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
1
1.6k
ミケル点とべズーの定理
unaoya
0
1k
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
700
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
920
導来代数幾何入門
unaoya
0
1.1k
Featured
See All Featured
4 Signs Your Business is Dying
shpigford
184
22k
Art, The Web, and Tiny UX
lynnandtonic
303
21k
The World Runs on Bad Software
bkeepers
PRO
70
11k
Fireside Chat
paigeccino
39
3.6k
Build The Right Thing And Hit Your Dates
maggiecrowley
37
2.9k
Save Time (by Creating Custom Rails Generators)
garrettdimon
PRO
32
1.6k
It's Worth the Effort
3n
187
28k
Java REST API Framework Comparison - PWX 2021
mraible
33
8.8k
Thoughts on Productivity
jonyablonski
70
4.8k
GraphQLの誤解/rethinking-graphql
sonatard
72
11k
No one is an island. Learnings from fostering a developers community.
thoeni
21
3.4k
The Myth of the Modular Monolith - Day 2 Keynote - Rails World 2024
eileencodes
26
3k
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