Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Features
Speaker Deck
PRO
Sign in
Sign up for free
Search
Search
楕円曲線の有理点と BSD 予想
Search
Naoya Umezaki
October 06, 2018
0
1.1k
楕円曲線の有理点と BSD 予想
MATHPOWER2018での講演スライド。 BSD予想についての解説。
Naoya Umezaki
October 06, 2018
Tweet
Share
More Decks by Naoya Umezaki
See All by Naoya Umezaki
証明支援系LEANに入門しよう
unaoya
0
820
ミケル点とべズーの定理
unaoya
0
920
すうがく徒のつどい@オンライン「ラマヌジャンのデルタ」
unaoya
0
670
合同式と幾何学
unaoya
0
2.2k
すうがく徒のつどい@オンライン「ヴェイユ予想とl進層のフーリエ変換」
unaoya
0
850
Egisonパターンマッチによる彩色
unaoya
1
600
関数等式と双対性
unaoya
1
780
直交多項式と表現論
unaoya
0
880
導来代数幾何入門
unaoya
0
980
Featured
See All Featured
Refactoring Trust on Your Teams (GOTO; Chicago 2020)
rmw
34
2.9k
Building Applications with DynamoDB
mza
94
6.3k
Understanding Cognitive Biases in Performance Measurement
bluesmoon
29
1.6k
It's Worth the Effort
3n
184
28k
Unsuck your backbone
ammeep
670
57k
Java REST API Framework Comparison - PWX 2021
mraible
30
8.5k
Facilitating Awesome Meetings
lara
54
6.3k
Being A Developer After 40
akosma
91
590k
Making Projects Easy
brettharned
116
6.1k
No one is an island. Learnings from fostering a developers community.
thoeni
21
3.2k
Building Better People: How to give real-time feedback that sticks.
wjessup
367
19k
Fantastic passwords and where to find them - at NoRuKo
philnash
51
3.1k
Transcript
ପԁۂઢͷ༗ཧͱBSD༧ ക࡚@unaoya ͢͏͕͘ͿΜ͔ɺཧۭؒ τ ´ oπoζ MATHPOWER2018 10/6
ฏํͱཱํ ฏํ 1, 4, 9, 16, 25, 36, 49, 64,
. . . ཱํ 1, 8, 27, 64, 125, 216, 343, 512, . . . ฏํͱཱํͷ͕ࠩ1 ฏํͱཱํʹڬ·Εͨ།Ұͷ26
ପԁۂઢ y2 = x3 + 1, (x, y) = (2,
3) y2 = x3 − 2, (x, y) = (3, 5) ༗ཧ x, y ࠲ඪ͕༗ཧͳ
༗ཧͷ܈ P Q R P+Q P, Q ͕༗ཧ ઢPQ ༗ཧ
R ༗ཧ P + Q ༗ཧ
༗ཧͷ܈ P Q 2P P ͕༗ཧ ઢ༗ཧ Q ༗ཧ 2P
༗ཧ
y2 = x3 + 1 P Q R P+Q P
= (−1, 0), Q = (0, 1) PQ : y = x + 1 (x + 1)2 = x3 + 1 x = −1, 0, 2 R = (2, 3), P + Q = (2, −3)
y2 = x3 + 1 P Q 2P P =
(2, 3) yy′ = 3x2 ઢ y = 2(x − 2) + 3 = 2x − 1 (2x − 1)2 = x3 + 1 x = 0, 2 Q = (0, −1), 2P = (0, 1)
y2 = x3 + 1 P Q R P +
Q y2 = x3 + 1ͷ༗ཧ (−1, 0), (0, ±1), (2, ±3), O ͷ6ݸɻ
y2 = x3 − 2 P = (3, 5) 2P
= (129/100, −383/1000) 3P = (164323/29241, −66234835/5000211) 4P = (2340922881/58675600, 113259286337279/44945509600) ༗ཧnP ͷΈ
y2 = x3 − 17x P = (−1, 4) 2P
= (1089/16, −35871/64) 3P = (−4169764/1329409, 7264943878/1532808577) 4P = (1416749814529/82350633024, − 1637173839697065089/23631996457631232)
y2 = x3 − 17x Q = (−4, 2) 2Q
= (81/16, 423/64) 3Q = (−36481/9409, −2520436/912673) 4Q = (119093569/11451456, − 1193164200991/38751727104)
y2 = x3 − 17x R = (0, 0) 2R
= O ༗ཧnP + mQ, nP + mQ + R Ͱશͯɻ
ϞʔσϧϰΣΠϢ֊ ༗ཧͷʢແݶ෦ͷʣ࠷খͷੜݩͷݸ 1. y2 = x3 + 1ϞʔσϧϰΣΠϢ֊0 2. y2
= x3 − 2nP ͷܗͳͷͰϞʔσϧ ϰΣΠϢ֊1 3. y2 = x3 − 17x nP + mQ ͷܗͳͷͰ ϞʔσϧϰΣΠϢ֊2
mod pͷͷݸ ପԁۂઢE ͷ mod p ͷͷݸNp (E)Λ ͑Δɻ
E : y2 = x3 + 1 N3 (E) mod
3Ͱ (x, y) = (0, 0), (1, 0), (0, 1), (1, 1) 02 ̸= 03 + 1 02 = 13 + 1 12 = 03 + 1 12 ̸= 13 + 1
E : y2 = x3 + 1 N3 (E) mod
2Ͱx = 0, 1, 2, y = 0, 1, 2 12 = 03 + 1, 22 = 03 + 1, 02 = 23 + 1 ͷ3ͭʹແݶԕΛՃ͑ͯ N3 (E) = 4
E : y2 = x3 + 1 ∏ p Np
(E) p Λߟ͑Δɻ N2 (E) 2 , N2 (E) 2 N3 (E) 3 , N2 (E) 2 N3 (E) 3 N5 (E) 5 , . . .
E : y2 = x3 + 1
E : y2 = x3 − 2
E : y2 = x3 − 17x
∏ Np(E)/p
Lؔ L(s, E) = ∏ p 1 1 − (1
+ p − Np (E))p−s + p1−2s ϦʔϚϯθʔλؔͷପԁۂઢ൛ ζ(s) = ∏ p 1 1 − p−s
Lؔ L(1, E) = ∏ p 1 1 − (1
+ p − Np (E))p−1 + p1−2 = ∏ p 1 1 − p−1 − 1 + Np (E)p−1 + p−1 = ∏ p 1 Np (E)/p
Birch and Swinnerton-Dyer༧ ▶ L(s, E)ͷs = 1Ͱͷॏෳͱ E ͷϞʔσϧϰΣΠϢ֊͕͍͠
▶ L(1, E) ̸= 0 ⇐⇒ ༗ཧ͕༗ݶ ෦తղܾ͋Γɻ શʹղ͍ͨΒ100ສυϧ.ɻ
ࢀߟจݙ 1. ాޱ༤Ұ, ༗ཧͷ 2. Birch and Swinnerton-Dyer, Notes on
elliptic curves. II.