SJ ¨ALVST ¨ANDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

SJ ¨ ALVST ¨ ANDIGA ARBETEN I MATEMATIK

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

The Sato-Tate Conjecture

av Johan Frisk

2012 - No 9

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM

The Sato-Tate Conjecture

Johan Frisk

Sj¨alvst¨andigt arbete i matematik 15 h¨ogskolepo¨ang, grundniv˚ a Handledare: Torsten Ekedahl/Rikard B¨ogvad

2012

N_p Zp

y²= x³+ 2x + 3

|E(C)|

Np

y²= x³+ Ax + B 4A³ 27B²6= 0

F p p

A B

4A³ 27B²= 0

Np

[p + 1 2pp, p + 1 + 2pp]

Np

Np p

1960

0 1 2 3 4 5 6 7 8 9 10

x

0 1 2 3 4 5 6 7 8 9 10

y

y²= x³+ x + 3 (mod 11)

17 y² = x³+ x + 3

x y 11 p = 11

[6, 18]

19 20

p = 19 [11, 27]

A

B p

(G,⇤) G ⇤

a, b2 G a ⇤ b G

(a⇤ b) ⇤ c = a ⇤ (b ⇤ c) a, b, c2 G e 2 G

e⇤ a = a ⇤ e a2 G

a2 G a ¹

a a⇤ a ¹= a ¹⇤ a = e

a⇤ b = b ⇤ a a, b2 G G

G G

|G|

(G,⇤) H G H

G H

G

F + ·

(F, +) 0

(F\{0}, ·) 16= 0

a· (b + c) = a · b + a · c a, b, c2 F

a2 F +

· ( a)

a ¹

+, ,⇤, /

/ a b = a + ( b) a/b = a⇤ b ¹

(G,⇤) a, b, c2 G a⇤ c = b ⇤ c ) a = b

a⇤ b = a ) b = e a⇤ b = e ) b = a ¹

+ ·

a2 F a· 0 = 0 a· b = 0 ) a = 0 b = 0

f (x) F n

c 2 F f (c) = 0 x c f (x)

f (c) = 0 , f(x) = (x c)g(x) g(x) n 1

f (x) n

n

t p > 2

Z^t Z^p t p

(Z^t, +) 0

Zp

0 1

a b > 0 q, r

a = bq+r, 0 r < b q r

a b

t t

Z^t={0, 1, . . . , t 1}

a, b c = a + b

5 c c = tq + r

0 r < t r2 Zt

⇤ a⇤ b a + b t

⇤

(a + b) + c = a + (b + c)

t 0

a⇤ 0 = 0 ⇤ a = a + 0 = a Zt ⇤ 0

0

(1, t 1), (2, t 2), ... (a, b) a + b = t a⇤ b = b ⇤ a = 0 b = a ¹ a = b ¹

(Z^t,⇤) (Zt, +)

p

⇥ Z^p\{0} 1

Z^p a⇥ b

a· b p

a 2 Zp\{0}

gcd(a, p) a p p

a, b gcd(a, b) = 1

m, n ma + nb = 1

a2 Z^p\{0} m

ma + np = 1 q, r

a = bq + r a kq = b(q k) + r k2 Z a

a kq b p

ma = ma + np np 1

o = m + lp l2 Z oa = ma + mlp = ma + (ml)p

oa 1 p

l 0 < o = m + lp < p) o 2 Zp\{0}

a ¹= o a

⇥ a ¹⇥ a = a ⇥ a ¹= 1

ax³+ bx²y + cxy²+ dy³+ ex²+ f xy + gy²+ hx + iy + j = 0 y² = x³+ ax²+ bx + c

Z^p p > 3

y²= x³+ Ax + B ab + by + c = 0 ax²+ bxy + cy²+ dx + ey + f = 0

C(F ) F

(x, y)

y²= x³+ ax²+ bx + c 4A³ 27B²6= 0 a, b, c2 F

F

R

4A³ 27B²6= 0 x2 R

4A³ 27B²6= 0 x³+ Ax + B

C E(C)

O

A, B, x2 R

-2 -1 0 1 2 3 x

-3 -2 -1 0 1 2 3

y

P

Q

P+Q R

-2 -1 0 1 2 3

x -3

-2 -1 0 1 2 3

y

P

-P

P Q

R

P + Q R

R = (x, y) P + Q = (x, y) = (x, y) R

R

P Q

R O

O

C O

O O

P + ( P ) =O P +O

P O P

P +O = ( P ) = P

P = (x₁, y₁), Q = (x₂, y₂)

x1= x2 P Q

O

P + Q + R =O x16= x²

R P + Q O

O P + Q + R =O

P + ( P ) +O = 0

P = Q P + P + Q =O

P Q

R R R ¹

P + P = Q P + P + Q =O

R

O

P = (x, y) C P = (x, y)

C y²= x³+ Ax + B, ( y)² = x³+ Ax + B F^p

P = (x₁, y₂), Q =

(x2, y2) C R C

4A³ 27B²6= 0 x³+Ax+B 3

x1, x2, x3 x³+Ax+b = (x x1)·(x x²)·(x x³)

y = kx+l y1= kx1+l

y2= kx2+ l k = y2 y1

x₂ x₁ l = kx1+ y1

R P

Q y

(kx + l)²= x³+ Ax + B

x³ k²x²+ (A 2kl)x + (B k²) = 0

x1 x2 x3

x³ k²x²+ (A 2kl)x + (B k²) = (x x1)· (x x2)· (x x3)

x³ k²x²+(A 2kl)x+(B k²) = x³ (x₁+x₂+x₃)x²+(x₁x₂+x₁x₃+x₂x₃)x x₁x₂x₃ x²

x1+ x2+ x3= k²) x³= k² x1 x2

x = x3 (x3, kx3+l) = (x3, kx3 kx1+

y1) C F

P + Q = R P = (x1, y1) Q = (x2, y2) R = (x3, (kx3 kx1+ y1)) x3

P + P

C F y² = x³+

Ax + B P = (x1, y2) Q = (x2, y2) C

O P + Q6= O S = P + Q

S = (x3, y3) x3= k² x1 x2

y3= k(x1 x3) y1

k = 8>

<

>: y₂ y₁ x2 x1

P6= Q 3x²₁+ A

2y1

P = Q

k P Q

P

y²= x³+ 3x + 7 Z¹¹ Z11

(1, 0), (5, 2), (5, 9), (8, 2), (8, 9), (9, 2), (9, 9), (10, 5), (10, 6)

O

y²= x³+ 3x + 7 F¹¹ O

O O

O

O O

O O

O O

O O

E(C)

y²= x³+ Ax + B |E(C)| = N^p+ 1

|E(C)| = 9 + 1 = 10

|E(C)|

p > 2 x > 0

x p y2 Z^p y²⌘ x (mod p)

p > 2 a 0

✓a p

◆

✓a p

◆

= 8>

<

>:

0 a⌘ 0 (mod p) 1

1 p + 1

2 Zp

x²= y² (x + y)(x y) = 0 x =±y 1

(Z^p, +) p 1

2 x, x x6= x

0 p 1

2 + 1 =p + 1 2

Np 1 =

p 1X

x=0

✓ 1 +

✓x³+ Ax + B p

◆◆

= p +

p 1X

x=0

✓x³+ Ax + B p

◆

y y²⌘ x³+ Ax + B (mod p) y y

y6= 0 y = 0

Np p 1 =

p 1X

x=0

✓x³+ Ax + B p

◆

p { 1, 0, 1}

Y =

p 1X

i=0

Xi

pXi(x)

A, B, x

p p + 1

2 Z^p

f (x) = x³+ Ax + B x = 0, 1 . . . , p 1 Z^p

f (x) P [Xi= 1] =

p+1 2

p ⇡ 1

2 i p

P [Xi= 1] + P [Xi= 0] = 1 P [X = 1]⇡ 1 0.5 = 0.5

4 f (x) = 0 3

P [Xi= 0] 3

p p

pXi(x)

p_X(x) = 8>

<

>:

0.5 x = 1 0.5 x = 1

0 x = 0

i

x³+ Ax + B x

Zp

{0, 1, . . . , p 1} p

z = f (x) x2 Z^p z

p f (x)

A, B, p

A, B, p y[p]

squares[p]

i = 1 . . . p y[i] 0 squares[i] 0

i = 0 . . . p 1

index i² + 1 squares[index] 1

x = 0 . . . p 1

index x³+ Ax + B p + 1 y[index] y[index] + 1

p 1, 2, . . . , p

Pp

i=1y[i]· squares[i]

p

f (x) = x³+ Ax + B p pX(1) x³ 2x + 7

x³+ 45x 22 x³+ 45x 22 x³ 5x + 19 x³+ 8x 13 x³ 4x + 8 x³+ 25x + 2

PX(1) 0.5

A B p

pY(x) Y =

p 1X

i=0

Xi

Y p ±1

Xi Y

pp

P [|Y | > 2pp] > 0 Y Xi

0

[1 2pp, 1 + 2pp]

pY(x) Z^p

X FX:= P r[X x] 1 < x < 1 fX(x) FX(x) =

Z x 1

fX(x) dx X

fX(x) X

cdf pdf

300000 Z^p p

p

p + 1 2pp N^p p + 1 + 2pp

2pp N^p (p + 1) 2pp

1Np (p + 1)

2pp  1

ap:= Np (p + 1) cp:= ap

2pp= Np (p + 1) 2pp

Np cp

N

Z

S ={(x, y) | x, y 2 Z^p} p² y²= x³+ Ax + B

Np O(p²) S

n + 1

2 y² Z^p

1

0 O(p) O(p)

x = 0, . . . , x 1

z = x³+ Ax + B z

1 0 z z = 0

(x, 0) z6= 0 y y

y²= ( y)²= z (x, y) (x, y)

z 0 y

(x, y)

O(p)

O(p) A, B, p

squares[p]

Np 0 i = 1 . . . p squares[i] 0

y = 0 . . . p 1 index y² (mod p) squares[index + 1] 1

x = 0 . . . p 1

z x³+ Ax + B (mod p) squares[z + 1] = 1

z = 0 Np N^p+ 1 Np N^p+ 2

Np

y

= x

+ 2x + 3

y²= x³+ 2x + 3 (mod p)

300000 Np Z^p

p 300000 p 44773

cp 5 L

300000 X

FX(x) fX(x)

100

X100 i=1

m(i) = 300000 m(i) =|bⁱ|

b_i={x 2 L| 1 + (i 1)h < x 1 + ih}

h =1 ( 1) 100 = 2

100

M (i) = Xi j=1

m(j)

i = 1, . . . , 100

fX FX

cp

pr(i) = M (i) 300000

xi xi= 1 + ih , i = 1, . . . , 100 F˜X(xi) = P r(X xⁱ) = M (i)

300000 F_X(x_i)

fX FX

fX(x) = d

dxFX(x) = lim

a!0

FX(x + a) FX(x) a

F˜X(x) FX(x) x1, . . . , x100

FX(xi+ a) FX(xi)

a ⇡ F˜X(xi+ a) F˜X(xi) a

a = h = 2 100 d

dxFX(xi)⇡ F˜X(xi+ h) F˜X(xi)

h =F˜X(xi+1) F˜X(xi)

h =

M (i+1) 300000

M (i) 300000

h =

M (i+1) M (i) 300000

h =

m(i+1) 300000

h = m(i + 1) 300000 ·100

2 = m(i + 1) 6000

cp

cp [ 1, 1] cp

✓p2 [0, ⇡] cos ✓p= cp

✓p

cp

3 fX(x)

2 2

f⇥(✓) = XN n=0

ancos n✓

fX(x)

2

{x¹, ˜F (x1)), . . . , (x100, ˜F (x100))} F⇥(✓) = Z ✓

0

f⇥(˜✓) d˜✓

F⇥(✓) = Z ✓

0

XN n=0

ancos n˜✓ d˜✓ =h a0✓˜i✓

0+ an

XN n=1

"

sin n˜✓ n

#✓

0

=

= a0✓ + XN n=1

an

n sin n✓

F⇥(⇡) = 1 ) a⁰⇡ + 0 + . . . = 1 ) a⁰= 1

⇡

F⇥(✓) = ✓

⇡+ XN n=1

bnsin n✓

{x¹, ˜F (x1)), . . . , (x100, ˜F (x100))}

x_i ✓_i2 [0, ⇡]

✓i = ⇡ cos ¹(xi)

F⇥ N = 12 12

b1, . . . , b12

x = [✓1, . . . , ✓100]^T y =h

F (x˜ 1), . . . , ˜F (x100)iT

X = 2 66 66 64

sin (1· ✓¹) sin (2· ✓¹) . . . sin (12· ✓¹) sin (1· ✓2) sin (2· ✓2) . . . sin (12· ✓2)

sin (1· ✓¹⁰⁰) sin (2· ✓¹⁰⁰) . . . sin (12· ✓¹⁰⁰) 3 77 77 75

1

⇡x + Xb = y z = y 1

⇡x

b Xb = z

b

kz Xbk²

(X^TX)b = X^Tz (28)

b = 2 66 66 66 66 66 66 66 66 66 4

0.00019087087 0.1594348064 0.00015471255 0.00013657114 0.00011571785 0.0000360403 0.0002155770 0.00005559181 0.0000210021 0.00007706562 0.0001150259 0.0000778209

3 77 77 77 77 77 77 77 77 77 5

b2

F_⇥(✓) = ✓

⇡ 0.1594348064 sin (2✓) = 1

⇡(✓ 0.5008792165 sin (2✓))

F⇥(✓) = 1

⇡

✓

✓ sin (2✓) 2

◆

2

dF⇥

d✓ = 1

⇡(1 cos 2✓) = 1

⇡(1 (cos²✓ sin²✓)) = 2

⇡sin²✓ fX(x)

dF⇥

dx = dF⇥

d✓

d✓

dx

✓(x) = ⇡ cos ¹(x) d✓

dx= 1 p1 x²

fX(x) = 2

⇡sin²✓(x)d✓

dx

sin ✓(x) = sin (⇡ cos ¹(x)) = sin ⇡· cos (cos ¹(x)) sin (cos ¹(x))· cos ⇡ =

= 0· x sin (cos ¹(x))· ( 1) = sin (cos ¹(x))

sin²✓(x) = sin²(cos ¹(x)) = 1 cos²(cos ¹(x)) = 1 x²

fX(x) = 2

⇡sin²✓(x)dx d✓ = 2

⇡ 1 x² · 1 p1 x² = 2

⇡

p1 x²

fx(x) f_x(x)

C

Zp p ap = Np (p + 1)

2pp [ 1, 1]

fX(x) = 2

⇡

p1 x²

✓p

fX(x) = 2

⇡

p1 x²

N⇥ N

2006

F^p

y = F (x)

x x = F ¹(y)

x y = F (x) x y

C P n n

Q = Pⁿ= nP = P + P + . . . + P

| {z }

n additions

n

n 2

n = d0+ 2d1+ 4d2+ . . . + 2^mdm)

nP = (d0+ 2d1+ 4d2+ . . . + 2^mdm)P = d0P + 2d1P + 4d2P + . . . + 2^mdmP

nP m = log2(n)

Q =O i = 0 di 2ⁱP = 2^{i 1}P + 2^{i 1}P

i > 0 Q di= 1

n P Q

n Q

P

n O(p

n)

C

hP i = {O, P, 2P, 3P, . . .}

E(C) P hP i

E(C) hP i

E(C)

H G |H|

|G|

E(C) C P2 E(C)

h =|E(C)|

|hP i|

h 4

E(C) |E(C)|

Fp p > 2²²⁴

|E(C)|

O(p) F^p x 2 F^p

p 1

3· 10⁹ 10⁵⁰

Fp p ⇡ 2²²⁴

⇡ 3 · 10⁹

E(C)

|E(C)|

[p + 1 2pp, p + 1 + 2pp] |E(C)| 4pp

hP i a2 hP i

a^N =O N =|hP i| N |E(C)|

a2 E(C) a^M = (a^N)^MN =O^MN =O M =|E(C)|

R2 E(C) m2 [p 2pp, p + 2pp]

R^m=O m =|E(C)|

m

P 2 E(C) mP p 2pp < m < p + 2pp

m a^m = O

mP m >=dp 2ppe m bp + 2ppc

l =dp 2ppe lP

(l+1)P, (l+2)P, . . . , (bp+2ppc)P

lP P 4pp

lP O(log₂l) = O(log₂p) < O(pp)

4pp lP

O(pp)

C y²=

x³+3x+7 F11 C

|E(C)| [6, 18]

P = (5, 2)

6P = 2P + 4P = 2(P + 2P ) 2P = (10, 5) , P + 2P = (10, 6) 6P = (10, 6) + (10, 6) = (5, 2) 7P, 8P, . . . , 18P

10P = 15P =O P

P = (8, 2) m2 [6, 12] P^m=O m = 10

E(C)

O(p⁴p)

|E(C)|

A, B, p C : y²= x³+Ax+B

F^p E(C) Np+1

h = 1

hP i 1 |E(C)|

p

1988

C : y²= x³+ Ax + B F^p r = 4A³ 27B²

|{p 2 P, p  n, p - r | |E(C) mod p| 2 P}|

D n

log²n

D C