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
1.3k
ミケル点とべズーの定理
unaoya
0
980
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
690
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
880
Egisonパターンマッチによる彩色
unaoya
1
620
関数等式と双対性
unaoya
1
810
直交多項式と表現論
unaoya
0
910
導来代数幾何入門
unaoya
0
1k
Featured
See All Featured
Thoughts on Productivity
jonyablonski
69
4.8k
A designer walks into a library…
pauljervisheath
207
24k
Testing 201, or: Great Expectations
jmmastey
45
7.6k
Put a Button on it: Removing Barriers to Going Fast.
kastner
60
4k
RailsConf & Balkan Ruby 2019: The Past, Present, and Future of Rails at GitHub
eileencodes
139
34k
Improving Core Web Vitals using Speculation Rules API
sergeychernyshev
18
1.1k
The Illustrated Children's Guide to Kubernetes
chrisshort
48
50k
Site-Speed That Sticks
csswizardry
10
770
Keith and Marios Guide to Fast Websites
keithpitt
411
22k
Building Applications with DynamoDB
mza
96
6.5k
Java REST API Framework Comparison - PWX 2021
mraible
33
8.8k
Large-scale JavaScript Application Architecture
addyosmani
512
110k
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.