1 n a ≡ b (mod n ) a b n gcd( a,b )=1 a b a,b a,b gcd( a,b ) Z , Q , R 12 126 105BSD 94 63Mordell 22 0 21 10 9 17

(1)

10

9

17

0

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2

1

%'&$($)'*$+

2

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6 3 Mordell

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9

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12

6

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12

B$C D

Z , Q , R

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D

P$R

a, b

U'_$`$a4b$R.\

gcd(a, b)

^{c']4^} ^D

gcd(a, b) = 1

^U.d'e'J

a

^d

b

f'g$h.E'ijc$k$l.d"hnm

D

P$R

a, b

^U'o.p.q

r R

n

c=sutwv$Mjljd=e'J

a ≡ b (mod n)

^d=x4e=J

a

^d

b

^f

n

^{\zy d4{=I}

W.|$c$k$ljd"hnm

D

(2)

0

a, b

^\

4a ³ − 27b ² 6 = 0

\ '^$[$R d"^.l

D

L$U.d'e'J

y ² = x ³ − ax − b (0.1)

E.F'H$I"M.l

E

\',.-$/'0 d"hnm D

m'M.l'Z$f'J

U.U \']Y^! " d#$.pYkYl!%$cYkYl

D&!'(

c$f J

E

)jE+*4y,jU+-.jp0/jl

d J

a, b

p=X4Z4RjUjd=e=J

x, y

^d1=E=X4Z

R.c$k$l.Fjm2

E

^).U3.U'V4W

E(Q)

p4",.\2'^

d=\56'^.l

D

0jU=P4R7jf=J98:<;jU4f =+%4E

Mordell

^EjF ^H ^I>?,

E ( Q )

^p

XA@BC c$kYlD d hnmEYZ.pF6A M.IGHYJIKJ U'PYR!7L!MN U

#+Oj\zV %=JK_P1RQ S4E+LMT MjI=h l+UVjU0WXjc4k4l

D

Wiles

^EjF=l

Fermat

^Y!Z ^U[!\K1 ^J

E!# ^ l]A^

–

^_!` ^UY!Z ^\[KJ

d E

F H I0aTb M D

0jE<# {=I fzJITJzU+c<d p+eTb MjIzh l p=J'W

|'R.Ufg$J

BSD

YZh$JIJ'Ufg.pi[\.Ujjk"MjI'h.l D j

l!mn c1 J

U*Yy.\o!m { I'` e2 R.\ iAp'R!"[Y^.l

$y.p=Jqr$Z7.Es4h.Itu"M.I=h.l

D

1

vxwzyz{x|z}

3

^~.U+'p

X, Y, Z

^U.f

X ² + Y ² = Z ² (1.1)

\'^

D

h

(1.1)

^{\=^ q}

r R

X, Y, Z

\.%'&$($)'*$+ d+

D

X Z Y

7

^:; ^J ^U

Brahmagupta

f0 R \ ^ I\ {

D

Ufg.f'J

x = X

Z , y = Y Z

^dsJ

d=E.F'H$I'J

C : x ² + y ² = 1 (1.2)

(3)

- 6

x y C : x ² + y ² = 1

` : y = t(x + 1) P

P 0

1:

^U']

) UY7

(x, y

^d1 E'XYZYR.cYk$l F.m 23

)

^{\\!$^} ^l.d ^hnm ^fg ^d

|='E2$l D

M.f.U.Fjm"E.{'I[$l

:

3

P 0 = ( − 1, 0)

^\.l.e

t

^U

`

^d

C

^f

P 0

^G.E1.m

1

³

P

^c$l

( 1)

^D

`

^U

y = t(x + 1)

^\

(1.2)

^{E/ {=I'J}

x ² + t ² (x + 1) ² = 1, (1 + t ² )x ² + 2t ² x + t ² − 1 = 0, (x + 1) (1 + t ² )x + t ² − 1

= 0.

M.xb"J3

P

^U.f'J

x = 1 − t ²

1 + t ² , y = 2t

1 + t ² (1.3)

d =]P Mjl

D<

c=J

t ∈ Q

^{2 b+'J}

x, y ∈ Q

^c4k4l

D

E

x, y ∈ Q

^2b'J

t = _x+1 ^y ∈ Q

^c$k$l

D

G).E.F'H$I'J

C

^).U

P 0

G.U'X$Z3.f'J.k4l

t ∈ Q

^E.F'H$I=Jh

(1.3)

c']"M.l

D

t ∈ Q

^\

t = _a ^b , a, b ∈ Z

^J

a > 0

^J

gcd(a, b) = 1

^d=xJ ^D ^L$U.d'e'J

x = a ² − b ²

a ² + b ² , y = 2ab a ² + b ²

c$k$l

D

gcd(a, b) = 1

c$k$l$xb"JYi4R

p

^p

2ab

^d

a ² + b ²

^\.d1'E'snt

vjH.d ^.ljd"J

p = 2

c2'M 2 b2=h

D

L4U.d'e=J

a, b

^fjd1=E

R.c$k$l$x b"J

a = 2a 1 + 1, b = 2b 1 + 1

dsJ4d"J

x = 2(a 1 − b 1 )(a 1 + b 1 + 1)

2a ² 1 + 2a 1 + 2b ² 1 + 2b 1 + 1 , y = (2a 1 + 1)(2b 1 + 1)

2a ² 1 + 2a 1 + 2b ² 1 + 2b 1 + 1 .

(4)

c'J

A = a 1 + b 1 + 1, B = a 1 − b 1

^ds ^'J

x = 2AB

A ² + B ² , y = A ² − B ² A ² + B ²

cYk t J

A ² + B ²

^f YR.cYkYl$xAb"J

2AB

A

²

+ B

²

, ^A _A

²₂

⁻ ₊ ^B _B

²₂ ^f ^d1 ^E ^b!"$R ^c

k$l

D

j'J

A, B

UW4p$Rjc$k$l

D

f=+%$xb J

2ab

^d

a ² + b ²

p=g4h E=i2 b0=J

a, b

^U+W ^p4R ^c4k ^l ^D ^L4Ujdze=J

_a

₂

² ₊ ^ab _b

₂

, ^a _a

²2

⁻ + ^b b

²²

f.d1'E b"$R.c$k$l

D

G)j.d%$l.d J

6$7

1.1. X ² + Y ² = Z ²

U q r

R<[

X, Y, Z

^czJ

X, Y, Z

^pzg ^hjEzi ^c4k ^l1zU4fzJ

a 6≡ b (mod 2) , gcd(a, b) = 1

^{\+'^ q}

r R

a > b

^EjF'H$I'J

X = a ² − b ² , Y = 2ab, Z = a ² + b ²

j$f'J

X = 2ab, Y = a ² − b ² , Z = a ² + b ²

d"].l

D

U$ZjUlm d${'I'J

X ⁴ − Y ⁴ = Z ² (1.4)

f q r

R<[ \ <2zh

dz\0F6j{ F m D

(1.4)

^p ^q

r

R[

X, Y, Z

^\H

.d"^jl

D

b=E'J q r

R[.U$c

X

p'_jE24l1'U4\.dzH.d4{'I F"h

D

L$Ujd'e=J

X, Y, Z

f'g4h.E=i.c4k$l

D

(Y ² ) ² + Z ² = (X ² ) ²

^d$M

'J$Z

1.1

^F.t J4g$h.E=i2.q

r R

A > B

^c'J

A 6≡ B (mod 2)

^p

{'I'J

Y ² = A ² − B ² , Z = 2AB, X ² = A ² + B ²

j$f'J

Z = A ² − B ² , Y ² = 2AB, X ² = A ² + B ²

d"].l D

N.U$W$J

X, Y

^f$R$J

Z

^{f $R.c$k$l} ^D ^F=HYI=J

X ⁴ − Y ⁴ = (X ² + Y ² )(X ² − Y ² ) = Z ²

EsYh.I J

X ² + Y ²

^d

X ² − Y ²

^f.d!1 ^EYR.cYk$l

D

gcd(X, Y ) = 1

^c

kYlYxAb J

X ² + Y ²

^d

X ² − Y ²

^U ^_Y`QaYbYR ^f

2

^cYkYl ^d ^pYxQl ^D

(5)

Z = 2Z 1 , X ² +Y ² = 2U , X ² − Y ² = 2V , Z 1 , U, V

^f ^q

r R J

gcd(U, V ) = 1

d'x'J

U V = Z 1 ²

^F.t ^J

U = X 1 ² , V = Y 1 ²

^d'x$l

D

L$U.d'e'J

X 1 ⁴ − Y 1 ⁴ = U ² − V ² = (U + V )(U − V ) = (XY ) ²

c$k$l

D

Y < X

c$k$l4xb"J

X 1 ² = U = X ² + Y ²

2 < X ² , X 1 < X

c4k4l D

Mjf

X, Y, Z

^p

(1.4)

^Ujq

r

R[jU4c

X

p=_jc4k$ljFjm E d'H

d=E

^.l

D

N.UYW.f'J

Y ² = 2AB , gcd(A, B) = 1

^c$k$l ^d'xb ^J

U, V

^\'g

hjE=i 2jq

r R d4{=I'J

A = U ² , B = 2V ²

^j4f

A = 2V ² , B = U ²

^d=x

$l D

Mj\

X ² = A ² + B ²

^{E/ {=I'J}

X ² = U ⁴ + 4V ⁴ = (U ² ) ² + (2V ² ) ²

\a.l D

$Z

1.1

\ m$^$M'J$g4h.E'i2.q

r R

a, b

^p ^{'I'J

U ² = a ² − b ² , 2V ² = 2ab, X = a ² + b ²

ckl

D

V ² = ab, gcd(a, b) = 1

^cklQx ^b ^J ^q

r R

X 1 , Y 1

^EF'H$IKJ

a = X 1 ² , b = Y 1 ²

^d'x$l

D

L$U.d'e=J

X 1 ⁴ − Y 1 ⁴ = a ² − b ² = U ²

c$k$l

D

X 1 ≤ X 1 ² = a ≤ a ² < a ² + b ² = X

c4k4l4xb J

M1

X, Y, Z

^p

(1.4)

^Ujq

r

R[jU4c

X

p=_.c4k4ljF m"E.d'H

d'E

^.l

D

G)j.d+%$I'J.U$Zj\a.l

:

6$7

1.2. X ⁴ − Y ⁴ = Z ²

f.q

r

R[j\2'h

D

{+$p$H$I'J

X ⁴ + Y ⁴ = Z ⁴

1$q

r

R[j\2'h

D

(6)

2

a, b

^\

4a ³ − 27b ² 6 = 0

\ '^$[$R d${'I'J

E : y ² = x ³ − ax − b

\$l

D

P 1 = (x 1 , y 1 ), P 2 = (x 2 , y 2 )

^\

E

^).U2$l

2

3 d"^.l

D

L$U

d'e'J

P 1

^d

P 2

^\c

`

^U.\

y = λx + µ

d"^.l

D

c'J

λ = y 2 − y 1

x 2 − x 1

, µ = y 1 − λx 1 = x 2 y 1 − x 1 y 2

x 2 − x 1

cYk$l

D

E

^d ^!

`

^d'U ³ ^f

P 1

^d

P 2

^G.EA1.m

1

³ ^k$l

D L M

\

P 3 = (x 3 , y 3 )

^{d"^.l} ^D ^L$U.d'e'J

x 1 , x 2 , x 3

^f

3 x ³ − ax − b = (λx + µ) ² ,

x ³ − λ ² x ² − (a + 2λµ)x − b − µ ² = 0

U.c$k$l

D

F'H$I'J d $$R.U#$.xb J

x 1 + x 2 + x 3 = λ ² =

y 2 − y 1

x 2 − x 1

² ,

x 3 =

y 2 − y 1

x 2 − x 1

²

− x 1 − x 2 (2.1)

c$k$l

D

j'J

y 3 = y 2 − y 1

x 2 − x 1

x 3 + x 2 y 1 − x 1 y 2

x 2 − x 1

(2.2)

c4k4l

D

x 3 , y 3

^\+h<

(2.1), (2.2)

^EjF ^H I+\ %jd=ezJ

2

³

P 1

^d

P 2

^U

P 1 + P 2

^\

P 1 + P 2 = (x 3 , − y 3 ) (2.3)

E.F'H$I$^.l

( 2)

^D

P 1 = P 2

^UdOefOJ

`

^\3

P 1

^Es ^l

E

^U ^d ^{^M} ^KJ

y 1 6 = 0

^2b'J

`

^U.f

y = λx + µ

(7)

x y

2:

^).U*$y

(8)

c$k$l c'J

2y dy

dx = 3x ² − a

c$k$l$xb J

λ = 3x ² 1 − a 2y 1

, µ = y 1 − λx 1 .

E

^d

`

^dOU ^3f ³

P 1

^G ^E ^1m

1

^3kQl

D

LOM\

P 3 = (x 3 , y 3 )

^d ^{^.l} ^D ^L$U.d'e=J

x 1 , x 3

^f

3 x ³ − ax − b = (λx + µ) ² , x ³ − λ ² x ² − (a + 2λµ)x − b − µ ² = 0

U.c$k t J

x 1

^f ^.c$k$l

D

F'H$I'J d$4R.U#$.xb"J

2x 1 + x 3 = λ ² =

3x ² 1 − a 2y 1

² ,

x 3 =

3x ² 1 − a 2y 1

²

− 2x 1 (2.4)

c$k$l

D

j'J

y 3 = 3x ² 1 − a 2y 1

(x 3 − x 1 ) + y 1 (2.5)

c4k l D

x 3 , y 3

^\+h<

(2.4), (2.5)

^EjF ^H ^{I+\ %<} ^d=ezJ

2

³

P 1

^U

2 2P 1

\

2P 1 = (x 3 , − y 3 ) (2.6)

E.F'H$I$^.l D _

E=J

P 1 = P 2

^xX

y 1 = 0

^U.d=e'J

2P 1 = P ∞ (P ^∞

^f@3

)

d"^.l

D

U.F.m E*$y.\$^.l

d'E F'H$I'J

E

^f

P ^∞

\ d"^

l ,jE24l

d'p"Mjl D

E=J

a, b

p=X$Z4R.Ujd'e=J

P 1 , P 2

U.p X$Z$R2b'J

P 3 = P 1 + P 2

U1"X$ZYR.E2$l

d'p$x

l D

{$pYH$I=J

E

^)jU X$Z3$SYT.U'V$W

E(Q)

^f

E

U4",.E!2$l

d'p$x$l

D

(9)

3 Mordell

Mordell

^f

1922

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D

6 7

3.1 (Mordell

² ⁶⁴⁷

). E(Q)

^fzX@0BC ^<,jc ^k ^l ^D ^{^2}

J$X@jU3

P 1 , . . . , P l ∈ E( Q )

^{p {=I'J .U}

P ∈ E( Q )

^f

P = n 1 P 1 + · · · + n l P l , n i ∈ Z

d"].l

D

{$p$H$I'J

T

^{\'X@ ,}

, r ≥ 0

^d${'I'J

E(Q) ∼ = T ×

r

z }| { Z × · · · × Z

c$k$l

D

).U$Z.U

r

^\

E(Q)

U4).d"hnm

D

r > 0

^c4kYl ^d$d

E(Q)

^p

@,.c$k$l

d'f$|.c$k4l

D

X@$RjU4"

T

fjU$Z.E.F'H$I=J.E+\"M.l

D

6Y7

3.2 (Nagell–Lutz

^2Y6$7

). a, b ∈ Z, D = 4a ³ − 27b ² 6 = 0

^d ^{^.l} ^D

P = (x, y)

^\

E(Q)

^U'X@$R ^{U3 d} ^{^$M} ^J

x, y ∈ Z

^c4k ^t ^J

y = 0

^c

k$l$xj4f

y

^f

D

U'b$Rjc$k$l

D

r = 0

^{d+2$l$T d}

r > 0

d2$l$Tj\$L'M.N'M4I F.m

D

3.3. E : y ² = x ³ − x

^{d"^.l}

D

P ^∞ , (0, 0), (1, 0), ( − 1, 0)

U

4

^3jf

E(Q)

^U ^jc4k4l ^D ^MPb ^GjU

E(Q)

^U ^jf ^{+2=h ^d=\

L m D

Mb U3 d 2Yl

E(Q)

^U

(x, y)

^pYk4H!.d ^{^.l.d} ^J

y 6 = 0

cYk$l

D

x = ^u _v , u, v ∈ Z, gcd(u, v) = 1

^J

v > 0

^d ^xAJ ^D ^| ^E'J

y = ^s _t , s, t ∈ Z, gcd(s, t) = 1

^J

t > 0

^d'xJ ^D ^L$Ujd'e'J

s ²

t ² = u(u ² − v ² ) v ³

c$k$l$p'J~ d1'E b"$R.c$k$l$xb J

s ² = u(u ² − v ² ), t ² = v ³

(10)

\+ajl

D

t ² = v ³

^U +~jU=i p=R"[j\ =M =J

v = V ² , t = V ³ , V ∈ Z

cYkYl

D

u

^d

u ² − v ²

^f ^gYhGE ⁱ ^cYkYl

D

s ² = u(u ² − v ² )

Û~ Û îAp

R " [\ KM OJ

u

¹

u ² − v ²

^1d ^1OE ^QRGckQlQx ^b ^J

u = U ² , u ² − v ² = W ² , U, W ∈ Z

^d'x$l

D

F'H$I=J

U ⁴ − V ⁴ = W ²

c k l D

{zx { J M f04Z

1.2

^E ^F ^H ^I ^J.q

r

R<[j\<2=h D

4EzJ

E(Q) = { P ^∞ , (0, 0), (1, 0), ( − 1, 0) }

c$k t J$f

0

^c$k$l ^D

3.4. E : y ² = x ³ − 5x

^{d"^jl} ^D

P 1 = 9

4 , 3 8

f

E(Q)

^Ujc$k4l ^D

Z

3.2

^EjF ^H ^I=J

P 1

^f ^@+4Rjc ^k4l

D

{+ p H IzJ

r

^f

1

^G)jc

k$l

D

[.f

r = 1

^c$k$l ^D

2P 1 , 3P 1

^{\$^$M=J}

2P 1 =

25921

144 , − 4172959 1728

, 3P 1 =

6197310729

2620825636 , 158681535363837 134170547609384

c$k$l

D

4

^wz}

a, b ∈ Z , D = 4a ³ − 27b ² 6 = 0

^d4{'I'J ⁺

E : y ² = x ³ − ax − b

^\

$l D

i$R

p

EU {'I=J$W.|

y ² ≡ x ³ − ax − b (mod p)

U[.U$Rj\

N p

^{d"^.l}

D

a p = p − N p

dsJ

D

L$Ujd'e'J$R

t

^U'X$Z#=R

Z E,p (t) = 1 − a p t + pt ² (1 − t)(1 − pt)

\

E mod p

U'8.9':$;.&4<'+ d"hnm

D

(11)

4.1. E : y ² = x ³ + x

^{d"^.l}

D

p = 3

^EX$h.I'J

0 ³ + 0 = 0, 1 ³ + 1 = 2, 2 ³ + 2 = 10 ≡ 1 (mod 3)

c$k t J

y ² ≡ 0 (mod 3) ⇐⇒ y ≡ 0 (mod 3), y ² ≡ 1 (mod 3) ⇐⇒ y ≡ 1, 2 (mod 3)

cYk t J$W |

y ² ≡ 2 (mod 3)

^f[.\ ^A2 ^h ^D ^{$p$H$I'J

N 3 = 3, a 3 = 0

^c$k ^t ^J

Z E, 3 (t) = 1 + 3t ² (1 − t)(1 − 3t)

c$k$l

D

p = 5

^EX$h.I'J

0 ³ + 0 = 0, 1 ³ + 1 = 2, 2 ³ + 2 ≡ 0, 3 ³ + 3 ≡ 0, 4 ³ + 4 ≡ 3 (mod 5)

c$k t J

y ² ≡ 0 (mod 5) ⇐⇒ y ≡ 0 (mod 5),

c4k t J Wj|0

y ² ≡ 2 (mod 5)

^s ^F

y ² ≡ 3 (mod 5)

^f=h ^MT1 ^[ ^\

2'h

D

{+$p$H$I'J

N 5 = 3, a 5 = 2

^{c4k t} ^J

Z E,5 (t) = 1 − 2t + 5t ²

(1 − t)(1 − 5t)

c$k$l

D

Artin

EjF=H4IYZP"M jUh.f=J

Hasse

E.FzH4I'_jEF 6 M

E'J

Weil

E.F'H$I=JW.U$R.E#'^.lh d${'I

M

D

6$7

4.2 (Hasse

^2$6$7

). | a p | ≤ 2 √ p

^{pCnt X}

D

{$p$H$I'J

Z E,p (t)

U".UI.f

1 − a p t + pt ² = (1 − ωt)(1 − ωt)

dp'R"[P"M.l

D

c=J

ω

^fU

√ p

U$i$Rjc$k$l

D

(12)

5 BSD

s

^\ ⁱ ^Rd{ ^J

t

^E

p ⁻ ^s

^\ ^{/{OIOJ$^} ^IUOiR

p

^EXhIU

Z E,p (p ⁻ ^s )

^U".U R.U .\.d'H$I=J

L E (s) = Y

p

1 1 − a p p ⁻ ^s + p ¹⁻² ^s

\ l D

L E (s)

^\

E

^U

Hasse–Weil

^:4;j& ^<z+ ^d ^h ^m ^D

1963

^jEzJ

Birch

d

Swinnerton-Dyer

^f

E(Q)

^U

r

^d

L E (s)

^U

s = 1

^E+s ^4l3jU

$R.EX$hjI'J.UYZ.\.{

:

>4?

5.1 (BSD

^> ^?

). r

^\

E(Q)

^U.d ^{^4M=J}

Hasse–Weil

#=R

L E (s)

^f=S4i ^njE+[ ^P"M4J

s = 1

E+s4hjI'J R

r

^U3

\X

D

U0Y<Z fITJzUzR<N E F HOI0L<M M Izh l pzJ c1R4<" <2

cd {'xab"M.I h2'h

D

_.Eab M { hcd.f

Coates

^d

Wiles

E.F'H$I

1977

jEab"M.U+$Z.c$k$l

:

6$7

5.2 (Coates–Wiles

^2$6$7

). E

^\

Q

⁾ ^M4R$yj\X

+ d4{ J

E( Q )

^U ^4\

r

^d ^{^} ^l ^D ^L4UjdzezJ

r > 0

^{2 b0}

L E (s)

f

s = 1

^c3j\X ^D

6

^x}

q r R

n

^E+U ^{=IzJ

3

~jU0Pzp=X4Z Rjc k4l FjmR2+<<jc0n p

n

U1'U$p$^.l.d'e'J

n

f'8.9=+.c$k$l.d"hnm

D

6.1. 6

f'W.|=R.c$k$l

D

['J

3 ² + 4 ² = 5 ² , 1

2 × 3 × 4 = 6.

5

p'W.|=R.c$k$l

d'f !$p {

(1225

)

^D ^[=J

9 ² + 40 ² = 41 ²

U~.\

6 ²

^c'sjHYI=J

3 2 ²

+ 20

3 ²

= 41

6 ²

, 1 2 × 3

2 × 20

3 = 5.

(13)

n

f'W.|'R.c4k$l.d"^.l ^ 2 J$X$Z$R

X < Y < Z

^c'J

X ² + Y ² = Z ² , 1

2 XY = n

\'^1=U$p$^.l.d ^.l

D

L$U.d'e=J

x = Z ²

4 , y = (Y ² − X ² )Z 8

ds'J

x ³ − n ² x = x(x − n)(x + n) = Z ² 4

Z ²

4 − n Z ² 4 + n

= Z ² 4

X ² + Y ²

4 − XY

2 X ² + Y ²

4 + XY 2

= Z ² 4

(Y − X) ² 4

(Y + X) ² 4

=

Z(Y − X)(Y + X) 8

²

= y ² .

{$p$H$I=J

E n : y ² = x ³ − n ² x

^f'X$Z3

P =

Z ²

4 , (Y ² − X ² )Z 8

\X

D

E n (Q)

U'X@4R.U3.f

P ^∞ , (0, 0), (n, 0), ( − n, 0)

U

4

³ ^{ ^x2 ^h ^d'p K"MYJ {Yp4H$I J

P

^f ^A@$R ^U3 ^c$kYl

d'p$x t J

E n (Q)

^U4f

r > 0

^c4k$l ^d'p$x$l ^D ^E'J

r > 0

2b'J

n

f'Wj|'R.c$k$l

d1 "M.l

D

G)jE.F'H$I'J

A

6.2.

^q

r R

n

^{EU {=I'J}

y ² = x ³ − n ² x

^c"Mjl

.\

E n

^d${ ^J

E n (Q)

^U4\

r

^{d"^.l} ^D ^L$U.d=e'J

n

p'Wj|'Rjc$k4l

%$U".f

r > 0

^c$k$l ^D

6.3. E 1 : y ² = x ³ − x

^EX$h ^I'J

E 1 (Q)

^UYf

3.3

^E

F'H$I

0

c$k$H$xb"J).U g.xb

1

f'W.|'R.c2=h

d'p$x$l

D

(14)

E n

^EXQh ^IYU

BSD

^Y!Z ^p ^{ ^h ^d ^{^YM} ^J

n

^p'WG| ^R ^cYkQl ^dYd

E n

^U

Hasse–Weil

^#'R

L E

n

(s)

^p

s = 1

^c3.\X ^d'f$|.E

2$l

D

> Wj|'Rj\'^.I+\4^.l

dD \'Wj|'R fg d hnm D

M.f

10

^:;

U! $E \ xYUYl

d'pYc$eYl 2!fg c$kYlYp'J

cA1YS.EYf[!\K M I hA2'h

.

^M.EXYh ^I ^J

1983

^E

Tunnell

^E

F'H$Iab M.U {"hcd.p$k$l

:

6$7

6.4 (Tunnell

^2$6$7

). n

^\ p'R.\2'h F.m 2$R d"^.l

D

.U

2

^.\$l

:

(A) n

f'W.|'R.c4k$l

;

(B) # { (x, y, z) ∈ Z ³ ; 2x ² + y ² + 8z ² = n }

= 2# { (x, y, z) ∈ Z ³ ; 2x ² + y ² + 32z ² = n } .

L4Ujd=e=J

(A)

^{2 b+}

(B)

^p+Cut ^X ^D ^<b'E=J

E n

^E+X4hjI4U

BSD

^Y<Z

p {"h d ^$M'J

(B)

^2b+

(A)

^{pCnt X} ^D

[1] J. W. S.

^'J ^/'J

1996

^J ^D

[2] J.

^{! A#"}

J.

^$ ^&% ^J!

9!7 /'KJ

1995

^J()*,+

-

./

D

[3]

^*01$J[\

!

2 jU'_3$Z$J

1995

^J54 ^&6 ⁷⁷ ^D

Mordell

^U04ZjU ⁸ 2 4WjU0F 64f=J

[1]

^d

[2]

^E94hjI4k4l ^D

[2]

^U+

p;:<$^$hjx1${ M2'h.p=J

[1]

^1>=?2@ j$c$x tA<$^J [5 { sP1${CB'h

&

c4k4l

D

Fermat

YZjEX4h.I4f'JD4

&

UCEFjU ;Fj\

4IC4h.I4k4l

[3]

^\C:G d=\+sH %=^.l

D

+4y.EjF=l=i p=R

"[.EX$hjI$f'J

[1]

^d

[2]

^E[5.p$k$l

D

1 n a ≡ b (mod n ) a b n gcd( a,b )=1 a b a,b a,b gcd( a,b ) Z , Q , R 12 126 105BSD 94 63Mordell 22 0 21 10 9 17

10

9

17

0

2

1

2

2

6

3 Mordell

9

4

10

5 BSD

12

6

12

Z , Q , R

a, b

gcd(a, b)

gcd(a, b) = 1

a

b

a, b

n

a ≡ b (mod n)

a

b

n

0

a, b

4a 3 − 27b 2 6 = 0

y 2 = x 3 − ax − b (0.1)

E

E

a, b

x, y

E

E(Q)

Mordell

E ( Q )

Wiles

Fermat

–

BSD

1

3

X, Y, Z

X 2 + Y 2 = Z 2 (1.1)

(1.1)

X, Y, Z

X Z Y

7

Brahmagupta

x = X

Z , y = Y Z

C : x 2 + y 2 = 1 (1.2)

- 6

x y C : x 2 + y 2 = 1

` : y = t(x + 1) P

P 0

1:

(x, y

)

:

P 0 = ( − 1, 0)

t

`

C

P 0

1

P

( 1)

`

y = t(x + 1)

(1.2)

x 2 + t 2 (x + 1) 2 = 1, (1 + t 2 )x 2 + 2t 2 x + t 2 − 1 = 0, (x + 1) (1 + t 2 )x + t 2 − 1

= 0.

P

4a ³ − 27b ² 6 = 0

y ² = x ³ − ax − b (0.1)

X ² + Y ² = Z ² (1.1)

C : x ² + y ² = 1 (1.2)

x y C : x ² + y ² = 1

x ² + t ² (x + 1) ² = 1, (1 + t ² )x ² + 2t ² x + t ² − 1 = 0, (x + 1) (1 + t ² )x + t ² − 1

x = 1 − t ²

1 + t ² , y = 2t

1 + t ² (1.3)

t = _x+1 ^y ∈ Q

t = _a ^b , a, b ∈ Z

x = a ² − b ²

a ² + b ² , y = 2ab a ² + b ²

a ² + b ²

2a ² 1 + 2a 1 + 2b ² 1 + 2b 1 + 1 , y = (2a 1 + 1)(2b 1 + 1)

2a ² 1 + 2a 1 + 2b ² 1 + 2b 1 + 1 .

A ² + B ² , y = A ² − B ² A ² + B ²

A ² + B ²

, ^A _A

⁻ ₊ ^B _B

a ² + b ²

_a

² ₊ ^ab _b

, ^a _a

⁻ + ^b b

X ² + Y ² = Z ²

X = a ² − b ² , Y = 2ab, Z = a ² + b ²

X = 2ab, Y = a ² − b ² , Z = a ² + b ²

X ⁴ − Y ⁴ = Z ² (1.4)

(Y ² ) ² + Z ² = (X ² ) ²

Y ² = A ² − B ² , Z = 2AB, X ² = A ² + B ²

Z = A ² − B ² , Y ² = 2AB, X ² = A ² + B ²

X ⁴ − Y ⁴ = (X ² + Y ² )(X ² − Y ² ) = Z ²

X ² + Y ²

X ² − Y ²

X ² + Y ²