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楕円曲線の有理点と BSD 予想
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Naoya Umezaki
October 06, 2018
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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
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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.