sede electrónica de muface

8/12/2019 Sede Electrnica de MUFACE

1/32

arXiv:0903

.4963v2

[math.NT

]27Apr2009

3

3

3

3

3

3

Q 3 3 y2 = x3 +D(ax+b)2 3 D= 1 D= 1

Q

2 3

E/k k n2

0 E(k)/nE(k)

H1(k, E[n])

Qv v

H1(k, E)[n]

Qv v

0

0 v

E(kv)/nE(kv)Q

vv

v

H1(kv, E[n]) v

H1(kv, E)[n] 0
http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2http://arxiv.org/abs/0903.4963v2


2/32

(k, E) := Ker

: H1(k, E[n])v

H1(kv, E)[n]

.

X(k, E) := Ker

H1(k, E)v

H1(kv, E)

.

0E(k)/nE(k) (k, E)X(k, E)[n]0 , (k, E) = E(k)/nE(k) X(k, E)[n] ,

nrk(E/k) = (k, E)E(k)

/nE(k)

X(k, E)[n] . rk(E/k) n n

X k Y k X k

(k, E)H1(k, E[n]) X Autk(X)=E[n] n X n

O E |n.O| EPn1. [C S] C E S [C1 S1] [C2 S2]

C1

=

S1

=

C2 S2

[C S] X =[EPn1] S Pn1

n E [C S] C S=Pn1


3/32

3

n= 3 [CP2] C E[3]

3

E Q 3

{O,T, T} T E

y2 =x3 + D(ax+ b)2

a b D Q E 16b3D2(4Da3 27b)

b

D

4Da3

27b= 0 3 T (0, bD) DQ2 E y2 =x3 + D(ax + b)2 a b D Z D 1 b > 0 b= b1b

33 b1 (a, b3) = 1

D =D0f2

D0 f Q y2 = x3 +D0(ax+b)2 a = f a b = f b (x, y) (x/u2,y/u3) (a, b) (ua, u3b) a b Z b = b1b

33 g = (b3, a)

(x, y) (xg2, yg3) (a, b) (a/g, b/g3) (a, b3) (a/g,b3/g) (a, b3) = 1 (a, b) (a, b) b > 0

D K K=Q(

D)

D Q D= 1 E E

3 3

E Q y2 =x3+D(ax+b)2 D T = (0, b

D)

3

3 E(Q) G3 K/K3 G3= Q

/Q3 D= 1 (O) = 1((0, b)) = 1/(2b) D= 1 ((x, y)) = y (ax + b)D


4/32

E y2 =x3+D(ax+b)2 D=3D

a= a

b= (27b 4a3D)/9 3

T = (0,bD) = (0, (27b 4a3D)

3D/9)

E E (P) =

x3 + 4D((a2/3)x2 + abx+ b2)

x2 ,

y(x3 4Db(ax + 2b))x3

P=O P=T (P) =O P =O P =T E E E E x 9 y 27

3 E

E

3

E E Q {O, T} {O, T} E(Q) G3 Ker() = ()

3 3

3 E E

3 3 y2 =x3 + D(ax + b)2 3 O

y2 =x3 +D(ax+b)2 E 3 D 3 E

D= 1 O (0, b) D=3 2(9b + 4a3) =t3 t= 0 O

P x(P) = t2

3 +

3

t2

4ab +

16

9a4

+4a2

3

O

Q = (x, y) 3 x([2]Q) = x(Q) = x(Q)

Ex(3x3 + 4Da2x2 + 12Dabx + 12Db2) = 0.

P(x) = 3x3+4Da2x2+12Dabx+12Db2 Disc(P) =48D2(27b+4Da3)2b2 x= 0 y2 =Db2 P(x) = 0

y2 =D3

(ax+ 3b)2.


5/32

3

D

E y2 =x3 + D(ax + b)2 E 3 y2 =x3 3D(ax + (27b 4a3D)/9)2 3

| ()|| ()|= 3r+ ,

r E E = 1 D= 1 D=3 = 0 Et E

E(Q)/3E(Q)Et(Q)/3Et(Q) (Z/3Z)r

. G= Et(Q) G G/3G G[3] 3 G Et(Q)/3Et(Q) D = 1 D =3 2(9b+ 4a3) D,n (a, b) 2(9b + 4a

3)

|E(Q)/3E(Q)|= 3r+D,1+D,3(a,b) . 3

3

|E(Q)/3E(Q)|= [E(Q) :(E(Q))][(E(Q)) :((E(Q)))].

B A (A)(B) AB+ Ker() A/B(B+ Ker())/B A/BKer()/(Ker() B) .

[

(A) :

(B)] =

[A: B]

[Ker(

) : Ker(

) B]

.

A=E(Q) B= (E(Q)) Ker() Q O T T (E(Q)) D=3 Q Ker() ={O, T} D=3 D=3 [Ker() : Ker() B] = 1 D =3 E y2 =x3 3(ax + b)2 E y2 =x3 + (3ax + (9b + 4a3))2 [Ker() : Ker() B] = 1 T (E(Q)) 3


6/32

T (E(Q)) 2(9b+ 4a3) (a, b) = 1

[(E(Q)) :((E(Q)))] =[E(Q) :(E(Q))]3D,3(1(a,b)) .

3r+D,1+D,3(a,b) =|E(Q)/3E(Q)|= [E(Q) :(E(Q))][(E(Q)) :((E(Q)))]= [E(Q) :(E(Q))][E(Q) :(E(Q))]3D,3((a,b)1) .

3 E(Q) (E(Q)) [E(Q) :(E(Q))] =|

()| [E(Q) :(E(Q))] =| ()| |

()|| (

)|= 3r+D,1+D,3 ,

() () 2 3 E

| ()|

D= 1

D = 1 y2 =x3 + (ax+ b)2 a b 3 E(Q) Q/Q3 (O) = 1 ((0, b)) = 1/(2b)

((x, y)) = y (ax+ b) E(Q)

0 pk p p p 0 min(vp(X), vp(Y), vp(Z)) = 0

uQ/Q3 uQ

uX3 + (1/u)Y3 + 2bZ3

2aXYZ= 0

u= 1 (X, Y , Z ) = (1, 1, 0) u= 1/(2b) (X, Y , Z ) = (0, 1, 1) y (ax + b) =uz3 zQ (X, Y , Z ) = (z2, x, z) (X, Y , Z ) Z= 0 (x, y) = (XY/Z2, (uX3 (1/u)Y3)/(2Z3)) u E(Q) z= X/Z


7/32

3

u u u1u2|(2b) u= u21u2 ui

u1X3 + u2Y3 + (2b/(u1u2))Z3 2aXYZ= 0. u= 1 u= 1/(2b) O

T = (0, b) u (x, y)E(Q) z Q uz3 =y (ax+ b) X=z2 Y =x Z=z

uX3 + (1/u)Y3 + 2bZ3 2aXYZ= 1u

(u2z6 + 2uz3(ax+ b) x3)

= 1

u((uz3 + ax+ b)2 x3 (ax+ b)2)

= 1

u(y2

(x3 + (ax+ b)2)) = 0,

zQ Z= 0 (X, Y , Z ) Z= 0 x=XY/Z2 y = (uX3 (1/u)Y3)/(2Z3) x3 + (ax + b)2 = (X3Y3 + Z2(bZ2 aXY)2)/Z6 2Z(bZ2 aXY) =uX3 + (1/u)Y3

x3 + (ax+ b)2 = ((uX3 + (1/u)Y3)2 4X3Y3)/(4Z6)= (uX3 (1/u)Y3)/(4Z6) = y2 ,

(x, y)E(Q) ((x, y)) = y (ax+ b) = (uX3 (1/u)Y3)/(2Z3) (bZ2 aXY)/Z2

= (1/(2Z3))(uX3 (1/u)Y3 2Z(bZ2 aXY))= (1/(2Z3))(2uX3) =u(X/Z)3 ,

u (x, y) u

u u u= u21u2 ui Y u1u2Y

u21u2X3 + u1u

22Y

3 + 2bZ3 2au1u2XY Z= 0.

gcd(X, Y , Z ) = 1

u1u2 2b

u1u2

p

p|u1u2 p2b X Y u1 u2 p|u1 pu2 p p 2b p|Z p2 u1 pu2 p|Y p3 u1 p u2


8/32

p|X gcd(X, Y , Z ) = 1 u1u2

C E sQ

(X :Y :Z)(sX : (1/s)Y :Z). C u us3

D= 1

D= 1 K = Q(D) 3 3 3 E 3 E D D =3D D= 1 Gal(K/Q) (

D) =D

N = NK/Q K Q uK [u] u K/K3

y2 =x3 + D(ax + b)2 3 E(Q) G3 K

/K3 [u] u u(u) = NK/Q(u)Q3 (O) = 1 ((x, y)) =

y (ax + b)D E(Q) T = (0, bD) /E(Q)

[u]G3K/K3 uK u= v2(v)

2v2X3 + 2Dv1Y

3 + 2b

v21 Dv22Z3 + 6v1X

2Y + 6v2DXY2 + 2a(X2Z DY2Z) = 0

v= v1+ v2

D u= 1 (X, Y , Z ) = (1, 0, 0) y (ax+

b)

D= v2(v)z3 z

Q (X, Y , Z ) = (z1, z2, 1)

z= z1+ z2D (X, Y , Z ) Z= 0

(x, y) =

v(v)

X2 DY2Z2

, v(v)(v(X+ YD)3)

Z3

u E(Q) z= (X+ Y

D)/Z

() = (+ ())/2


9/32

3

a = a

D b = b

D y2 = x3 + (ax+ b)2 y(ax+ b) = y(ax+b)

D =uz3 (X, Y , Z ) = (z2, x, z) uX3 +

(1/u)Y3

+ 2bDZ3

2aDXY Z = 0 (X, Y , Z ) (x, y) = (XY/Z2, (uX3 (1/u)Y3)/(2Z3)) (x, y) E(K) (x, y)E(Q)

u u [u] u(u) =3 Q (u2/3)(u2/3) = 1 u2 = 3v/(v) v K u = (u/)3(v)/v = (u/(v))3v2(v) u u= v2(v) vK u(u) =3 = v(v) v vZK

x

Q XY/Z2

Q Y /Z = (X/Z)

Q y Q u3 (3/u)()3 Q = X/Z u v2(v) v(v) v3 (3/(v3(v)2))()3 Q =v3 r= (/(v(v)))3 Q r()Q =s + t

D r() =s + t

D r(s t

D)

t(r+ 1) = 0 r=1 = t= 0 t= 0 = v3 Q u= v2(v) = (/(2()))3 [u]

t= 0 G3 = u3 (u3) +

2a

D()+2b

D= 0 = v(v) (v3(v3))/(2D)+a()+b/(v(v)) = 0 v v =v1+v2

D = x1+x2

D

2v2x31+ 2Dv1x

32+ 6v1x

21x2+ 6v2Dx1x

22+ 2a(x

21 Dx22) + 2b/(v21 Dv22) = 0,

x1= X/Z x2= Y /Z X Y Z Z

2v2X3 + 2Dv1Y

3 + (2b/(v21 Dv22))Z3 + 6v1X2Y + 6v2DXY2 + 2a(X2Z DY2Z) = 0. (X, Y , Z ) x y D= 1

(v(X+ YD)3 (v)(X YD)3)/D+ 2aZ(X+ Y

D)(X Y

D) + 2b/(v(v))Z3 = 0.

[u] u v v3 K v vr rQ


10/32

() [u] () [1/u] () v v1 (u) = u1(v(v))3 [(u)] () v (v)

C E sQ

(X :Y :Z)(sX :sY : (1/s2)Z). C u us9

G3

G3

x y y2 =x3 +D(ax+ b)2 x= m/d2 y = n/d3 gcd(m, d) = gcd(n, d) = 1 d >0 f = gcd(n d(am+bd2)D, n+d(am+bd2)D)

f g g D f= f gd d d2 =gZK D = 1 g= 1 d= Z

f = f1q3

f1 gf1q2 | 2b g |

gcd(D, 2b) p|f1 p K/Q g f1

D = 1 D= 1 K f= Fd F Z d

K Z 1 d f

d

p p (p)

p = p(p) p2 = pZK p d

d

d =

pS0p

S0 p g =

pS0

p p p f= F

p|gp g| D

f

2

=F

2

g

x y m/d2 n/d3

(n d(am+ bd2)

D)(n + d(am+ bd2)

D) =n2 Dd2(am+ bd2)2 =m3 . F2g|m3 p|g p p v1 p


11/32

3

v= vp(f) 3vp(m) = 2vp(f)

3vp(m) =vp(f) =v = vp(f2)/2 =vp(F

2g)/2 =vp(F2g) = 1 + 2vp(F)

p| g

g

vp(F) 1 (mod 3)

vp(F) 1 p| F g| F F = f g f Z p f 3vp(m) =vp(f) 3|vp(f)

f2g3 | m3 g| m f2 | (m/g)3 f = f1q3 f1 p vp(m/g) vp(q) + 2vp(f1)/3 0 vp(f1) 2 2vp(f1)/3= vp(f1) f1q2 |m/g f2 =f2g3 =f21 g3q6 3|vp(f) vp(f1) = 0 f1 D g

(nd(am+ bd2)D)/(f g) D f g|2 gcd(n, d(am+ bd2)) f g = gf1q3 |2 gcd(n, d(am+ bd2))

2bd3

= 2d(am+bd2

) 2adm gf1q2

| m 2bd3

0 (mod gf1q2

)

d n u v un+ vd3 = 1 2b= 2bvd3 + 2bun gf1q

2 |2bd3 gf1q2 |2n gf1q2 |2b

[u] () G3

v K uZK = v

2(v)q3 q K gcd(v, (v)) = 1 f1= NK/Q(v) 2b K/Q

f= gcdn d(am+ bd2)D, n + d(am + bd2)D f1 g q Z d K f = f1q3gd f1 g| gcd(D, 2b) gZK = d2 gf1q2 | 2b f= f1(qd)

3

a= (n d(am+bd2)

D)/f a+ = (n+d(am+bd2)

D)/f

gcd(a, a+) = 1 aa+ = m3/f2

f1

f2 |m3 f1(qd)2 |m gf1q

2 |m Z aa+= f1(m/(gf1q

2))3ZK .

f

a+= (a)

a a = fq3 a+ = f+q

3+ f

gcd(f, f+) = 1 f+ =(f) q+ =(q)

ff+(qq+)3 = f1(m/(gf1q

2))3ZK f1 K f f+

qq+ = m/(gf1q2)ZK

ff+= f1ZK f|f1


12/32

3

y (ax + b)D= (n d(am + bd2)D)/d3 ((x, y))ZK=fa= f1(qd)

3fq3= f1f(qdq)

3 =v2(v)q31

q1 v= f

gcd(v, (v) ) = 1 v Z 1 v2(v) = uq3 v Cl(K) (Cl(K)/Cl(K)3)[+ 2] G G G G[] G

3 3 uK uZK=q3 3 K S3(K)

S3(K) G3

1U(K)/U(K)3 S3(K)Cl(K)[3]1 . I(K) K Gi3 I(K)/I(K)3

1S3(K)G3Gi3C l(K)/Cl(K)3 1 . i G3 G

i3 [u]

uZK G3 Gi3 [u] G

i3

uZK=q3

q [u]

S3(K)

Gi3 Cl(K)/Cl(K)3

a

Cl(K)/Cl(K)3 q K a= q3 a ZK

a 1

G3 Gi3 a

Cl(K)/Cl(K)3 NK/Q(a) = a Q aNK/Q(a) a (NK/Q(a))

3

a

aNK/Q(a) Gi3

i =i 3 E Gi3 |

()|=| (i)||S3(K) ()|. D


13/32

3

() S3(K) = 1

3 N() = () N() =n3 =v2(v)q3 v= q= 1/n

| ()|

3 E

3

3

[] S3(K) K

v = TS [] TG

TGT TSS3(K) T =S3(K) () T TS=TG T =TG TS/TG 3 TG TS [] F3

RS

S3(K)/TS

RG

TS/TG

f 2b f =

1isp

vii 1 vi 2 pi pi

pi v v=

1isp

xivii 1xi1

3s s Cl(K)

v

v = uq3 q u K [u] u G3 S3(K) u

[S] RS RS S3(K)/TS SK v= uS [S]RS vSel(i)

v /Sel(i) [G]RG RG TS/TG G K


14/32

v= uSR

v (i)

v / (i) IG v

IG

(i

)

| Sel()|=| Sel(i)||TS| |IG| TG (i) T TG T TS IG (i) Sel(i) |IG||TG| | | ()| | | Sel()| | ()| |IG||TG| =| Sel()| | ()| | ()| |IG||TG| Sel()/IGTG 3

u1X3 + u2Y

3 + u3Z3 cXY Z = 0

3

| ()|

R Qp

F(X, Y , Z ) =u1X3 + u2Y

3 + u3Z3 cXY Z = 0.

D = 1 u3= 2b/(u1u2) c= 2a

X Y Z p u1 u2 u3 c p p

min(vp(u1), vp(u2), vp(u3), vp(c)) = 0.

min(vp(u1), vp(u2), vp(u3), vp(c)) = 0

min(vp(u1), vp(u2), vp(u3))>0 Qp vp(c) > 0 min(vp(u1), vp(u2), vp(u3)) = 0

vp(c) = 0

max(vp(u1), vp(u2), vp(u3))2 .


15/32

3

vp(c) > 0 max(vp(u1), vp(u2), vp(u3)) 2 0 =vp(u1)vp(u2)vp(u3)2 v= (vp(u2), vp(u3))

v

= (1, 2)

Qp

vp(c) = 0 vp(u1u2u3)2 min(vp(u1), vp(u2), vp(u3)) = 0 vp(c)> 0

u1X3 +u2Y

3 +u3Z3

cXY Z = 0 c= 0 c= 0

c= 0 g = gcd(u1, u2, u3, c) (u1, u2, u3, c) (u1/g,u2/g,u3/g,c/g)

gcd(u1, u2, u3, c) = 1 p|c

u1

u2

u3

c

p

vp(c) = 0 max(vp(u1), vp(u2), vp(u3))2

vp(c) = 0 min(vp(u1), vp(u2), vp(u3)) > 0 p ui 0 =vp(u1)vp(u2)vp(u3)2

vp(u2) = 1 vp(u3) = 2 (u1, u2, u3, c)

(u2/p2, u3/p

2, pu1,c/p),

vp(u1u2u3)2

c= 0

g = gcd(u1, u2, u3) (u1, u2, u3) (u1/g,u2/g,u3/g) gcd(u1, u2, u3) = 1

u1 u2 u3 max(vp(u1), vp(u2), vp(u3))2

p|u1u2u3 ui 0 =vp(u1)vp(u2)vp(u3)2

vp(u2) = 1 vp(u3) = 2 (u1, u2, u3)

(u2/p2, u3/p

2, pu1),

vp(u1u2u3)2

p

min(vp(u1), vp(u2), vp(u3)) = 0

p c c= 0 vp(u1u2u3)2


16/32

p min(vp(u1), vp(u2), vp(u3))>0 p

p u1X3 + u2Y

3 +u3Z3

cXY Z = 0

p p p= 3 vp(u1) = vp(u2) = vp(u3) = 0 vp(27u1u2u3 c3) = 0 Qp

Fp Z= 0 u1X

3 + u2Y3 = 0 3u1X

2 = 0 3u2Y2 = 0 p= 3

vp(ui) = 0 X = Y = 0 Z= 0 Z= 1 p= 3 Fp Z= 1 3u1X

2 cY = 0 3u2Y2 cX= 0 3u3 cXY = 0 c= 0 u3 = 0 vp(u3)1 Y = 3u1X

2/c X = 3u2Y2/c = 27u21u2X

4/c3 X = 0 X = Y = 0 u3 = 0 X

3 = c3/(27u21u2) 3u3 = cXY = 3u1X

3 = c3/(9u1u2) 27u1u2u3

c3 = 0 Fp 1

3 22 > 0 p Fp p p= 3 Zp p

p|u1u2u3 p= 3 p p vp(u1u2u3) > 0 vp(27u1u2u3c3) > 0 p = 3 p= 3 vp(u1u2u3)>0 vp(u1)> 0 p min(v

p(u

2), vp

(u3)) = 0

0vp(u1)vp(u2)vp(u3) p p p= 3 vp(u1u2u3)> 0 0vp(u1)vp(u2)vp(u3) Qp

vp(c) = 0 vp(c)>0 vp(u1) =vp(u2) = 0 u1/u2 p F

p

vp(c) > 0 vp(u1) = 0 vp(u2) = vp(u3) = 1 u2/u3 p Fp

p

p= 3 vp(u1u2u3) > 0 vp(c) > 0 vp(u1u2u3)2 vp(u1) = vp(u2) = 0 vp(u3)2 vp(u1) = 0 vp(u2) =vp(u3) = 1

p = 3 v3(u1u2u3)>0 v3(c) = 0 p= 3


17/32

3

p|(27u1u2u3 c3) p= 3 p 3 p|(27u1u2u3 c3) pu1u2u3

p p p= 3vp(u1u2u3) = 0 vp(27u1u2u3 c3)> 0 Qp u2/u1 Fp

p= 3 p= 3 v3(c) = 0 3 v3(u1u2u3) > 0 v3(c) > 0 v3(c) =v3(u1u2u3) = 0

v3(c) = 0 Q3

v3(c)>0

P0 = (X0, Y0, Z0) k1 v3(F(P0))2k min(v3(F

X(P0)), v3(F

Y(P0)), v3(F

Z(P0))) = k

F 3 P0 k= 1 3 P = (X, Y , Z ) F(P) = 0 P P0 (mod 3k)

F(P) = u1X3 + u2Y

3 + u3Z3 cXY Z FX(P) = 3u1X2

cY Z v3(c) >0 3 F(P0)

0 (mod 32k)

min(v3(FX(P0)), v3(F

Y(P0)), v3(F

Z(P0))) =k.

k= 1 k= 2 k= 2

P0 = (X0, Y0, Z0)

v3(F(P0))3 min(v3(FX(P0)), v3(FY(P0)), v3(FZ(P0))) = 2, F 3 P0 P1 P0 (mod 3) v3(F(P1)) 3 min(v3(F

X(P1)), v3(F

Y(P1)), v3(F

Z(P1))) = 2 3

P = (X, Y , Z ) F(P) = 0 P

P0 (mod 3)

(X, Y , Z ) 3 min(v3(X), v3(Y), v3(Z)) = 0

p = 3 3 v3(c) >0 v3(u1u2u3)2 0 =v3(u1)v3(u2)v3(u3)


18/32

v3(c) 2 v3(u1u2u3) = 0 Q3 ui uj (mod 9) i=j

v3(c)2 ui 3 v3(u2) = 0

v3(u3) > 0

Q3

u1 u2(mod 9) v3(u3) = 1 v3(c) 2 ui 3 v3(u2) =

v3(u3) = 1 3 Q3 u2/3 u3/3 (mod 9)

v3(c) = 1 ui 3 v3(u2) = 0 v3(u3)>0 Q3 u1 u2 (mod 9) s1 s2 {1, 1} cs1u1+s2u2+ s1s2u3 (mod 9)

v3(c) = 1 ui 3 v3(u2) = v3(u3) = 1 Q3

v3(c) = 1 ui 3 v3(u1u2u3) = 0

p= 3 v3(c)> 0 v3(u1u2u3) = 0

ui uj (mod 9) i=j Q3 ui uj (mod 9) i=j u1u2u3 1 (mod 9)

Q3 s1 =1 s2 =1 cs1u1+ s2u2+ s1s2u3 (mod 27)

v3(c) >0 v3(c) = 1 v3(c) 2

v3(c)2

(X, Y , Z ) min(v3(X), v3(Y), v3(Y)) = 0 3 si

3| c s1 u2u3 (mod 3) s2 u1u3(mod 3)

F(X, Y , Z ) = u1X3 + u2Y

3 + u3Z3 cXY Z = 0,

u1X3 + u2Y

3 + (2b/(u1u2))Z3

2aXYZ= 0

D= 1

D= 1 D= 1


19/32

3

D = 1 u= u21u2 ui D= 1

F(X, Y , Z ) = 0 v= v1+ v2

D

F(X, Y , Z ) = 2v2X3 + 2Dv1Y

3 + (2b/(v21 Dv22))Z3+ 6v1X

2Y + 6v2DXY2 + 2a(X2Z DY2Z)

= (v(X+ Y

D)3 (v)(X Y

D)3)/

D

+ 2aZ(X+ Y

D)(X Y

D) + (2b/(v(v)))Z3 ,

p= 3 vp(v(v)) = 0 vp(2b) = 0 vp(27b 4a3D) = 0 Qp

v v/nZK nZ n=1 [u] () [u] = [v2(v)] v v(v) v (v) D 0 (mod 4) v= v1+ v2

D v2Z

D= 1 v(v)|2b 3h(K) v(v)|(2b)h(K) v(v) 2b 3|h(K)

v 2v1Z 2v2Z 2b/(v(v))

v = v1 +v2

D v(v) p dp Qp d2p = D Qp u1X

3 +u2Y3 +u3Z

3

cXY Z = 0 u1 = v1+v2dp u2 = v1

v2dp

u3= (2b/v(v))dp c= 2adp

Qp u1X

3 +u2Y3 +u3Z

3 cXY Z = 0 D= 1 p vp(v(v)) = 0 vp(2b/(v(v)))0


20/32

p vp(v(v)) = 0 F(X, Y , Z ) = 0

F(X, Y , Z ) = (v(X+ Y

D)3

(v)(X

Y

D)3)/

D+ u3Z

3

+ 2aZ(X+ YD)(X YD), u3= 2b/(v(v)) vp(u3)0 p Kp = Qp(

D)

2 Qp

Fp2

D Fp2

Fp2 0 K=Q(

D) Kp (mod p) Kp K Q Q

p= 3 X Y (v)/v((X+ YD)/(X YD))3 (modp)

(v)/v Fp2

p1 (mod 3) p2 (mod 3) v(p21)/3 1 (modp) D = 1 p vp(2b) > 0 vp(v(v)) = 0 p= 3 p= 2 vp(2b)2

p

vp(2a) = 0 vp(2a)>0 (v)/v p F

p2

p = 2 v2(2b) 3 2 v2(a)> 0 v2(b)2 v2(2b) = 3 v2(a) = 0 v = v1+ v2

D = (w1+w2

D)/2 w1

w2 Z w1w2 (mod 2) v w1w21(mod 2) w1 w2 w1w2 (mod 4) D = 1 p = 2 D 5 (mod 8) v2(2b) 3 w1 = 2v1 w2= 2v2

w1 2 (mod 4) w2 0 (mod 4) w1 0 (mod 4) w2 2 (mod 4) Q2

w1 w2 1 (mod 2) Q2 v2(2b)4 v2(a)>0


21/32

3

D = 1 p vp(2b) = vp(v(v)) = 0 vp(27b4Da3) > 0 p= 3

p

(v)/v

F

p2

p vp(v(v)) = 0

D= 1 p= 3 D2 (mod 3) u1= 2v2 u2= 2v1D u3= 2b/(v(v))

v3(2a) = 0 Q3 v3(2a)2 Q3 v3(u1)2 v3(u2)

2 ui uj (mod 9) i=j u32(u1 u2)(mod 9)

v3(2a) = 1 v3(2b)> 0 v3(u1u2)

1 Q3

v3(u1)2 v3(u2)2 v3(2a + 2b) = 1 v3(2a) = 1 v3(2b) > 0 v3(u1u2) = 0 Q3

u1 u2 (mod 9) 2a + b u1 u2 (mod 9)

D= 1 p= 3 D2 (mod 3) u1 = 2v2 u2 = 2v1D u3 = 2b/(v(v)) v3(2a) = 1 v3(u1) = v3(u2) =v3(2b) = 0 u4= u3+ 2a v3(u4) = 0

ui uj (mod 9) i=j i, j = 1 2 4 Q3

Q3

2a+ u4 = 4a+ u3 3D(mod 27)

D= 1 p p= 3 Qp

D

= 1 p= 3 3|D u3= 2b/(v(v))

v3(2a) = 0

v3(2v2)>0 v3(2v2) =v3(2a + u3) = 0


22/32

v3(2a)2

D3 (mod 9) v3(u3) = 0

D3 (mod 9) v3(u3)> 0 v3(2v2)>0 D6 (mod 9) v3(2v2)2

D6 (mod 9) v3(2v2) =v3(u3) = 1 D6 (mod 9) v3(2v2) = 0 u3 2v2 (mod 9)

u3 2v1D (mod 27) v3(2a) = 1

v3(2a) = 1 u4 = u3+ 2a

D

3 (mod 9) v3(u4) = 0

D3 (mod 9) v3(u4)>0 v3(2v2)> 0 D6 (mod 9) v3(2v2)2

D6 (mod 9) v3(2v2) =v3(u4) = 1 D6 (mod 9) v3(2v2) = 0 u4 2v2 (mod 9)

v3(u3) = 1 s =1 2v1(D/3) s(u3/32a(D/3))(mod 9) 2v1s2a/3 (mod 3)

v3(u3) = 1 s =1 2v1(D/3) s(u3/32a(D/3))(mod 9) 2v1s2a/3 (mod 3) v3(2v2) = 0 D3 (mod 9)

v3(u3) = 1 s =1 2v1(D/3) s(u3/32a(D/3))(mod 27) 2v1s2a/3 (mod 3) v3(2v2) = 0 D6 (mod 9)

v3(u3) = 1 s =

1 2v1s

2a/3 (mod 3) v3(2v2) > 0

t {1, 0, 1} r {1, 0, 1} 2v1(D/3)s(u3/3 2a(D/3)) 6v2(D+ 3)t 9(2v1s 2a/3)st2

3r(D(2v1+ a/3) + 6rDv1+ 6at2) (mod 81).

y2 = x3 + (kp)2 k = 1 2 4 Ekp y

2 =x3 + (kp)2 p k= 1 2 4

k

2

2 y2 =x3 + (8kp)2 y2 = x3 + (kp)2 p = 2 3 p = 2 0 (k, p) = (1, 3) (2, 3) 1 (2, 1) (3, 3) (k, p) = (4, 3) 0 3 T = (0, kp) (k, p) = (4, 2)


23/32

3

6

(8, 24) p5 3 T = (0, kp)

u1X

3 +u2Y3 +u3Z

3 = 0 u1u2| 2kp u3 = 2kp/(u1u2) u1u2 u1 u2 (1, 1, 2kp) (1, 2, kp) (1, p, 2k) (1, 2p, k) (2, p , k) u = 1

k= 1 u= 2p u= 4p2 3 u= 2 4 p p2 4p 2p2

k = 2 u = 4p u = 2p2 3 u= 2 4 2p 4p2 u= p p2 u 4p 2p2 [p] [p2] 4 2

k= 4 p p2 2k = 8 3 u = 2 4 u = 2p 4p2 u = 4p 2p2

u1 = 1 u2 = 2 u3 = kp

X3 + 2Y3 +kpZ3 = 0 k= 4 p2 (mod 3) 2(p1)/3 1 (mod 3) kp 4 (mod 9)

k = 4 2 (0, 1, 2) 2 k = 1 k = 2 2 (0, 0, 1) (0, 1, 1) F2 Q2

p

p

(0, 0, 1)

p 2 Fp 3 (0, 0, 0) 3 ui uj (mod 9) i= j kp 1 2 (mod 9) kp 4 (mod 9) 3kp


24/32

p 5 k = 1 2 4 E y2 =x3 + (kp)2

k = 1 p

4 (mod 9) p

1 7 (mod 9) 2(p1)/3

1

(modp) () ={1, 2p, 4p2} | ()|= 3 k= 1 p2 (mod 9) | ()|= 9 k = 2 p 2 (mod 9) p 1 4 (mod 9) 2(p1)/3 1

(modp) () ={1, 4p, 2p2} | ()|= 3 k= 4 () ={1, p , p2} | ()|= 3

| ()| = 3 9

X3 + 2Y3 + kpZ3 = 0 | ()|= 9

x3 + 2y3 x3 + 4y3 ()

E y2 =x327(kp)2 D=3 b= 3kp = (1 + 3)/2 k|4 k 8 k

2 3 p p 1 (mod 9) k 4 (mod 9)

4|k

k= 1

2

4

k= 4 p 1 (mod 9) 2v1 =1 2v2 = 1 2b= 6kp u3= 2b/(v(v)) = 6kp

2 4k 2 F4 1 4| k 2

p p1 (mod 3) p p (v1+ v2dp)X

3 + (v1 v2dp)Y3 + 6kpZ3 = 0 p (0, 0, 1) (v1+ v2dp)/(v1 v2dp)2/= (modp) Fp

(p1)/3 1 (modp)

p1 (mod 9) p2 (mod 3) p p Fp2 p

2 1 (mod 9) p 1 (mod 9) p2 (mod 3) p p 1 (mod 9) 3 2a = 0 3 6kp 3 (mod 27) kp 4 (mod 9) p 1 (mod 9)


25/32

3

p 1 (mod 3) p = () = (w1+w2

3)/2 12 () 3|w2 p1 (mod 3) = (w1+w23)/2 () =p k 8k 3k pk 2 3 p 4|k 4k 2|w2Z (w1/(2k))(p1)/3 1(modp) 3|w2 3w2 pk2 + 3 (mod 9)

2v1 = w1 2v2 = w2 u3 = 2b/(v(v)) = 6k 2 2 4| k 4 k (w1+ w2

3)/2 = 1 F4 2|w2 p (v1+ v2dp)X3 +(v1v2dp)Y3 + 6kdpZ3 = 0 v21v22d2p = p dp vp(v1

v2dp) = 1 v2dp

v1 (modp) p

(0, 1, 0) (v1+ v2dp)/(6kdp) F

p v1 v2dp (mod p)

v1/(3kdp) =w1/(6kdp) Fp (w1/(6kdp))

2 =w21/(108k2) (w1/(2k))

2 w1/(2k)

3 2a = 0 3k v3(u3) = 1 3 3| w2 6k 3w1 (mod 27) 2k w1(mod 9) w21 + 3w

22 = 4p 3 w2 w

21 4p3 (mod 9)

2k w1 (mod 9) w214k2 (mod 9) 3w1 4p4k2 + 3 (mod 9) pk2 + 3 (mod 9)

p 5 k = 1 2 4 E y2 =x3 27(kp)2

k = 1 k = 2 p 2 (mod 3) p 1 (mod 3) p k2 + 3(mod 9) 2(p1)/3 1 (mod p) ()

k= 4 p2 5 (mod 9) ()

| ()|= 1 3 k= 1 k = 2 k= 4 p8 (mod 9)

|

()|= 1 3 9 k= 4 p1 (mod 3) 2b= 6kp 2 3

f1

= 1 p

2 (mod 3) f1

= 1 p p

1 (mod 3) v 1 2 () =p

/ () () () ={1} |

()|= 1

() () () ={1, , ()} | ()|= 3


26/32

() (

) (

) ={1, , 2} | (

)|= 3

(

) (

) (

) =

{j

, j

, j

(), 0j2} | ()|= 9

k= 1 p4 (mod 9) k= 2 p7(mod 9) p = m2 + 3n2 m n = m +n

3 v1 = m v2 = n | (

)|= 3

k = 4 p 8 (mod 9)

| ()|= 3 k = 4 p4 7 (mod 9) 4p =m2 + 27n2 m

n = (m+ 3n3)/2

v1 = m/2 v2 = 3n/2 | ()| = 3

p5 (mod 9) p1 7 (mod 9) 2(p1)/3 1 (mod p) Ep y

2 = x3 +p2 0 p

2 (mod 9)

1 p4 8 (mod 9) 0 1 p1 7(mod 9) 0 1 2

p 2 (mod 9) p 1 4 (mod 9) 2(p1)/3 1 (mod p) E2p y

2 = x3 + 4p2 0 p 5 7 8(mod 9)

0

1 p1 4 (mod 9) 0 1 2

p2 5 (mod 9) E4p y2 =x3 + 16p2 0 p4 7 8 (mod 9) 0 1 p1 (mod 9) 0 1 2

| ()|| (

)|= 3r+1

k= 1 p= 61 79 113 131 149 151 163 293 k = 2 p= 29 83 137 139 173 181 199 k= 4 p= 41 59 101 131 137 2


27/32

3

0

1

1 0 1 2 0 2

k = 1

p2 (mod 9) k = 4 p 4 7 (mod 9) 1

k= 4 p1 (mod 9) 0 2 2 k= 1 k= 2

k= 4 2

P = (x, y) y2 =x3 +p2 p= 1759 1

x=2424795595146084331000753504998742211139235353063551062176562878606796987394973602467684

y= 8643240396318605197724619647046515784779281219388876514209037894857

5362134274928159502186511847328850266140274118035321166956948248

p = 9511 y2 = x3 +p2 2

P1 = (210, 9011) (P1, P2) P2= (x, y)

x= 32701984517186448621442294824950874787830128281

456289760665179363242981599270033206574137600

y= 92890043770264171014255964610503972850176417273682124237369198272789821

9746778232027925565271633950191532413151456450450966045051557376000

3

p 9

2p

3

2 p 1 2(p1)/3 1 (mod p) 2 2(p1)/3 1 (mod p)


28/32


29/32

3

k= 1 p4 (mod 9) k= 2 p7 (mod 9) p= m2 + 3n2

nX3

3mY3 + 3kZ3 + 3mX2Y

9nXY2 = 0

p8 (mod 9) X3 + 3Y3 + 24pZ3 3X2Y 9XY2 = 0

p4 7 (mod 9) 4p= m2 + 27n2

nX3 mY3 + 8Z3 + mX2Y 9nXY2 = 0

| ()|= 9 | ()|= 3

| ()|= 3 y2 =x3 + (kp)2 k= 3 9 k 3 3 p= 2 p= 3 (k, p) = (3, 2) 1 (3, 3) 3 T = (0, kp) (k, p) = (9, 3) 6 (18, 81) p

5

u1X

3 + u2Y3 +u3Z

3 = 0 u1u2 | 2kp u3 = 2kp/(u1u2) u1u2 u1 u2 (1, 1, 2kp) (1, 2, kp) (1, 3, 2kp/3)(1, 6,kp/3) (1, p, 2k) (1, 2p, k) (1, 3p, 2k/3) (1, 6p, k/3) (2, 3,kp/3) (2, p , k) (2, 3p, k/3) (3, p, 2k/3) (3, 2p, k/3) (6,p,k/3) u = 1

k = 3 (1, 2, 3p) (1, 3, 2p) (1, 6, p) (1, p, 6) (1, 2p, 3) (1, 3p, 2) (1, 6p, 1) (2, 3, p) (2, p, 3) (2, 3p, 1) (3, p, 2) (3, 2p, 1) (6, p, 1) (1, 6p, 1) u = 6p u = (6p)2

3

ui

(1, 2, 3p) u= 2 4 3p 9p2 12p 18p2 (1, 3, 2p) u= 3 9 2p 4p2 18p 12p2 (1, 6, p) u= 6 36

p p2 36p 6p2 (2, 3, p) u= 12 18 4p 2p2 9p 3p2

p2 (mod 3) 27


30/32

p 1 (mod 3) 2 3 Fp 27 2 3 9

2 3 p p 2 3 6 3/2 Fp p2 (mod 3)

p1 (mod 3) 2 3 p

2 3 Fp (1, 6, p) (2, 3, p) 6 + 3 = 9 u

2 3 Fp 6 3/2

(1, 2, 3p)

(1, 3, 2p)

9

u 2 3 Fp

24 + 3 = 27 u

p2 (mod 3) | ()|= 3 9 27

| ()|= 3 27

| ()|= 27

rk(E) = 2 x y Q p= x3 + 6y3

x y Q p= 2x3 + 3y3

x

y

Q

p= 4x3

+ 12y3

x y Q p= 9x3 + 18y3 p1 (mod 3) 2 3 Fp | ()|= 3 9

| ()| = 9 rk(E) = 1 (a, b) =

(1, 6) (2, 3) (4, 12) (9, 18) x y Q p= ax3 + by3 b/a Fp

p 1 (mod 3) 2 3 Fp C0

rk(E) = 3 C0

| ()| = 9 rk(E) = 1 p = x3 + 6y3

p= 2x3 + 3y3 p= 4x3 + 12y3 p= 9x3 + 18y3 x y Q

p2 (mod 3) p1 (mod 3) 2 3 Fp | ()| = 1 3rk(E)+1 =| ()| p 2 (mod 3) E +1 E |

()| = 3 27 p 1 (mod 3) E 1 E | ()|= 9


31/32

3

2

3 p1 (mod 3) 2

3 | (

)|= 1 3 3 C0

E

C0 | ()|= 1 | ()|= 9 rk(E) = 1

C0 | ()|= 3 | ()|= 3

rk(E) = 1 | ()|= 27 rk(E) = 3

C0 p p= 3889 2 1 (X, Y , Z ) = (91, 211, 19) (2, 3, p) | ()| 9 C0 rk(E) = 3

k = 9 (1, 2, 9p) (1, 3, 6p) (1, 6, 3p) (1, p, 18)

(1, 2p, 9) (1, 3p, 6) (1, 6p, 3) (2, 3, 3p) (2, p, 9) (2, 3p, 3) (3, p, 6) (3, 2p, 3) (6, p, 3) (3, 2p, 3) u= 18p u= 12p2 3 ui (1, 2, 9p) u= 2 4 (1, 3, 6p) u= 3 9 6p 36p2 (1, 6, 3p) u = 6 36 3p 9p2 (1, 9, 2p) u = 2p 4p2 (1, 18, p) u = p p2 (2, 3, 3p) u = 12 18 12p 18p2 (2, 9, p) u = 4p 2p2 (3, 6, p) u= 9p 3p2 36p 6p2

2 3 (3, 6, p) 3

p 2 (mod 3) 7 9 u

p1 (mod 3) 3/2 3 6 p1 4 7 (mod 9) 7 9 u 3 u 3

3 p 4 (mod 9) 3 (1, 3, 6p) (1, 9, 2p) p 2 (mod 9) 6 (1, 6, 3p) (2, 9, p) p 1 (mod 9) 3/2 (1, 18, p) (2, 3, 3p)

1


32/32

p

|x3 y2|

p

3

3

sede electrónica de muface

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