 1.3.10)

1.3.10 [1, 0, 0]

bX aY = 0

f(1, X, Y) :

f(1, X, Y) = (bX aY) + h2(X, Y) + +hd(X, Y),

hi(X, Y) i i = 2, 3, , d

h2(at, bt) = t2 h2(a, b) = 0.

C k.

C . L ( k, ) . 1(2 + 1(1 = 3 > 2 1.3.14 L C L C, C . C , , .

: C .

( 2 = f(X), f(X) ) ( 1.3.15) . . f . ( [S1], .3, 26).

1. 2 = 2 ( + 1) . = (0, 0) . f = (.

2. 2 = 3 . , = (0, 0) . f .

1.4.3 Cf f(X0, X1, X2) = i Xj ( aij = aji) det(aij) .( [1], 43, 45).

Cf , P[x0, x1, x2]. , 1.3.12,

(P) = (P) = (P) = 0.

. ,

(P) = 2xj, i = 0, 1, 2

- det(aij) = 0.

1.4.4 Cf , f(X0, X1, X2) ( k[X0, X1, X2].

H(X0, X1, X2)

( [1], 44, 46).

f d H 3(d 2).

1.4.5 - 3 . .

, , - [1, 0, 0] Y =0, .

f(1, X, Y) = Y + h2(X, Y) +h3(X, Y) = Y +bXY + aY2 + h3(X, Y)

, = 0, h2(X, 0) = 0.

f(W, X, Y) = W2Y + aY2W + bWXY + h3(X, Y).

[0, 0, 1] , W = 0, .

1.4.6 - C

WY2 + a1WXY + a3W2Y = X3 + a2X2W + a4XW2 + a6W3.

( Weierstrass).

, .

k 2, :

W = X3 + X2W + XW2 + W3

b2 = + 4a2,b4 = a1a3 + 2a4,b6 = + 4a6.

b8 = a6 a1a3a4 + 4a2a6 + a2

4b8 = b2b6 b4.

= +

W2 = X3 + X2W + XW2 + W3 k 2 3

c4 = 24b4 c6 = + 36b2b4 216b6

= X +

W2 = 3 + W2 W3.

b2, b4, b6, b8 c4, c6 a1, a2, a3, a4, a6 .

:

1.4.7 - C Weierstrass

WY2 = X3 + bW2X + cW3

2 = f(),

f(X) ( K[X], , .

f(X) = X3 + bX + c D(f) f D(f) = 4b3 27c2 = 0.

1.4.8 [1, a, 0] C WY2 = X3 + bW2X + cW3 - a X3 + bX + c. ( [1], 48, 49).

1.4.9 , 1.4.8 Y2 = X3 + bX + c D(f) ( 0 D(f) = 4b3 27c2.

5.

, , E - Q . , ( 1.4.7 ) :

: WY2 = f(X)

f(X) = X3 + bW2X + cW3 b, c ( Q.

, ( 1.4.8) D(f) = 4b3 27c2 ( 0.

1.5.1 .

1.5.2

a X + b Y + c = 0 a, b, c ( Q.

1.5.3 P = (x1, y1), Q = (x2, y2) - E

L : (y1 y2) X + (x2 x1) Y + (x1y2 y1x2) = 0.

L E PQ . , P PP.

PQ P Q .

, - .

:

1.5.4 C - (non-singular) . P Q C P+Q O PQ C.

.

. P .

, 1.3.18, - .

1.5.5 - (non-singular) ( ) .

1.5.6 E Q (Q).

1.5.7 ( Mordell) E(Q) Q . ( [1], 5).

1.5.8 . .

1.5.9 ( Lutz-Nagell) . P = (x, y) E(Q) x y y = 0 2 y | D(f). ( [1], 1, 64)

Lutz-Nagell:

1.5.10 y | D(f) y2 | D(f). ( [S2], 7.2, 221)

1.5.11 P, Q, R . P, Q, R P + Q + R = .

, P + Q + R = O ( P + Q = R ( R = PQ ( P, Q, R .

, , : Y2 = X3 + bX + c P, Q . P+Q .

, . = [0, 0, 1]. P = (x, y) P = (x, y).

, = (x1, y1), Q = (x2, y2), PQ = (x3, y3). P+Q = PQ = (x3, y3). P, Q P+Q.

P Q , , L : (y1 y2) X + (x2 x1) Y + (x1y2 y1x2) = 0.

:

: x1 ( x2. L Y = X + = = y1 x1 = y2 x2. L E P, Q. PQ Y E.

(X + )2 = X3 + bX + c

X3 2 X2 + (b 2)X + (c 2) = 0

x x1, x2, x3 :

X3 2 X2 + (b 2)X + (c 2) = (X x1) (X x2) (X x3)(1.5.12)

, : x1 + x2 + x3 = (2)

x3 = 2 x1 x2 y3 = x3 + (1.5.13)

,

x3 = x1 x2 y3 = (x3 x1) + y1(1.5.14)

: x1 = x2. x = x1, P, P , 1.5.11, . , Q = P Q = P.

Q = P P+Q = P + (P) = O.

Q = P y2 = y1. P + Q = P + P = 2P. 2P PP = (x3, y3), x PP. :

Y2 = X3 + bX + c = = . (1.5.12) Y2 X3 + bX + c :

x3 = 2x1 + =

(1.5.15)

( [S1], .4, 31)

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