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
数論幾何と分岐
Search
Naoya Umezaki
June 26, 2018
0
1.4k
数論幾何と分岐
ある企業の研究者の方に自分の研究の概要を説明したものです。
Naoya Umezaki
June 26, 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
740
ミケル点とべズーの定理
unaoya
0
900
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
660
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
590
関数等式と双対性
unaoya
1
770
直交多項式と表現論
unaoya
0
870
導来代数幾何入門
unaoya
0
970
Featured
See All Featured
We Have a Design System, Now What?
morganepeng
51
7.5k
Templates, Plugins, & Blocks: Oh My! Creating the theme that thinks of everything
marktimemedia
30
2.3k
The Cost Of JavaScript in 2023
addyosmani
48
7.6k
Making Projects Easy
brettharned
116
6.1k
Exploring the Power of Turbo Streams & Action Cable | RailsConf2023
kevinliebholz
31
4.7k
Build your cross-platform service in a week with App Engine
jlugia
229
18k
[RailsConf 2023 Opening Keynote] The Magic of Rails
eileencodes
28
9.4k
Helping Users Find Their Own Way: Creating Modern Search Experiences
danielanewman
29
2.5k
StorybookのUI Testing Handbookを読んだ
zakiyama
28
5.6k
Keith and Marios Guide to Fast Websites
keithpitt
411
22k
Performance Is Good for Brains [We Love Speed 2024]
tammyeverts
8
700
Put a Button on it: Removing Barriers to Going Fast.
kastner
60
3.8k
Transcript
زԿͱذ ക࡚ ౦ژେֶཧՊֶݚڀՊ August 8, 2014 ക࡚ زԿͱذ
زԿͱʁ ݚڀରɿํఔࣜͷղશମͷͳ͢ਤܗɻ ྫ y = x2, x2 + y2 =
1, y2 = x(x − 1)(x − 2) ͍Ζ͍ΖͳՃݮΛͭͷू߹ʢશମɺ༗ཧશମɺ࣮ શମɺෳૉશମͳͲʣͰํఔࣜΛߟ͑Δɻ ྫɻϑΣϧϚʔ༧ xn + yn = 1 ͷ༗ཧղɻ ക࡚ زԿͱذ
ෳૉۂઢͷذ ෳૉۂઢͷྫ P1ɿෳૉฏ໘ʹҰແݶԕΛ͚ͭՃ͑ͨͷɻٿ໘ͱಉ͡ ͔ͨͪɻ ക࡚ زԿͱذ
ෳૉۂઢͷذ y2 = x(x − 1)(x − 2) Ұൠʹ x
ͷΛܾΊΔͱ y ͷ͕;͖ͨͭ·Δɻٿ໘;ͨ ͭͱ͍͍ͩͨಉ͡ɻ ͱ͜Ζ͕ x = 0, 1, t, ∞ ͰॏղΛͭɺͭ·Γ y ͷͻ ͱͭɻ͜ͷΑ͏ͳΛذͱ͍͏ɻ ͜ͷۂઢͷղશମʢʹແݶԕΛ͚ͭՃ͑ͨͷʣͷ͔ͨͪ ʁٿ໘ೋͭΛΈ߹ΘͤͯɺυʔφπܕΛͭ͘Δɻ ക࡚ زԿͱذ
ക࡚ زԿͱذ
छ ۂઢͷෆมྔɿ݀ͷʢछ gʣʹΑ͓͓ͬͯ·͔ʹྨ͢Δɻ ക࡚ زԿͱذ
Hurwitz ͷެࣜ ೋͭͷۂઢ Y → X ͷؒͷछͷެࣜ 2g(Y) − 2
= d(2g(X) − 2) + ∑ P (eP − 1) g ͕݀ͷɺ2g − 2 ΛΦΠϥʔʢߴ࣍ݩͷਤܗʹ͍ͨͯ͠ ఆٛͰ͖ΔʣͱΑͿɻd ͕Ұൠతͳͷ্ʹ͋Δͷɺ P ذɺeP ذͷେ͖͞ʢղͷॏෳʣ ɻ લͷྫͰɺd = 2, P = 0, 1, 2, ∞, eP = 2, g(X) = 0 ͳͷͰ g(Y) = 1 ͱͳΔɻ ͱ͘ʹɺ͜Ε͔Β P1 ্ෆذɺҰͰذ͢Δඃ෴ଘࡏ͠ͳ ͍͜ͱ͕Θ͔Δɻ2g − 2 = −2d + 1 ͱ͢Δͱ g = −d + 3 2 < 0 ͱ ͳΔͷͰɻ ക࡚ زԿͱذ
ͷذ ૉશମʹۂઢ ༗ཧʹํఔࣜͷղΛ͚ͭՃ֦͑ͯେ͢Δʢ࣮͔ΒෳૉΛͭ ͘ΔΑ͏ʹʣ ذΛݟΔ͜ͱͰ్தʹ͋Δ֦େΛ͠Δ͜ͱ͕Ͱ͖Δɻ ్தͰذͯͨ͠Βɺ্·Ͱ͍ͬͯذɻ ྫɺQ(ζ5)ɿ༗ཧશମʹ x5 = 1
ͷղΛ͚ͭՃ͑ͨମɻ͜͜ͰͲ Μͳೋ࣍ํఔ͕ࣜղ͚Δ͔ʁQ(ζ5) Ͱ 5 ͚ͩذɺx2 = n n ͕ 5 ͰΘΕͳ͚Εղ͚ͳ͍ʂ ക࡚ زԿͱذ
༗ݶମ Λૉ p ͰΘͬͨ͋·Γͷͳ͢ू߹ Fp Λߟ͑Δɻ͜ΕՃ ݮআͰด͡Δɻ F3 = {0,
1, 2}, F5 = {0, 1, 2, 3, 4} F3 Ͱ 2 × 2 = 1 ͱͳΓɺ1/2 = 2 ͱͳΔɻ ͞ΒʹҰมํఔࣜͷղʢͨͱ͑ x2 = −1 ͷղͳͲʣΛͯ͢ ͚ͭ͘Θ͑ͨͷΛΛ ¯ Fp ͱ͔͘ɻ͜Ε p ͝ͱʹଘࡏɻෳૉ ͷྨࣅɻ ക࡚ زԿͱذ
༗ݶମ্ͷۂઢͷذ ༗ݶମ্ͷۂઢͷྫɻ P1ɿ ¯ Fp શମͱແݶԕʢٿ໘ͷྨࣅʣ yp − y =
x x Λ P1 ͷ࠲ඪͱΈͯɺͦͷ্ͷඃ෴ͱߟ͑Δɻ ͨͱ͑ x = 0 ͩͱ y = 0, 1, 2, . . . , p − 1 ͕ղɻ ذ͢Δ͔ʁ ॏղ͕ଘࡏ͢ΔͳΒɺඍͱͷڞ௨Ҽࢠ͋Δɻඍ͢Δͱ pyp−1 − 1 = −1 ͰɺͲ͜ফ͑ͳ͍ɻͭ·Γ x = ∞ Ҏ֎Ͱ ذ͠ͳ͍ɻ P1 ্ҰͰذ͢Δඃ෴͕ଘࡏɻHurwitz ͷެ͕ࣜͳΓͨͨͳ͍ʂ ക࡚ زԿͱذ
Grothendieck-Ogg-Shafarevich ެࣜ ༗ݶମ্ͷۂઢͰذͷ༷ࢠΛΑΓਂ͘ଊ͑Δඞཁ͕͋Δɻ ذͷΑ͏͢Λ͋ΒΘ͋ͨ͢Β͍͠ෆมྔɿSwan ಋख SwP ʢSerreʣΛఆٛɻ Grothendieck-Ogg-Shafarevich ެࣜ χc(U,
F) = rankFχc(U, Q ) − ∑ P SwPF F ͕ඃ෴ɺχc(U, F) ͕ΦΠϥʔɻ ͞Βʹ͜ΕΒͷߴ࣍ݩԽɻ ʢมํఔࣜͷΛ૿ͯ͠ਤܗΛ ߟ͑Δɻ ʣ ߴ࣍ݩͷਤܗʹମ͢Δ Swan ಋखͷఆٛɺGOS ެࣜɻ ʢՃ౻-ࡈ౻ʣ ക࡚ زԿͱذ
ݱࡏͷݚڀ ෳૉͷઢܗඍํఔࣜʢD Ճ܈ʣͷෆ֬ఆಛҟͱ༗ݶମ্ͷ ذͷྨࣅɻ D Ճ܈ͷΦΠϥʔʹղͷ࣍ݩ ྫɻexp z ෳૉฏ໘্ਖ਼ଇͰ z
= ∞ Ͱෆ֬ఆಛҟΛͭ D Ճ܈ʹ͓͍ͯಛੑαΠΫϧ͕ॏཁͳෆมྔɻ ͜ͷྨࣅΛ༗ݶମͷํఔࣜͷͳ͢ਤܗʹରͯ͠ఆٛ͠ ͍ͨɻͦΕΛͬͯΦΠϥʔͷܭࢉͳͲΛߦ͏ɻ ʢݱࡏਐߦதʣ ക࡚ زԿͱذ