Upgrade to Pro
— share decks privately, control downloads, hide ads and more …
Speaker Deck
Sign up for free
Menu
Search
Features
All features
Private URLs
Password Protection
Custom URLS
Scheduled publishing
Remove Branding
Restrict embedding
Deck Collections
Notes
Features
All features
Private URLs
Password Protection
Custom URLS
Scheduled publishing
Remove Branding
Restrict embedding
Deck Collections
Notes
Explore
Featured decks
Featured speakers
Programming
Technology
Storyboards
Explore
Featured decks
Featured speakers
Programming
Technology
Storyboards
Pricing
Search
Sign in
Sign up for free
数論幾何と分岐
Search
Naoya Umezaki
June 26, 2018
1.5k
0
Share
Embed
Copy iframe code
Copy JS code
Copy link
Start on current slide
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
More Decks by Naoya Umezaki
See All by Naoya Umezaki
証明支援系LEANに入門しよう
unaoya
2
5.3k
ミケル点とべズーの定理
unaoya
0
1.2k
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
850
合同式と幾何学
unaoya
0
2.3k
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
1k
Egisonパターンマッチによる彩色
unaoya
1
690
関数等式と双対性
unaoya
1
910
直交多項式と表現論
unaoya
0
1k
導来代数幾何入門
unaoya
0
1.4k
Featured
See All Featured
Facilitating Awesome Meetings
lara
57
7.2k
Discover your Explorer Soul
emna__ayadi
2
1.3k
How to build an LLM SEO readiness audit: a practical framework
nmsamuel
2
940
Agile Actions for Facilitating Distributed Teams - ADO2019
mkilby
0
300
Data-driven link building: lessons from a $708K investment (BrightonSEO talk)
szymonslowik
1
1.4k
Six Lessons from altMBA
skipperchong
29
4.5k
Bioeconomy Workshop: Dr. Julius Ecuru, Opportunities for a Bioeconomy in West Africa
akademiya2063
PRO
1
390
SERP Conf. Vienna - Web Accessibility: Optimizing for Inclusivity and SEO
sarafernandez
2
1.6k
Measuring Dark Social's Impact On Conversion and Attribution
stephenakadiri
2
300
How to Ace a Technical Interview
jacobian
280
24k
Documentation Writing (for coders)
carmenintech
77
5.6k
Build The Right Thing And Hit Your Dates
maggiecrowley
39
3.5k
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